text:
@testset "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n" begin
    (A, B, a, b, c, d, e, f, g, m, n, p, x, ) = @variables A B a b c d e f g m n p x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^(m/2)*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(e + f*x)^2*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2), x, 3, -(a*cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(15*c*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))/(6*c*f)]
    @test_int [cos(e + f*x)^2*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2), x, 3, -(a*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(10*c*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(5*c*f)]
    @test_int [cos(e + f*x)^2*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2), x, 3, -(a*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(6*c*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(4*c*f)]
    @test_int [cos(e + f*x)^2*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)), x, 3, -(a*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(3*c*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2))/(3*c*f)]
    @test_int [(cos(e + f*x)^2*sqrt(a + a*sin(e + f*x)))/sqrt(c - c*sin(e + f*x)), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*a*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(3/2), x, 5, (-2*a*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(5/2), x, 5, (cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*(c - c*sin(e + f*x))^(3/2)) + (a*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(7/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(4*a*c*f*(c - c*sin(e + f*x))^(5/2))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2), x, 4, (-4*a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(105*c*f*sqrt(a + a*sin(e + f*x))) - (2*a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))/(21*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(9/2))/(7*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2), x, 4, -(a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(15*c*f*sqrt(a + a*sin(e + f*x))) - (2*a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(15*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(6*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2), x, 4, (-2*a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(15*c*f*sqrt(a + a*sin(e + f*x))) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(5*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/(5*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)), x, 3, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(6*a*f*sqrt(c - c*sin(e + f*x))) + (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(4*a*f)]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/sqrt(c - c*sin(e + f*x)), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*a*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(3/2), x, 6, (-4*a^2*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x))) - (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(5/2), x, 6, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c*f*(c - c*sin(e + f*x))^(3/2)) + (4*a^2*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(7/2), x, 6, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c*f*(c - c*sin(e + f*x))^(5/2)) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*(c - c*sin(e + f*x))^(3/2)) - (a^2*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(9/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(6*a*c*f*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(11/2), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(8*a*c*f*(c - c*sin(e + f*x))^(9/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(48*a*c^2*f*(c - c*sin(e + f*x))^(7/2))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2), x, 5, -(a^3*cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(35*c*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))/(14*c*f) - (3*a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(9/2))/(28*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(9/2))/(8*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2), x, 5, (-2*a^3*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(35*c*f*sqrt(a + a*sin(e + f*x))) - (4*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(35*c*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(7*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2))/(7*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2), x, 4, (c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(15*a*f*sqrt(c - c*sin(e + f*x))) + (2*c*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(15*a*f) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2))/(6*a*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)), x, 3, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(10*a*f*sqrt(c - c*sin(e + f*x))) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(5*a*f)]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/sqrt(c - c*sin(e + f*x)), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(4*a*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(3/2), x, 7, (-8*a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (4*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x))) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c*f*sqrt(c - c*sin(e + f*x))) - (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(5/2), x, 7, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(c*f*(c - c*sin(e + f*x))^(3/2)) + (12*a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (6*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*sqrt(c - c*sin(e + f*x))) + (3*a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c^2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(7/2), x, 7, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(2*c*f*(c - c*sin(e + f*x))^(5/2)) - (3*a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c^2*f*(c - c*sin(e + f*x))^(3/2)) - (6*a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (3*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^3*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2), x, 7, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c*f*(c - c*sin(e + f*x))^(7/2)) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c^2*f*(c - c*sin(e + f*x))^(5/2)) + (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^3*f*(c - c*sin(e + f*x))^(3/2)) + (a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(11/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(8*a*c*f*(c - c*sin(e + f*x))^(9/2))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(13/2), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(10*a*c*f*(c - c*sin(e + f*x))^(11/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(80*a*c^2*f*(c - c*sin(e + f*x))^(9/2))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(9/2), x, 6, (-4*a^4*cos(e + f*x)*(c - c*sin(e + f*x))^(11/2))/(315*c*f*sqrt(a + a*sin(e + f*x))) - (4*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(11/2))/(105*c*f) - (a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(11/2))/(15*c*f) - (4*a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(11/2))/(45*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(11/2))/(10*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(7/2), x, 6, (-8*a^4*cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(315*c*f*sqrt(a + a*sin(e + f*x))) - (4*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))/(63*c*f) - (2*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(9/2))/(21*c*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(9/2))/(9*c*f) - (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(9/2))/(9*c*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2), x, 5, (c^3*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(35*a*f*sqrt(c - c*sin(e + f*x))) + (c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*sqrt(c - c*sin(e + f*x)))/(14*a*f) + (3*c*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*(c - c*sin(e + f*x))^(3/2))/(28*a*f) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*(c - c*sin(e + f*x))^(5/2))/(8*a*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2), x, 4, (4*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(105*a*f*sqrt(c - c*sin(e + f*x))) + (2*c*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*sqrt(c - c*sin(e + f*x)))/(21*a*f) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*(c - c*sin(e + f*x))^(3/2))/(7*a*f)]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)), x, 3, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(15*a*f*sqrt(c - c*sin(e + f*x))) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2)*sqrt(c - c*sin(e + f*x)))/(6*a*f)]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/sqrt(c - c*sin(e + f*x)), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(5*a*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(3/2), x, 8, (-16*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (8*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x))) - (2*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c*f*sqrt(c - c*sin(e + f*x))) - (2*a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c*f*sqrt(c - c*sin(e + f*x))) - (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(4*c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(5/2), x, 8, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(c*f*(c - c*sin(e + f*x))^(3/2)) + (32*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (16*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*sqrt(c - c*sin(e + f*x))) + (4*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c^2*f*sqrt(c - c*sin(e + f*x))) + (4*a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c^2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(7/2), x, 8, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(2*c*f*(c - c*sin(e + f*x))^(5/2)) - (2*a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(c^2*f*(c - c*sin(e + f*x))^(3/2)) - (24*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (12*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^3*f*sqrt(c - c*sin(e + f*x))) - (3*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c^3*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(9/2), x, 8, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(3*c*f*(c - c*sin(e + f*x))^(7/2)) - (2*a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c^2*f*(c - c*sin(e + f*x))^(5/2)) + (2*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(c^3*f*(c - c*sin(e + f*x))^(3/2)) + (8*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (4*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^4*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2), x, 8, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(4*c*f*(c - c*sin(e + f*x))^(9/2)) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*c^2*f*(c - c*sin(e + f*x))^(7/2)) + (a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c^3*f*(c - c*sin(e + f*x))^(5/2)) - (a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^4*f*(c - c*sin(e + f*x))^(3/2)) - (a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(13/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(10*a*c*f*(c - c*sin(e + f*x))^(11/2))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(15/2), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(12*a*c*f*(c - c*sin(e + f*x))^(13/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(120*a*c^2*f*(c - c*sin(e + f*x))^(11/2))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(17/2), x, 4, (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(14*a*c*f*(c - c*sin(e + f*x))^(15/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(84*a*c^2*f*(c - c*sin(e + f*x))^(13/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(9/2))/(840*a*c^3*f*(c - c*sin(e + f*x))^(11/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(5/2))/sqrt(a + a*sin(e + f*x)), x, 2, -(cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(4*c*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(3/2))/sqrt(a + a*sin(e + f*x)), x, 2, -(cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*c*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*sqrt(c - c*sin(e + f*x)))/sqrt(a + a*sin(e + f*x)), x, 2, -(cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*c*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [cos(e + f*x)^2/(sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))), x, 2, -((cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))))]
    @test_int [cos(e + f*x)^2/(sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)), x, 4, -((cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))))]
    @test_int [cos(e + f*x)^2/(sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)), x, 2, cos(e + f*x)/(c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2))]

    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(7/2))/(a + a*sin(e + f*x))^(3/2), x, 8, (16*c^4*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (8*c^3*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))) + (2*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a*f*sqrt(a + a*sin(e + f*x))) + (2*c*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*a*f*sqrt(a + a*sin(e + f*x))) + (cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(4*a*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(3/2), x, 7, (8*c^3*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (4*c^2*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))) + (c*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a*f*sqrt(a + a*sin(e + f*x))) + (cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*a*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x))^(3/2), x, 6, (4*c^2*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*c*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))) + (cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*a*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*sqrt(c - c*sin(e + f*x)))/(a + a*sin(e + f*x))^(3/2), x, 5, (2*c*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))), x, 4, (cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2)), x, 3, (atanh(sin(e + f*x))*cos(e + f*x))/(a*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2)), x, 4, cos(e + f*x)/(2*a*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*a*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(9/2))/(a + a*sin(e + f*x))^(5/2), x, 9, (-80*c^5*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (40*c^4*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (10*c^3*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (10*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*a^2*f*sqrt(a + a*sin(e + f*x))) - (5*c*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(4*a^2*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(7/2))/(a + a*sin(e + f*x))^(5/2), x, 8, (-32*c^4*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (16*c^3*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (4*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (4*c*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*a^2*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(5/2), x, 7, (-12*c^3*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (6*c^2*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (3*c*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*a^2*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [(cos(e + f*x)^2*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x))^(5/2), x, 6, (-4*c^2*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*c*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) - (cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [(cos(e + f*x)^2*sqrt(c - c*sin(e + f*x)))/(a + a*sin(e + f*x))^(5/2), x, 5, -((c*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x))), x, 2, -(cos(e + f*x)/(a*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2)), x, 4, -cos(e + f*x)/(2*a*c*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*a^2*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2)), x, 5, -cos(e + f*x)/(2*a*c*f*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2)) + cos(e + f*x)/(2*a^2*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*a^2*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 5, (2^(3/2 + n)*c^2*cos(e + f*x)^3*HypergeometricFunctions._₂F₁((1/2)*(3 + 2*m), (1/2)*(-1 - 2*n), (1/2)*(5 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 + n))/(f*(3 + 2*m))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^3, x, 4, -((2^(3/2 + m)*a^4*c^3*cos(e + f*x)^9*HypergeometricFunctions._₂F₁(9/2, -(1/2) - m, 11/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-4 + m))/(9*f))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^2, x, 4, -((2^(3/2 + m)*a^3*c^2*cos(e + f*x)^7*HypergeometricFunctions._₂F₁(7/2, -(1/2) - m, 9/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-3 + m))/(7*f))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^1, x, 4, -((2^(3/2 + m)*a^2*c*cos(e + f*x)^5*HypergeometricFunctions._₂F₁(5/2, -(1/2) - m, 7/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-2 + m))/(5*f))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^0, x, 3, -((2^(3/2 + m)*a*cos(e + f*x)^3*HypergeometricFunctions._₂F₁(3/2, -(1/2) - m, 5/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-1 + m))/(3*f))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^1, x, 3, -((2^(3/2 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -(1/2) - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(c*f))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^2, x, 4, (2^(3/2 + m)*HypergeometricFunctions._₂F₁(-(1/2), -(1/2) - m, 1/2, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(1 + m))/(a*c^2*f)]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^3, x, 4, (2^(3/2 + m)*HypergeometricFunctions._₂F₁(-(3/2), -(1/2) - m, -(1/2), (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)^3*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(2 + m))/(3*a^2*c^3*f)]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2), x, 5, If(13>=8, (768*c^3*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (192*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*sqrt(c - c*sin(e + f*x)))/(a*f*(9 + 2*m)*(35 + 24*m + 4*m^2)) + (24*c*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(3/2))/(a*f*(63 + 32*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(5/2))/(a*f*(9 + 2*m)), (768*c^3*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*(63 + 32*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (192*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*sqrt(c - c*sin(e + f*x)))/(a*f*(315 + 286*m + 84*m^2 + 8*m^3)) + (24*c*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(3/2))/(a*f*(63 + 32*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(5/2))/(a*f*(9 + 2*m)))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(3/2), x, 4, If(13>=8, (64*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(7 + 2*m)*(15 + 16*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (16*c*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*sqrt(c - c*sin(e + f*x)))/(a*f*(35 + 24*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(3/2))/(a*f*(7 + 2*m)), (64*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(105 + 142*m + 60*m^2 + 8*m^3)*sqrt(c - c*sin(e + f*x))) + (16*c*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*sqrt(c - c*sin(e + f*x)))/(a*f*(35 + 24*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(3/2))/(a*f*(7 + 2*m)))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*sqrt(c - c*sin(e + f*x)), x, 3, (8*c*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*sqrt(c - c*sin(e + f*x)))/(a*f*(5 + 2*m))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/sqrt(c - c*sin(e + f*x)), x, 2, (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(3 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(3/2), x, 4, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1, 3/2 + m, 5/2 + m, (1 + sin(e + f*x))/2)*(a + a*sin(e + f*x))^(1 + m))/(a*c*f*(3 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(5/2), x, 4, (cos(e + f*x)*HypergeometricFunctions._₂F₁(2, 3/2 + m, 5/2 + m, (1 + sin(e + f*x))/2)*(a + a*sin(e + f*x))^(1 + m))/(2*a*c^2*f*(3 + 2*m)*sqrt(c - c*sin(e + f*x)))]

    #= The*same*rules*should*be*used*to*integrate*the*following*two*problems: =#
    @test_int [(cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/sqrt(c - c*sin(e + f*x)), x, 2, (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(3 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(cos(e + f*x)^2*(c + c*sin(e + f*x))^m)/sqrt(a - a*sin(e + f*x)), x, 2, (2*cos(e + f*x)*(c + c*sin(e + f*x))^(1 + m))/(c*f*(3 + 2*m)*sqrt(a - a*sin(e + f*x)))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 5), x, 4, If(13>=8, (cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-2 - m))/(a*c^3*f*(7 + 2*m)*(15 + 16*m + 4*m^2)), (cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-2 - m))/(a*c^3*f*(105 + 142*m + 60*m^2 + 8*m^3)))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 4), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-3 - m))/(a*c*f*(5 + 2*m)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-2 - m))/(a*c^2*f*(15 + 16*m + 4*m^2))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 3), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c - c*sin(e + f*x))^(-2 - m))/(a*c*f*(3 + 2*m))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 2), x, 5, (2^(-(1/2) - m)*cos(e + f*x)^3*HypergeometricFunctions._₂F₁((1/2)*(3 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(f*(3 + 2*m))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 1), x, 5, (2^(1/2 - m)*c*cos(e + f*x)^3*HypergeometricFunctions._₂F₁((1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(f*(3 + 2*m))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 0), x, 5, (2^(3/2 - m)*c^2*cos(e + f*x)^3*HypergeometricFunctions._₂F₁((1/2)*(-1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(f*(3 + 2*m))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m - 1), x, 5, (2^(5/2 - m)*c^3*cos(e + f*x)^3*HypergeometricFunctions._₂F₁((1/2)*(-3 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(f*(3 + 2*m))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(m/2)*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2), x, 8, (2*a*c^4*(g*cos(e + f*x))^(5/2))/(3*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a*c^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a*c^3*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(7*f*g*sqrt(a + a*sin(e + f*x))) + (10*a*c^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(77*f*g*sqrt(a + a*sin(e + f*x))) + (2*a*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(33*f*g*sqrt(a + a*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2))/(11*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2), x, 7, (22*a*c^3*(g*cos(e + f*x))^(5/2))/(45*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*a*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*a*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(105*f*g*sqrt(a + a*sin(e + f*x))) + (2*a*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(21*f*g*sqrt(a + a*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(9*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2), x, 6, (2*a*c^2*(g*cos(e + f*x))^(5/2))/(5*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (6*a*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (6*a*c*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(35*f*g*sqrt(a + a*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(7*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)), x, 5, (2*a*c*(g*cos(e + f*x))^(5/2))/(5*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (6*a*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x)))/sqrt(c - c*sin(e + f*x)), x, 4, -((2*a*(g*cos(e + f*x))^(5/2))/(3*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (2*a*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(3/2), x, 4, (4*a*(g*cos(e + f*x))^(5/2))/(f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (6*a*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(5/2), x, 5, (4*a*(g*cos(e + f*x))^(5/2))/(5*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (6*a*(g*cos(e + f*x))^(5/2))/(5*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (6*a*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(7/2), x, 6, (4*a*(g*cos(e + f*x))^(5/2))/(9*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) - (2*a*(g*cos(e + f*x))^(5/2))/(15*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (2*a*(g*cos(e + f*x))^(5/2))/(15*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (2*a*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^(9/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2))/(13*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2)) - (2*a*(g*cos(e + f*x))^(5/2))/(39*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) - (2*a*(g*cos(e + f*x))^(5/2))/(65*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (2*a*(g*cos(e + f*x))^(5/2))/(65*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (2*a*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(65*c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2), x, 8, (14*a^2*c^3*(g*cos(e + f*x))^(5/2))/(45*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (14*a^2*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a^2*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(15*f*g*sqrt(a + a*sin(e + f*x))) + (2*a^2*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(33*f*g*sqrt(a + a*sin(e + f*x))) - (14*a^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(99*f*g*sqrt(a + a*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(11*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2), x, 7, (14*a^2*c^2*(g*cos(e + f*x))^(5/2))/(45*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (14*a^2*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a^2*c*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(15*f*g*sqrt(a + a*sin(e + f*x))) - (2*a^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(9*f*g*sqrt(a + a*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2))/(9*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)), x, 6, -((2*a^2*c*(g*cos(e + f*x))^(5/2))/(5*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (6*a^2*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (6*a*c*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(35*f*g*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(7*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/sqrt(c - c*sin(e + f*x)), x, 5, -((14*a^2*(g*cos(e + f*x))^(5/2))/(15*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (14*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(5*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(3/2), x, 5, (4*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(f*g*(c - c*sin(e + f*x))^(3/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(3*c*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (14*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(5/2), x, 5, (4*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(5*f*g*(c - c*sin(e + f*x))^(5/2)) - (28*a^2*(g*cos(e + f*x))^(5/2))/(5*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (42*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(7/2), x, 6, (4*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(9*f*g*(c - c*sin(e + f*x))^(7/2)) - (28*a^2*(g*cos(e + f*x))^(5/2))/(45*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(15*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (14*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(9/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(13*f*g*(c - c*sin(e + f*x))^(9/2)) - (28*a^2*(g*cos(e + f*x))^(5/2))/(117*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(195*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(195*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (14*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(195*c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x))^(11/2), x, 8, (4*a*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(17*f*g*(c - c*sin(e + f*x))^(11/2)) - (28*a^2*(g*cos(e + f*x))^(5/2))/(221*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(663*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(1105*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (14*a^2*(g*cos(e + f*x))^(5/2))/(1105*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (14*a^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(1105*c^5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2), x, 9, (154*a^3*c^3*(g*cos(e + f*x))^(5/2))/(585*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (154*a^3*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(195*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*a^3*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(195*f*g*sqrt(a + a*sin(e + f*x))) + (2*a^3*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(39*f*g*sqrt(a + a*sin(e + f*x))) - (14*a^3*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(117*f*g*sqrt(a + a*sin(e + f*x))) - (2*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(13*f*g) - (2*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/(13*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2), x, 8, -((14*a^3*c^2*(g*cos(e + f*x))^(5/2))/(45*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (14*a^3*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a^2*c^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(15*f*g*sqrt(c - c*sin(e + f*x))) - (2*a*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(33*f*g*sqrt(c - c*sin(e + f*x))) + (14*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(99*f*g*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(11*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)), x, 7, -((22*a^3*c*(g*cos(e + f*x))^(5/2))/(45*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (22*a^3*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (22*a^2*c*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(105*f*g*sqrt(c - c*sin(e + f*x))) - (2*a*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(21*f*g*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(9*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/sqrt(c - c*sin(e + f*x)), x, 6, -((22*a^3*(g*cos(e + f*x))^(5/2))/(15*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (22*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (22*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(35*f*g*sqrt(c - c*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(7*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(3/2), x, 6, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(f*g*(c - c*sin(e + f*x))^(3/2)) + (154*a^3*(g*cos(e + f*x))^(5/2))/(15*c*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (154*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(5*c*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(5/2), x, 6, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(5*f*g*(c - c*sin(e + f*x))^(5/2)) - (44*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(5*c*f*g*(c - c*sin(e + f*x))^(3/2)) - (154*a^3*(g*cos(e + f*x))^(5/2))/(15*c^2*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (154*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(7/2), x, 6, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(9*f*g*(c - c*sin(e + f*x))^(7/2)) - (44*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(45*c*f*g*(c - c*sin(e + f*x))^(5/2)) + (308*a^3*(g*cos(e + f*x))^(5/2))/(45*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (154*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(13*f*g*(c - c*sin(e + f*x))^(9/2)) - (44*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(117*c*f*g*(c - c*sin(e + f*x))^(7/2)) + (308*a^3*(g*cos(e + f*x))^(5/2))/(585*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (154*a^3*(g*cos(e + f*x))^(5/2))/(195*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (154*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(195*c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(11/2), x, 8, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(17*f*g*(c - c*sin(e + f*x))^(11/2)) - (44*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(221*c*f*g*(c - c*sin(e + f*x))^(9/2)) + (308*a^3*(g*cos(e + f*x))^(5/2))/(1989*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) - (154*a^3*(g*cos(e + f*x))^(5/2))/(3315*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (154*a^3*(g*cos(e + f*x))^(5/2))/(3315*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (154*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(3315*c^5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(13/2), x, 9, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(21*f*g*(c - c*sin(e + f*x))^(13/2)) - (44*a^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(357*c*f*g*(c - c*sin(e + f*x))^(11/2)) + (44*a^3*(g*cos(e + f*x))^(5/2))/(663*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2)) - (22*a^3*(g*cos(e + f*x))^(5/2))/(1989*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) - (22*a^3*(g*cos(e + f*x))^(5/2))/(3315*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) - (22*a^3*(g*cos(e + f*x))^(5/2))/(3315*c^5*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (22*a^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(3315*c^6*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2), x, 10, -((154*a^4*c^3*(g*cos(e + f*x))^(5/2))/(585*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (154*a^4*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(195*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (22*a^3*c^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(195*f*g*sqrt(c - c*sin(e + f*x))) - (2*a^2*c^3*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(39*f*g*sqrt(c - c*sin(e + f*x))) - (14*a*c^3*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(585*f*g*sqrt(c - c*sin(e + f*x))) + (14*c^3*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2))/(195*f*g*sqrt(c - c*sin(e + f*x))) + (22*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(195*f*g) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2))/(15*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2), x, 9, -((14*a^4*c^2*(g*cos(e + f*x))^(5/2))/(39*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (14*a^4*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(13*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a^3*c^2*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(13*f*g*sqrt(c - c*sin(e + f*x))) - (10*a^2*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(143*f*g*sqrt(c - c*sin(e + f*x))) - (14*a*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(429*f*g*sqrt(c - c*sin(e + f*x))) + (14*c^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2))/(143*f*g*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(13*f*g)]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)), x, 8, -((2*a^4*c*(g*cos(e + f*x))^(5/2))/(3*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (2*a^4*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (2*a^3*c*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(7*f*g*sqrt(c - c*sin(e + f*x))) - (10*a^2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(77*f*g*sqrt(c - c*sin(e + f*x))) - (2*a*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(33*f*g*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(7/2))/(11*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/sqrt(c - c*sin(e + f*x)), x, 7, -((22*a^4*(g*cos(e + f*x))^(5/2))/(9*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) + (22*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (22*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(21*f*g*sqrt(c - c*sin(e + f*x))) - (10*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(21*f*g*sqrt(c - c*sin(e + f*x))) - (2*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(9*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(3/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(f*g*(c - c*sin(e + f*x))^(3/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(c*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (66*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (66*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(7*c*f*g*sqrt(c - c*sin(e + f*x))) + (30*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(7*c*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(5/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(5*f*g*(c - c*sin(e + f*x))^(5/2)) - (12*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(c*f*g*(c - c*sin(e + f*x))^(3/2)) - (154*a^4*(g*cos(e + f*x))^(5/2))/(5*c^2*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (462*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (66*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(5*c^2*f*g*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(7/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(9*f*g*(c - c*sin(e + f*x))^(7/2)) - (4*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(3*c*f*g*(c - c*sin(e + f*x))^(5/2)) + (44*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(3*c^2*f*g*(c - c*sin(e + f*x))^(3/2)) + (154*a^4*(g*cos(e + f*x))^(5/2))/(9*c^3*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (154*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(3*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(9/2), x, 7, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(13*f*g*(c - c*sin(e + f*x))^(9/2)) - (20*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(39*c*f*g*(c - c*sin(e + f*x))^(7/2)) + (44*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(39*c^2*f*g*(c - c*sin(e + f*x))^(5/2)) - (308*a^4*(g*cos(e + f*x))^(5/2))/(39*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (154*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(13*c^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2), x, 8, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(17*f*g*(c - c*sin(e + f*x))^(11/2)) - (60*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(221*c*f*g*(c - c*sin(e + f*x))^(9/2)) + (220*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(663*c^2*f*g*(c - c*sin(e + f*x))^(7/2)) - (308*a^4*(g*cos(e + f*x))^(5/2))/(663*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (154*a^4*(g*cos(e + f*x))^(5/2))/(221*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (154*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(221*c^5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(13/2), x, 9, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(21*f*g*(c - c*sin(e + f*x))^(13/2)) - (20*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(119*c*f*g*(c - c*sin(e + f*x))^(11/2)) + (220*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(1547*c^2*f*g*(c - c*sin(e + f*x))^(9/2)) - (220*a^4*(g*cos(e + f*x))^(5/2))/(1989*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(663*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(663*c^5*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (22*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(663*c^6*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(15/2), x, 10, (4*a*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(5/2))/(25*f*g*(c - c*sin(e + f*x))^(15/2)) - (4*a^2*(g*cos(e + f*x))^(5/2)*(a + a*sin(e + f*x))^(3/2))/(35*c*f*g*(c - c*sin(e + f*x))^(13/2)) + (44*a^3*(g*cos(e + f*x))^(5/2)*sqrt(a + a*sin(e + f*x)))/(595*c^2*f*g*(c - c*sin(e + f*x))^(11/2)) - (44*a^4*(g*cos(e + f*x))^(5/2))/(1105*c^3*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(3315*c^4*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(5525*c^5*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (22*a^4*(g*cos(e + f*x))^(5/2))/(5525*c^6*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (22*a^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5525*c^7*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/sqrt(a + a*sin(e + f*x)), x, 6, (22*c^3*(g*cos(e + f*x))^(5/2))/(15*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (22*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(35*f*g*sqrt(a + a*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(7*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))/sqrt(a + a*sin(e + f*x)), x, 5, (14*c^2*(g*cos(e + f*x))^(5/2))/(15*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (14*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*c*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*f*g*sqrt(a + a*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)))/sqrt(a + a*sin(e + f*x)), x, 4, (2*c*(g*cos(e + f*x))^(5/2))/(3*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))), x, 3, (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)), x, 4, (2*(g*cos(e + f*x))^(5/2))/(f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)), x, 5, (2*(g*cos(e + f*x))^(5/2))/(5*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (2*(g*cos(e + f*x))^(5/2))/(5*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)), x, 6, (2*(g*cos(e + f*x))^(5/2))/(9*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (2*(g*cos(e + f*x))^(5/2))/(15*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (2*(g*cos(e + f*x))^(5/2))/(15*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(a + a*sin(e + f*x))^(3/2), x, 7, -((22*c^4*(g*cos(e + f*x))^(5/2))/(a*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (66*c^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (66*c^3*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(7*a*f*g*sqrt(a + a*sin(e + f*x))) - (30*c^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(7*a*f*g*sqrt(a + a*sin(e + f*x))) - (4*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(3/2), x, 6, -((154*c^3*(g*cos(e + f*x))^(5/2))/(15*a*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (154*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (22*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*a*f*g*sqrt(a + a*sin(e + f*x))) - (4*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(f*g*(a + a*sin(e + f*x))^(3/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x))^(3/2), x, 5, -((14*c^2*(g*cos(e + f*x))^(5/2))/(3*a*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (14*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (4*c*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(f*g*(a + a*sin(e + f*x))^(3/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)))/(a + a*sin(e + f*x))^(3/2), x, 4, -((4*c*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)))) - (6*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))), x, 4, (-2*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2)), x, 5, (-2*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2)) + (2*(g*cos(e + f*x))^(5/2))/(a*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(a*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2)), x, 6, (-2*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2)) + (6*(g*cos(e + f*x))^(5/2))/(5*a*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (6*(g*cos(e + f*x))^(5/2))/(5*a*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (6*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*a*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2)), x, 7, -((2*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))) + (10*(g*cos(e + f*x))^(5/2))/(9*a*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (2*(g*cos(e + f*x))^(5/2))/(3*a*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (2*(g*cos(e + f*x))^(5/2))/(3*a*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(3*a*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(9/2))/(a + a*sin(e + f*x))^(5/2), x, 8, (418*c^5*(g*cos(e + f*x))^(5/2))/(5*a^2*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (1254*c^5*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (1254*c^4*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(35*a^2*f*g*sqrt(a + a*sin(e + f*x))) + (114*c^3*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(7*a^2*f*g*sqrt(a + a*sin(e + f*x))) + (76*c^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)) - (4*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(a + a*sin(e + f*x))^(5/2), x, 7, (154*c^4*(g*cos(e + f*x))^(5/2))/(5*a^2*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (462*c^4*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (66*c^3*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*a^2*f*g*sqrt(a + a*sin(e + f*x))) + (12*c^2*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(a*f*g*(a + a*sin(e + f*x))^(3/2)) - (4*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(5/2), x, 6, (154*c^3*(g*cos(e + f*x))^(5/2))/(15*a^2*f*g*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (154*c^3*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (44*c^2*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)) - (4*c*(g*cos(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x))^(5/2), x, 5, (28*c^2*(g*cos(e + f*x))^(5/2))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) + (42*c^2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (4*c*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(5*f*g*(a + a*sin(e + f*x))^(5/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)))/(a + a*sin(e + f*x))^(5/2), x, 5, -((4*c*(g*cos(e + f*x))^(5/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))) + (6*c*(g*cos(e + f*x))^(5/2))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) + (6*c*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x))), x, 5, (-2*(g*cos(e + f*x))^(5/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x))) - (2*(g*cos(e + f*x))^(5/2))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) - (2*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2)), x, 6, (2*(g*cos(e + f*x))^(5/2))/(f*g*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2)) - (6*(g*cos(e + f*x))^(5/2))/(5*c*f*g*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x))) - (6*(g*cos(e + f*x))^(5/2))/(5*a*c*f*g*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) - (6*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*a^2*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2)), x, 7, (-2*(g*cos(e + f*x))^(5/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2)) - (2*(g*cos(e + f*x))^(5/2))/(a*f*g*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2)) + (6*(g*cos(e + f*x))^(5/2))/(5*a^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (6*(g*cos(e + f*x))^(5/2))/(5*a^2*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (6*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*a^2*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2)), x, 8, -((2*(g*cos(e + f*x))^(5/2))/(5*f*g*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2))) - (14*(g*cos(e + f*x))^(5/2))/(5*a*f*g*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2)) + (14*(g*cos(e + f*x))^(5/2))/(9*a^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2)) + (14*(g*cos(e + f*x))^(5/2))/(15*a^2*c*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (14*(g*cos(e + f*x))^(5/2))/(15*a^2*c^2*f*g*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) - (14*g*sqrt(cos(e + f*x))*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*a^2*c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (2^(9/4 + n)*c*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(5 + 4*m), (1/4)*(-1 - 4*n), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(f*g*(5 + 4*m))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^3, x, 4, -((2^(9/4 + m)*a^4*c^3*(g*cos(e + f*x))^(17/2)*HypergeometricFunctions._₂F₁(17/4, -(1/4) - m, 21/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-4 + m))/(17*f*g^7))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^2, x, 4, -((2^(9/4 + m)*a^3*c^2*(g*cos(e + f*x))^(13/2)*HypergeometricFunctions._₂F₁(13/4, -(1/4) - m, 17/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-3 + m))/(13*f*g^5))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^1, x, 4, -((2^(9/4 + m)*a^2*c*(g*cos(e + f*x))^(9/2)*HypergeometricFunctions._₂F₁(9/4, -(1/4) - m, 13/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-2 + m))/(9*f*g^3))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^0, x, 3, -((2^(9/4 + m)*a*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁(5/4, -(1/4) - m, 9/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-1 + m))/(5*f*g))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^1, x, 4, -((2^(9/4 + m)*g*sqrt(g*cos(e + f*x))*HypergeometricFunctions._₂F₁(1/4, -(1/4) - m, 5/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^m)/(c*f))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^2, x, 4, (2^(9/4 + m)*g^3*HypergeometricFunctions._₂F₁(-(3/4), -(1/4) - m, 1/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(1 + m))/(3*a*c^2*f*(g*cos(e + f*x))^(3/2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^3, x, 4, (2^(9/4 + m)*g^5*HypergeometricFunctions._₂F₁(-(7/4), -(1/4) - m, -(3/4), (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(2 + m))/(7*a^2*c^3*f*(g*cos(e + f*x))^(7/2))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2), x, 4, -((2^(9/4 + m)*a^3*c^2*(g*cos(e + f*x))^(15/2)*HypergeometricFunctions._₂F₁(15/4, -(1/4) - m, 19/4, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-3 + m)*sqrt(c - c*sin(e + f*x)))/(15*f*g^6))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(3/2), x, 4, -((2^(9/4 + m)*a^2*c*(g*cos(e + f*x))^(11/2)*HypergeometricFunctions._₂F₁(11/4, -(1/4) - m, 15/4, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-2 + m)*sqrt(c - c*sin(e + f*x)))/(11*f*g^4))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(1/2), x, 4, -((2^(9/4 + m)*a*(g*cos(e + f*x))^(7/2)*HypergeometricFunctions._₂F₁(7/4, -(1/4) - m, 11/4, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-1 + m)*sqrt(c - c*sin(e + f*x)))/(7*f*g^2))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(1/2), x, 4, -((2^(9/4 + m)*a*cos(e + f*x)*(g*cos(e + f*x))^(3/2)*HypergeometricFunctions._₂F₁(3/4, -(1/4) - m, 7/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-1 + m))/(3*f*sqrt(c - c*sin(e + f*x))))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(3/2), x, 4, (2^(9/4 + m)*g^2*cos(e + f*x)*HypergeometricFunctions._₂F₁(-(1/4), -(1/4) - m, 3/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^m)/(c*f*sqrt(g*cos(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(5/2), x, 4, (2^(9/4 + m)*g^4*cos(e + f*x)*HypergeometricFunctions._₂F₁(-(5/4), -(1/4) - m, -(1/4), (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(1 + m))/(5*a*c^2*f*(g*cos(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))]

    #= The*same*rules*should*be*used*to*integrate*the*following*two*problems: =#
    @test_int [((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/sqrt(c - c*sin(e + f*x)), x, 4, -((2^(9/4 + m)*a*cos(e + f*x)*(g*cos(e + f*x))^(3/2)*HypergeometricFunctions._₂F₁(3/4, -(1/4) - m, 7/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(a + a*sin(e + f*x))^(-1 + m))/(3*f*sqrt(c - c*sin(e + f*x))))]
    @test_int [((g*cos(e + f*x))^(3/2)*(c + c*sin(e + f*x))^m)/sqrt(a - a*sin(e + f*x)), x, 4, -((2^(9/4 + m)*c*cos(e + f*x)*(g*cos(e + f*x))^(3/2)*HypergeometricFunctions._₂F₁(3/4, -(1/4) - m, 7/4, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/4) - m)*(c + c*sin(e + f*x))^(-1 + m))/(3*f*sqrt(a - a*sin(e + f*x))))]

    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 3), x, 4, (2^(-(3/4) - m)*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(5 + 4*m), (1/4)*(11 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(c^2*f*g*(5 + 4*m))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 2), x, 4, (2^(1/4 - m)*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(5 + 4*m), (1/4)*(7 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(c*f*g*(5 + 4*m))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 1), x, 4, (2^(5/4 - m)*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(3 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*g*(5 + 4*m))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 0), x, 4, (2^(9/4 - m)*c*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(-1 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*g*(5 + 4*m))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m - 1), x, 4, (2^(13/4 - m)*c^2*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(-5 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*g*(5 + 4*m))]
    @test_int [(g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m - 2), x, 4, (2^(17/4 - m)*c^3*(g*cos(e + f*x))^(5/2)*HypergeometricFunctions._₂F₁((1/4)*(-9 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(-(1/4) + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*g*(5 + 4*m))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*with*m, n*and*p*symbolic=#

    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (2^(1/2 + n + p/2)*c*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(1 - 2*n - p), (1/2)*(1 + 2*m + p), (1/2)*(3 + 2*m + p), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((1/2)*(1 - 2*n - p))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(f*g*(1 + 2*m + p))]

    @test_int [(g*cos(e + f*x))^(1 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(m-1), x, 4, -((g*log(1 - sin(e + f*x))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^m)/((g*cos(e + f*x))^(2*m)*(c*f)))]

    @test_int [(g*cos(e + f*x))^(5 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 3, -((8*a^3*(g*cos(e + f*x))^(6 - 2*m)*(a + a*sin(e + f*x))^(-3 + m)*(c - c*sin(e + f*x))^n)/(f*g*(3 - m + n)*(4 - m + n)*(5 - m + n))) - (4*a^2*(g*cos(e + f*x))^(6 - 2*m)*(a + a*sin(e + f*x))^(-2 + m)*(c - c*sin(e + f*x))^n)/(f*g*(4 - m + n)*(5 - m + n)) - (a*(g*cos(e + f*x))^(6 - 2*m)*(a + a*sin(e + f*x))^(-1 + m)*(c - c*sin(e + f*x))^n)/(f*g*(5 - m + n))]
    @test_int [(g*cos(e + f*x))^(3 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 2, -((2*a^2*(g*cos(e + f*x))^(4 - 2*m)*(a + a*sin(e + f*x))^(-2 + m)*(c - c*sin(e + f*x))^n)/(f*g*(2 - m + n)*(3 - m + n))) - (a*(g*cos(e + f*x))^(4 - 2*m)*(a + a*sin(e + f*x))^(-1 + m)*(c - c*sin(e + f*x))^n)/(f*g*(3 - m + n))]
    @test_int [(g*cos(e + f*x))^(1 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 1, -((a*(g*cos(e + f*x))^(2 - 2*m)*(a + a*sin(e + f*x))^(-1 + m)*(c - c*sin(e + f*x))^n)/(f*g*(1 - m + n)))]
    @test_int [(g*cos(e + f*x))^(-1 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (HypergeometricFunctions._₂F₁(1, -m + n, 1 - m + n, (1/2)*(1 - sin(e + f*x)))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/((g*cos(e + f*x))^(2*m)*(2*f*g*(m - n)))]
    @test_int [(g*cos(e + f*x))^(-3 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (c*HypergeometricFunctions._₂F₁(2, -1 - m + n, -m + n, (1/2)*(1 - sin(e + f*x)))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/((g*cos(e + f*x))^(2*m)*(4*f*g^3*(1 + m - n)))]
    @test_int [(g*cos(e + f*x))^(-5 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (c^2*HypergeometricFunctions._₂F₁(3, -2 - m + n, -1 - m + n, (1/2)*(1 - sin(e + f*x)))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 + n))/((g*cos(e + f*x))^(2*m)*(8*f*g^5*(2 + m - n)))]

    @test_int [(g*cos(e + f*x))^(-1 - m - m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^m, x, 3, (atanh(sin(e + f*x))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^m)/((g*cos(e + f*x))^(2*m)*(f*g))]

    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n + 3), x, 4, (2^(3 - m/2 + n/2)*c^3*(g*cos(e + f*x))^(-m - n)*HypergeometricFunctions._₂F₁((1/2)*(-4 + m - n), (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((m - n)/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(f*g*(m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n + 2), x, 4, (2^(2 - m/2 + n/2)*c^2*(g*cos(e + f*x))^(-m - n)*HypergeometricFunctions._₂F₁((1/2)*(-2 + m - n), (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((m - n)/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(f*g*(m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n + 1), x, 4, (2^(1 - m/2 + n/2)*c*(g*cos(e + f*x))^(-m - n)*HypergeometricFunctions._₂F₁((m - n)/2, (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((m - n)/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(f*g*(m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 0), x, 1, ((g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(f*g*(m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 1), x, 2, ((g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(f*g*(2 + m - n)) + ((g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(c*f*g*(m - n)*(2 + m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 2), x, 3, ((g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 + n))/(f*g*(4 + m - n)) + (2*(g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(c*f*g*(2 + m - n)*(4 + m - n)) + (2*(g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(c^2*f*g*(m - n)*(2 + m - n)*(4 + m - n))]
    @test_int [(g*cos(e + f*x))^(-1 - m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 3), x, 4, ((g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-3 + n))/(f*g*(6 + m - n)) + (3*(g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 + n))/(c*f*g*(4 + m - n)*(6 + m - n)) + (6*(g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(c^2*f*g*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (6*(g*cos(e + f*x))^(-m - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n)/(c^3*f*g*(m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#
    #==#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*with*m, n*and*p*symbolic=#

    @test_int [(g*sec(e + f*x))^p*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 5, (2^(1/2 + n - p/2)*c*cos(e + f*x)*HypergeometricFunctions._₂F₁((1/2)*(1 + 2*m - p), (1/2)*(1 - 2*n + p), (1/2)*(3 + 2*m - p), (1/2)*(1 + sin(e + f*x)))*(g*sec(e + f*x))^p*(1 - sin(e + f*x))^((1/2)*(1 - 2*n + p))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(f*(1 + 2*m - p))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^1*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^2)/(2*d) + (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 3, (a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (a*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 3, -((csc(c + d*x)^2*(a + a*sin(c + d*x))^2)/(2*a*d))]
    @test_int [cos(c + d*x)*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x)^2)/(2*d)) - (a*csc(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x)^3)/(3*d)) - (a*csc(c + d*x)^4)/(4*d)]

    @test_int [cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, (a^2*sin(c + d*x)^3)/(3*d) + (a^2*sin(c + d*x)^4)/(2*d) + (a^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 4, (a^2*sin(c + d*x)^2)/(2*d) + (2*a^2*sin(c + d*x)^3)/(3*d) + (a^2*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 3, (a^2*log(sin(c + d*x)))/d + (2*a^2*sin(c + d*x))/d + (a^2*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x))/d) + (2*a^2*log(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 4, -((2*a^2*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/(2*d) + (a^2*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 3, -((csc(c + d*x)^3*(a + a*sin(c + d*x))^3)/(3*a*d))]
    @test_int [cos(c + d*x)*csc(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x)^2)/(2*d)) - (2*a^2*csc(c + d*x)^3)/(3*d) - (a^2*csc(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x)^3)/(3*d)) - (a^2*csc(c + d*x)^4)/(2*d) - (a^2*csc(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x)^4)/(4*d)) - (2*a^2*csc(c + d*x)^5)/(5*d) - (a^2*csc(c + d*x)^6)/(6*d)]

    @test_int [cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 4, (a^3*sin(c + d*x)^4)/(4*d) + (3*a^3*sin(c + d*x)^5)/(5*d) + (a^3*sin(c + d*x)^6)/(2*d) + (a^3*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, (a^3*sin(c + d*x)^3)/(3*d) + (3*a^3*sin(c + d*x)^4)/(4*d) + (3*a^3*sin(c + d*x)^5)/(5*d) + (a^3*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, -((a + a*sin(c + d*x))^4/(4*a*d)) + (a + a*sin(c + d*x))^5/(5*a^2*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 3, (a^3*log(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (3*a^3*sin(c + d*x)^2)/(2*d) + (a^3*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x))/d) + (3*a^3*log(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (a^3*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 4, -((3*a^3*csc(c + d*x))/d) - (a^3*csc(c + d*x)^2)/(2*d) + (3*a^3*log(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 4, -((3*a^3*csc(c + d*x))/d) - (3*a^3*csc(c + d*x)^2)/(2*d) - (a^3*csc(c + d*x)^3)/(3*d) + (a^3*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)*csc(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 3, -((csc(c + d*x)^4*(a + a*sin(c + d*x))^4)/(4*a*d))]
    @test_int [cos(c + d*x)*csc(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 4, (csc(c + d*x)^4*(a + a*sin(c + d*x))^4)/(20*a*d) - (csc(c + d*x)^5*(a + a*sin(c + d*x))^4)/(5*a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x)^3)/(3*d)) - (3*a^3*csc(c + d*x)^4)/(4*d) - (3*a^3*csc(c + d*x)^5)/(5*d) - (a^3*csc(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^8*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x)^4)/(4*d)) - (3*a^3*csc(c + d*x)^5)/(5*d) - (a^3*csc(c + d*x)^6)/(2*d) - (a^3*csc(c + d*x)^7)/(7*d)]

    @test_int [cos(c + d*x)*sin(c + d*x)^4*(a + a*sin(c + d*x))^4, x, 4, (a^4*sin(c + d*x)^5)/(5*d) + (2*a^4*sin(c + d*x)^6)/(3*d) + (6*a^4*sin(c + d*x)^7)/(7*d) + (a^4*sin(c + d*x)^8)/(2*d) + (a^4*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^4, x, 4, (a^4*sin(c + d*x)^4)/(4*d) + (4*a^4*sin(c + d*x)^5)/(5*d) + (a^4*sin(c + d*x)^6)/d + (4*a^4*sin(c + d*x)^7)/(7*d) + (a^4*sin(c + d*x)^8)/(8*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^4, x, 4, (a + a*sin(c + d*x))^5/(5*a*d) - (a + a*sin(c + d*x))^6/(3*a^2*d) + (a + a*sin(c + d*x))^7/(7*a^3*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1*(a + a*sin(c + d*x))^4, x, 4, -((a + a*sin(c + d*x))^5/(5*a*d)) + (a + a*sin(c + d*x))^6/(6*a^2*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x))^4, x, 3, (a^4*log(sin(c + d*x)))/d + (4*a^4*sin(c + d*x))/d + (3*a^4*sin(c + d*x)^2)/d + (4*a^4*sin(c + d*x)^3)/(3*d) + (a^4*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^4, x, 4, -((a^4*csc(c + d*x))/d) + (4*a^4*log(sin(c + d*x)))/d + (6*a^4*sin(c + d*x))/d + (2*a^4*sin(c + d*x)^2)/d + (a^4*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x))^4, x, 4, -((4*a^4*csc(c + d*x))/d) - (a^4*csc(c + d*x)^2)/(2*d) + (6*a^4*log(sin(c + d*x)))/d + (4*a^4*sin(c + d*x))/d + (a^4*sin(c + d*x)^2)/(2*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, log(1 + sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d) + sin(c + d*x)^2/(2*a*d) - sin(c + d*x)^3/(3*a*d) + sin(c + d*x)^4/(4*a*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, -(log(1 + sin(c + d*x))/(a*d)) + sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d) + sin(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, log(1 + sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d) + sin(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, -(log(1 + sin(c + d*x))/(a*d)) + sin(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - log(1 + sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - log(sin(c + d*x))/(a*d) + log(1 + sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d) + log(sin(c + d*x))/(a*d) - log(1 + sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) + csc(c + d*x)^2/(2*a*d) - csc(c + d*x)^3/(3*a*d) - log(sin(c + d*x))/(a*d) + log(1 + sin(c + d*x))/(a*d)]

    @test_int [cos(c + d*x)*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 4, -((4*log(1 + sin(c + d*x)))/(a^2*d)) + (3*sin(c + d*x))/(a^2*d) - sin(c + d*x)^2/(a^2*d) + sin(c + d*x)^3/(3*a^2*d) - 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 4, (3*log(1 + sin(c + d*x)))/(a^2*d) - (2*sin(c + d*x))/(a^2*d) + sin(c + d*x)^2/(2*a^2*d) + 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 4, -((2*log(1 + sin(c + d*x)))/(a^2*d)) + sin(c + d*x)/(a^2*d) - 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, log(1 + sin(c + d*x))/(a^2*d) + 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 3, log(sin(c + d*x))/(a^2*d) - log(1 + sin(c + d*x))/(a^2*d) + 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 4, -(csc(c + d*x)/(a^2*d)) - (2*log(sin(c + d*x)))/(a^2*d) + (2*log(1 + sin(c + d*x)))/(a^2*d) - 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 4, (2*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a^2*d) + (3*log(sin(c + d*x)))/(a^2*d) - (3*log(1 + sin(c + d*x)))/(a^2*d) + 1/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 4, -((3*csc(c + d*x))/(a^2*d)) + csc(c + d*x)^2/(a^2*d) - csc(c + d*x)^3/(3*a^2*d) - (4*log(sin(c + d*x)))/(a^2*d) + (4*log(1 + sin(c + d*x)))/(a^2*d) - 1/(d*(a^2 + a^2*sin(c + d*x)))]

    @test_int [cos(c + d*x)*sin(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 4, -((10*log(1 + sin(c + d*x)))/(a^3*d)) + (6*sin(c + d*x))/(a^3*d) - (3*sin(c + d*x)^2)/(2*a^3*d) + sin(c + d*x)^3/(3*a^3*d) + 1/(2*a*d*(a + a*sin(c + d*x))^2) - 5/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 4, (6*log(1 + sin(c + d*x)))/(a^3*d) - (3*sin(c + d*x))/(a^3*d) + sin(c + d*x)^2/(2*a^3*d) - 1/(2*a*d*(a + a*sin(c + d*x))^2) + 4/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 4, -((3*log(1 + sin(c + d*x)))/(a^3*d)) + sin(c + d*x)/(a^3*d) + 1/(2*a*d*(a + a*sin(c + d*x))^2) - 3/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 4, log(1 + sin(c + d*x))/(a^3*d) - 1/(2*a*d*(a + a*sin(c + d*x))^2) + 2/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 3, sin(c + d*x)^2/(2*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 3, log(sin(c + d*x))/(a^3*d) - log(1 + sin(c + d*x))/(a^3*d) + 1/(2*a*d*(a + a*sin(c + d*x))^2) + 1/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 4, -(csc(c + d*x)/(a^3*d)) - (3*log(sin(c + d*x)))/(a^3*d) + (3*log(1 + sin(c + d*x)))/(a^3*d) - 1/(2*a*d*(a + a*sin(c + d*x))^2) - 2/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 4, (3*csc(c + d*x))/(a^3*d) - csc(c + d*x)^2/(2*a^3*d) + (6*log(sin(c + d*x)))/(a^3*d) - (6*log(1 + sin(c + d*x)))/(a^3*d) + 1/(2*a*d*(a + a*sin(c + d*x))^2) + 3/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 4, -((6*csc(c + d*x))/(a^3*d)) + (3*csc(c + d*x)^2)/(2*a^3*d) - csc(c + d*x)^3/(3*a^3*d) - (10*log(sin(c + d*x)))/(a^3*d) + (10*log(1 + sin(c + d*x)))/(a^3*d) - 1/(2*a*d*(a + a*sin(c + d*x))^2) - 4/(d*(a^3 + a^3*sin(c + d*x)))]

    @test_int [cos(c + d*x)*sin(c + d*x)^5/(a + a*sin(c + d*x))^4, x, 4, (10*log(1 + sin(c + d*x)))/(a^4*d) - (4*sin(c + d*x))/(a^4*d) + sin(c + d*x)^2/(2*a^4*d) + 1/(3*a*d*(a + a*sin(c + d*x))^3) - 5/(2*d*(a^2 + a^2*sin(c + d*x))^2) + 10/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^4/(a + a*sin(c + d*x))^4, x, 4, -((4*log(1 + sin(c + d*x)))/(a^4*d)) + sin(c + d*x)/(a^4*d) - 1/(3*a*d*(a + a*sin(c + d*x))^3) + 2/(d*(a^2 + a^2*sin(c + d*x))^2) - 6/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^3/(a + a*sin(c + d*x))^4, x, 4, log(1 + sin(c + d*x))/(a^4*d) + 1/(3*a*d*(a + a*sin(c + d*x))^3) - 3/(2*d*(a^2 + a^2*sin(c + d*x))^2) + 3/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)*sin(c + d*x)^2/(a + a*sin(c + d*x))^4, x, 3, sin(c + d*x)^3/(3*a*d*(a + a*sin(c + d*x))^3)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1/(a + a*sin(c + d*x))^4, x, 4, 1/(3*a*d*(a + a*sin(c + d*x))^3) - 1/(2*d*(a^2 + a^2*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1/(a + a*sin(c + d*x))^4, x, 3, log(sin(c + d*x))/(a^4*d) - log(1 + sin(c + d*x))/(a^4*d) + 1/(3*a*d*(a + a*sin(c + d*x))^3) + 1/(2*d*(a^2 + a^2*sin(c + d*x))^2) + 1/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^2/(a + a*sin(c + d*x))^4, x, 4, -(csc(c + d*x)/(a^4*d)) - (4*log(sin(c + d*x)))/(a^4*d) + (4*log(1 + sin(c + d*x)))/(a^4*d) - 1/(3*a*d*(a + a*sin(c + d*x))^3) - 1/(d*(a^2 + a^2*sin(c + d*x))^2) - 3/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^3/(a + a*sin(c + d*x))^4, x, 4, (4*csc(c + d*x))/(a^4*d) - csc(c + d*x)^2/(2*a^4*d) + (10*log(sin(c + d*x)))/(a^4*d) - (10*log(1 + sin(c + d*x)))/(a^4*d) + 1/(3*a*d*(a + a*sin(c + d*x))^3) + 3/(2*d*(a^2 + a^2*sin(c + d*x))^2) + 6/(d*(a^4 + a^4*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x))^(1/2), x, 4, -((2*sqrt(a)*atanh(sqrt(a + a*sin(c + d*x))/sqrt(a)))/d) + (2*sqrt(a + a*sin(c + d*x)))/d]

    #= ::Subsubsection:: =#
    #=m<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)*sin(e+f*x)^n*(a+a*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^4, x, 3, (a^4*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (4*a^4*sin(c + d*x)^(2 + n))/(d*(2 + n)) + (6*a^4*sin(c + d*x)^(3 + n))/(d*(3 + n)) + (4*a^4*sin(c + d*x)^(4 + n))/(d*(4 + n)) + (a^4*sin(c + d*x)^(5 + n))/(d*(5 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^3, x, 3, (a^3*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (3*a^3*sin(c + d*x)^(2 + n))/(d*(2 + n)) + (3*a^3*sin(c + d*x)^(3 + n))/(d*(3 + n)) + (a^3*sin(c + d*x)^(4 + n))/(d*(4 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x, 3, (a^2*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (2*a^2*sin(c + d*x)^(2 + n))/(d*(2 + n)) + (a^2*sin(c + d*x)^(3 + n))/(d*(3 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^1, x, 3, (a*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (a*sin(c + d*x)^(2 + n))/(d*(2 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n/(a + a*sin(c + d*x))^1, x, 2, (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a*d*(1 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n/(a + a*sin(c + d*x))^2, x, 2, (HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^2*d*(1 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n/(a + a*sin(c + d*x))^3, x, 2, (HypergeometricFunctions._₂F₁(3, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^3*d*(1 + n))]
    @test_int [cos(c + d*x)*sin(c + d*x)^n/(a + a*sin(c + d*x))^4, x, 2, (HypergeometricFunctions._₂F₁(4, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^4*d*(1 + n))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 8, (a*x)/16 - (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^3*sin(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 7, (a*x)/8 - (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, (a*x)/8 - (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, (a*x)/2 - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 7, (-a)*x - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d - (a*cot(c + d*x))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 5, (-a)*x + (a*atanh(cos(c + d*x)))/(2*d) - (a*cot(c + d*x))/d - (a*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 5, (a*atanh(cos(c + d*x)))/(2*d) - (a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 6, (a*atanh(cos(c + d*x)))/(8*d) - (a*cot(c + d*x)^3)/(3*d) + (a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + a*sin(c + d*x)), x, 7, (a*atanh(cos(c + d*x)))/(8*d) - (a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^5)/(5*d) + (a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 12, (a^2*x)/8 - (2*a^2*cos(c + d*x)^3)/(3*d) + (3*a^2*cos(c + d*x)^5)/(5*d) - (a^2*cos(c + d*x)^7)/(7*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a^2*cos(c + d*x)^3*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 5, (3*a^2*x)/16 - (a^2*cos(c + d*x)^5)/(10*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (cos(c + d*x)^3*(a + a*sin(c + d*x))^3)/(6*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 5, (a^2*x)/4 - (2*a^2*cos(c + d*x)^3)/(3*d) + (a^2*cos(c + d*x)^5)/(5*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(4*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d), (a^2*x)/4 - (a^2*cos(c + d*x)^3)/(6*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(4*d) - (cos(c + d*x)^3*(a + a*sin(c + d*x))^2)/(5*d) - (cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x)))/(10*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 9, a^2*x - (a^2*atanh(cos(c + d*x)))/d + (a^2*cos(c + d*x))/d - (a^2*cos(c + d*x)^3)/(3*d) + (a^2*cos(c + d*x)*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 8, -((a^2*x)/2) - (2*a^2*atanh(cos(c + d*x)))/d + (2*a^2*cos(c + d*x))/d - (a^2*cot(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 7, -2*a^2*x - (a^2*atanh(cos(c + d*x)))/(2*d) + (a^2*cos(c + d*x))/d - (2*a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 8, (-a^2)*x + (a^2*atanh(cos(c + d*x)))/d - (a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)*csc(c + d*x))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 9, (5*a^2*atanh(cos(c + d*x)))/(8*d) - (2*a^2*cot(c + d*x)^3)/(3*d) - (3*a^2*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 10, (a^2*atanh(cos(c + d*x)))/(4*d) - (2*a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)^5)/(5*d) + (a^2*cot(c + d*x)*csc(c + d*x))/(4*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 12, (3*a^2*atanh(cos(c + d*x)))/(16*d) - (2*a^2*cot(c + d*x)^3)/(3*d) - (2*a^2*cot(c + d*x)^5)/(5*d) + (3*a^2*cot(c + d*x)*csc(c + d*x))/(16*d) - (5*a^2*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a^2*cot(c + d*x)*csc(c + d*x)^5)/(6*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 15, (5*a^3*x)/16 - (4*a^3*cos(c + d*x)^3)/(3*d) + (a^3*cos(c + d*x)^5)/d - (a^3*cos(c + d*x)^7)/(7*d) + (5*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (5*a^3*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (a^3*cos(c + d*x)^3*sin(c + d*x)^3)/(2*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 6, (7*a^3*x)/16 - (4*a^3*cos(c + d*x)^3)/(3*d) + (3*a^3*cos(c + d*x)^5)/(5*d) + (7*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (7*a^3*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (a^3*cos(c + d*x)^3*sin(c + d*x)^3)/(6*d), (7*a^3*x)/16 - (7*a^3*cos(c + d*x)^3)/(24*d) + (7*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^3*(a + a*sin(c + d*x))^2)/(10*d) - (cos(c + d*x)^3*(a + a*sin(c + d*x))^3)/(6*d) - (7*cos(c + d*x)^3*(a^3 + a^3*sin(c + d*x)))/(40*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 12, (13*a^3*x)/8 - (a^3*atanh(cos(c + d*x)))/d + (a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/d + (13*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 10, (a^3*x)/2 - (3*a^3*atanh(cos(c + d*x)))/d + (3*a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/(3*d) - (a^3*cot(c + d*x))/d + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 10, -((5*a^3*x)/2) - (5*a^3*atanh(cos(c + d*x)))/(2*d) + (3*a^3*cos(c + d*x))/d - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)*csc(c + d*x))/(2*d) + (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 10, -3*a^3*x + (a^3*atanh(cos(c + d*x)))/(2*d) + (a^3*cos(c + d*x))/d - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 11, (-a^3)*x + (13*a^3*atanh(cos(c + d*x)))/(8*d) - (a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/d - (11*a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 12, (7*a^3*atanh(cos(c + d*x)))/(8*d) - (4*a^3*cot(c + d*x)^3)/(3*d) - (a^3*cot(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 14, (7*a^3*atanh(cos(c + d*x)))/(16*d) - (4*a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)^5)/(5*d) + (7*a^3*cot(c + d*x)*csc(c + d*x))/(16*d) - (17*a^3*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a^3*cot(c + d*x)*csc(c + d*x)^5)/(6*d)]

    @test_int [cos(c + d*x)^2*csc(c + d*x)^0*(a + a*sin(c + d*x))^4, x, 6, (21*a^4*x)/16 - (7*a^4*cos(c + d*x)^3)/(8*d) + (21*a^4*cos(c + d*x)*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^3*(a + a*sin(c + d*x))^3)/(6*d) - (3*cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x))^2)/(10*d) - (21*cos(c + d*x)^3*(a^4 + a^4*sin(c + d*x)))/(40*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^4, x, 15, (5*a^4*x)/2 - (a^4*atanh(cos(c + d*x)))/d + (a^4*cos(c + d*x))/d - (7*a^4*cos(c + d*x)^3)/(3*d) + (a^4*cos(c + d*x)^5)/(5*d) + (5*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) - (a^4*cos(c + d*x)^3*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^4, x, 12, (17*a^4*x)/8 - (4*a^4*atanh(cos(c + d*x)))/d + (4*a^4*cos(c + d*x))/d - (4*a^4*cos(c + d*x)^3)/(3*d) - (a^4*cot(c + d*x))/d + (23*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^4*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 6, (3*x)/(8*a) + cos(c + d*x)/(a*d) - (2*cos(c + d*x)^3)/(3*a*d) + cos(c + d*x)^5/(5*a*d) - (3*cos(c + d*x)*sin(c + d*x))/(8*a*d) - (cos(c + d*x)*sin(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 6, -((3*x)/(8*a)) - cos(c + d*x)/(a*d) + cos(c + d*x)^3/(3*a*d) + (3*cos(c + d*x)*sin(c + d*x))/(8*a*d) + (cos(c + d*x)*sin(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, x/(2*a) + cos(c + d*x)/(a*d) - cos(c + d*x)^3/(3*a*d) - (cos(c + d*x)*sin(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, -(x/(2*a)) - cos(c + d*x)/(a*d) + (cos(c + d*x)*sin(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 3, -(x/a) - atanh(cos(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, atanh(cos(c + d*x))/(a*d) - cot(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 5, -(atanh(cos(c + d*x))/(2*a*d)) + cot(c + d*x)/(a*d) - (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 5, atanh(cos(c + d*x))/(2*a*d) - cot(c + d*x)/(a*d) - cot(c + d*x)^3/(3*a*d) + (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 6, -((3*atanh(cos(c + d*x)))/(8*a*d)) + cot(c + d*x)/(a*d) + cot(c + d*x)^3/(3*a*d) - (3*cot(c + d*x)*csc(c + d*x))/(8*a*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 6, (3*atanh(cos(c + d*x)))/(8*a*d) - cot(c + d*x)/(a*d) - (2*cot(c + d*x)^3)/(3*a*d) - cot(c + d*x)^5/(5*a*d) + (3*cot(c + d*x)*csc(c + d*x))/(8*a*d) + (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 12, -((27*x)/(8*a^2)) - (4*cos(c + d*x))/(a^2*d) + (2*cos(c + d*x)^3)/(3*a^2*d) + (11*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) + (cos(c + d*x)*sin(c + d*x)^3)/(4*a^2*d) - (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 9, (3*x)/a^2 + (3*cos(c + d*x))/(a^2*d) - cos(c + d*x)^3/(3*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(a^2*d) + (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 8, -((5*x)/(2*a^2)) - (2*cos(c + d*x))/(a^2*d) + (cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 3, (2*x)/a^2 + cos(c + d*x)/(a^2*d) + (2*cos(c + d*x))/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 5, -(atanh(cos(c + d*x))/(a^2*d)) + (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, If(13<9, 9, 7), If(13<9, (2*atanh(cos(c + d*x)))/(a^2*d) - (3*cot(c + d*x))/(a^2*d) + (2*cot(c + d*x))/(a^2*d*(1 + sin(c + d*x))), (2*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a^2*d) - (2*cot(c + d*x))/(a^2*d*(1 + csc(c + d*x))))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 9, -((5*atanh(cos(c + d*x)))/(2*a^2*d)) + (2*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d) + (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 11, (3*atanh(cos(c + d*x)))/(a^2*d) - (3*cot(c + d*x))/(a^2*d) - cot(c + d*x)^3/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(a^2*d) - (2*cos(c + d*x))/(a^2*d*(1 + sin(c + d*x)))]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 9, -((11*x)/(2*a^3)) - (3*cos(c + d*x))/(a^3*d) + (cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (2*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x))^2) - (19*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 4, (3*x)/a^3 + (3*cos(c + d*x))/(a^3*d) - cos(c + d*x)^3/(3*d*(a + a*sin(c + d*x))^3) + (2*cos(c + d*x)^3)/(a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 3, -(x/a^3) - (7*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x))) + (2*cos(c + d*x))/(3*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 7, -(atanh(cos(c + d*x))/(a^3*d)) + (2*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x))^2) + (5*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, If(13<9, 13, 10), If(13<9, (3*atanh(cos(c + d*x)))/(a^3*d) - (14*cot(c + d*x))/(3*a^3*d) + (2*cot(c + d*x))/(3*a^3*d*(1 + sin(c + d*x))^2) + (3*cot(c + d*x))/(a^3*d*(1 + sin(c + d*x))), (3*atanh(cos(c + d*x)))/(a^3*d) - cot(c + d*x)/(a^3*d) + (2*cot(c + d*x))/(3*a^3*d*(1 + csc(c + d*x))^2) - (13*cot(c + d*x))/(3*a^3*d*(1 + csc(c + d*x))))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 11, -((11*atanh(cos(c + d*x)))/(2*a^3*d)) + (3*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d) + (2*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x))^2) + (17*cos(c + d*x))/(3*a^3*d*(1 + sin(c + d*x)))]

    @test_int [cos(e + f*x)^2*sin(e + f*x)^1/(a + a*sin(e + f*x))^6, x, 5, (2*cos(e + f*x))/(9*a*f*(a + a*sin(e + f*x))^5) - (19*cos(e + f*x))/(63*a^2*f*(a + a*sin(e + f*x))^4) + (2*cos(e + f*x))/(105*f*(a^2 + a^2*sin(e + f*x))^3) + (4*cos(e + f*x))/(315*f*(a^3 + a^3*sin(e + f*x))^2) + (4*cos(e + f*x))/(315*f*(a^6 + a^6*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 7, -((76*a*cos(c + d*x))/(495*d*sqrt(a + a*sin(c + d*x)))) - (38*a*cos(c + d*x)*sin(c + d*x)^3)/(693*d*sqrt(a + a*sin(c + d*x))) + (2*a*cos(c + d*x)*sin(c + d*x)^4)/(99*d*sqrt(a + a*sin(c + d*x))) + (152*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3465*d) + (2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(11*d) - (76*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(1155*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 4, -((8*a^2*cos(c + d*x)^3)/(63*d*(a + a*sin(c + d*x))^(3/2))) - (2*a*cos(c + d*x)^3)/(21*d*sqrt(a + a*sin(c + d*x))) + (4*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(21*d) - (2*cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(9*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 3, -((8*a^2*cos(c + d*x)^3)/(105*d*(a + a*sin(c + d*x))^(3/2))) - (2*a*cos(c + d*x)^3)/(35*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(7*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 5, -((2*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (2*a*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 4, -((sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (3*a*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 5, (5*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) - (a*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 6, (3*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) + (3*a*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(3*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 8, -((1724*a^2*cos(c + d*x))/(6435*d*sqrt(a + a*sin(c + d*x)))) - (862*a^2*cos(c + d*x)*sin(c + d*x)^3)/(9009*d*sqrt(a + a*sin(c + d*x))) - (38*a^2*cos(c + d*x)*sin(c + d*x)^4)/(1287*d*sqrt(a + a*sin(c + d*x))) + (3448*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(45045*d) + (6*a*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(143*d) - (1724*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(15015*d) + (2*cos(c + d*x)*sin(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(13*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 5, -((64*a^3*cos(c + d*x)^3)/(385*d*(a + a*sin(c + d*x))^(3/2))) - (48*a^2*cos(c + d*x)^3)/(385*d*sqrt(a + a*sin(c + d*x))) - (6*a*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(77*d) + (4*cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(33*d) - (2*cos(c + d*x)^3*(a + a*sin(c + d*x))^(5/2))/(11*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 4, -((64*a^3*cos(c + d*x)^3)/(315*d*(a + a*sin(c + d*x))^(3/2))) - (16*a^2*cos(c + d*x)^3)/(105*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(21*d) - (2*cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(9*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 6, -((2*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (2*a^2*cos(c + d*x))/(5*d*sqrt(a + a*sin(c + d*x))) + (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*d) + (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 5, -((3*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (11*a^2*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) + (5*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d) - (cot(c + d*x)*(a + a*sin(c + d*x))^(3/2))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 6, (a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) + (13*a^2*cos(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (3*a*cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (cot(c + d*x)*csc(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 6, (13*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) + (5*a^2*cot(c + d*x))/(24*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(3*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 6, -((4*cos(c + d*x))/(45*d*sqrt(a + a*sin(c + d*x)))) - (2*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sin(c + d*x)^4)/(9*d*sqrt(a + a*sin(c + d*x))) + (8*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*a*d) - (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*a^2*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 4, -((22*a*cos(c + d*x)^3)/(105*d*(a + a*sin(c + d*x))^(3/2))) + (12*cos(c + d*x)^3)/(35*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(7*a*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 2, (2*a*cos(c + d*x)^3)/(15*d*(a + a*sin(c + d*x))^(3/2)) - (2*cos(c + d*x)^3)/(5*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 4, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(sqrt(a)*d)) + (2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 4, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(sqrt(a)*d) - cot(c + d*x)/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 5, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(4*sqrt(a)*d) + cot(c + d*x)/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/sqrt(a + a*sin(c + d*x)), x, 6, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(8*sqrt(a)*d) + cot(c + d*x)/(8*d*sqrt(a + a*sin(c + d*x))) + (cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 8, (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (344*cos(c + d*x))/(105*a*d*sqrt(a + a*sin(c + d*x))) - (16*cos(c + d*x)*sin(c + d*x)^2)/(35*a*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sin(c + d*x)^3)/(7*a*d*sqrt(a + a*sin(c + d*x))) + (76*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*a^2*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 5, -((2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d)) + (18*cos(c + d*x))/(5*a*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)^3)/(5*a*d*sqrt(a + a*sin(c + d*x))) - (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*a^2*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 4, (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (10*cos(c + d*x))/(3*a*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*a^2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 6, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d)) + (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 6, (3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d) - (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - cot(c + d*x)/(a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 8, -((11*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(3/2)*d)) + (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) + (5*cot(c + d*x))/(4*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 9, (23*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*a^(3/2)*d) - (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (9*cot(c + d*x))/(8*a*d*sqrt(a + a*sin(c + d*x))) + (7*cot(c + d*x)*csc(c + d*x))/(12*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*sqrt(a + a*sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^3*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^3*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^3*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^4)/(4*d) + (a*sin(c + d*x)^5)/(5*d) - (a*sin(c + d*x)^6)/(6*d) - (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^4)/(4*d) - (a*sin(c + d*x)^5)/(5*d) - (a*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, -((a*cos(c + d*x)^4)/(4*d)) + (a*sin(c + d*x)^3)/(3*d) - (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^0*(a + a*sin(c + d*x)), x, 3, (2*(a + a*sin(c + d*x))^3)/(3*a^2*d) - (a + a*sin(c + d*x))^4/(4*a^3*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 4, (a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/(2*d) - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (a*log(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 3, -((a*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (a*log(sin(c + d*x)))/d - (a*sin(c + d*x))/d]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^3*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, sin(c + d*x)^3/(3*a*d) - sin(c + d*x)^4/(4*a*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 5, sin(c + d*x)^2/(2*a*d) - sin(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^0/(a + a*sin(c + d*x)), x, 2, sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - log(sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 5, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)^2/(2*a*d) - csc(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)^3/(3*a*d) - csc(c + d*x)^4/(4*a*d)]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^3*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 9, (3*a*x)/128 - (a*cos(c + d*x)^5)/(5*d) + (2*a*cos(c + d*x)^7)/(7*d) - (a*cos(c + d*x)^9)/(9*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(128*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 9, (3*a*x)/128 - (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)^7)/(7*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(128*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 8, (a*x)/16 - (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)^7)/(7*d) + (a*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 7, (a*x)/16 - (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 8, (3*a*x)/8 - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 9, -((3*a*x)/2) - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) - (3*a*cot(c + d*x))/(2*d) + (a*cos(c + d*x)^2*cot(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 9, -((3*a*x)/2) + (3*a*atanh(cos(c + d*x)))/(2*d) - (3*a*cos(c + d*x))/(2*d) - (3*a*cot(c + d*x))/(2*d) + (a*cos(c + d*x)^2*cot(c + d*x))/(2*d) - (a*cos(c + d*x)*cot(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 9, a*x + (3*a*atanh(cos(c + d*x)))/(2*d) - (3*a*cos(c + d*x))/(2*d) + (a*cot(c + d*x))/d - (a*cos(c + d*x)*cot(c + d*x)^2)/(2*d) - (a*cot(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 7, a*x - (3*a*atanh(cos(c + d*x)))/(8*d) + (a*cot(c + d*x))/d - (a*cot(c + d*x)^3)/(3*d) + (3*a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + a*sin(c + d*x)), x, 6, -((3*a*atanh(cos(c + d*x)))/(8*d)) - (a*cot(c + d*x)^5)/(5*d) + (3*a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + a*sin(c + d*x)), x, 7, -((a*atanh(cos(c + d*x)))/(16*d)) - (a*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)*csc(c + d*x))/(16*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + a*sin(c + d*x)), x, 8, -((a*atanh(cos(c + d*x)))/(16*d)) - (a*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)^7)/(7*d) - (a*cot(c + d*x)*csc(c + d*x))/(16*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + a*sin(c + d*x)), x, 9, -((3*a*atanh(cos(c + d*x)))/(128*d)) - (a*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)^7)/(7*d) - (3*a*cot(c + d*x)*csc(c + d*x))/(128*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (a*cot(c + d*x)*csc(c + d*x)^5)/(16*d) - (a*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 16, (9*a^2*x)/256 - (2*a^2*cos(c + d*x)^5)/(5*d) + (4*a^2*cos(c + d*x)^7)/(7*d) - (2*a^2*cos(c + d*x)^9)/(9*d) + (9*a^2*cos(c + d*x)*sin(c + d*x))/(256*d) + (3*a^2*cos(c + d*x)^3*sin(c + d*x))/(128*d) - (3*a^2*cos(c + d*x)^5*sin(c + d*x))/(32*d) - (3*a^2*cos(c + d*x)^5*sin(c + d*x)^3)/(16*d) - (a^2*cos(c + d*x)^5*sin(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 13, (3*a^2*x)/64 - (2*a^2*cos(c + d*x)^5)/(5*d) + (3*a^2*cos(c + d*x)^7)/(7*d) - (a^2*cos(c + d*x)^9)/(9*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(64*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(32*d) - (a^2*cos(c + d*x)^5*sin(c + d*x))/(8*d) - (a^2*cos(c + d*x)^5*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 14, (11*a^2*x)/128 - (2*a^2*cos(c + d*x)^5)/(5*d) + (2*a^2*cos(c + d*x)^7)/(7*d) + (11*a^2*cos(c + d*x)*sin(c + d*x))/(128*d) + (11*a^2*cos(c + d*x)^3*sin(c + d*x))/(192*d) - (11*a^2*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (a^2*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 6, (a^2*x)/8 - (a^2*cos(c + d*x)^5)/(15*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(12*d) - (cos(c + d*x)^5*(a + a*sin(c + d*x))^2)/(7*d) - (cos(c + d*x)^5*(a^2 + a^2*sin(c + d*x)))/(21*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 11, (3*a^2*x)/4 - (a^2*atanh(cos(c + d*x)))/d + (a^2*cos(c + d*x))/d + (a^2*cos(c + d*x)^3)/(3*d) - (a^2*cos(c + d*x)^5)/(5*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 13, -((9*a^2*x)/8) - (2*a^2*atanh(cos(c + d*x)))/d + (2*a^2*cos(c + d*x))/d + (2*a^2*cos(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^2*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 12, -3*a^2*x + (a^2*atanh(cos(c + d*x)))/(2*d) + (a^2*cos(c + d*x)^3)/(3*d) - (2*a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)*csc(c + d*x))/(2*d) - (a^2*cos(c + d*x)*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 12, -((a^2*x)/2) + (3*a^2*atanh(cos(c + d*x)))/d - (2*a^2*cos(c + d*x))/d - (a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)*csc(c + d*x))/d - (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 13, 2*a^2*x + (9*a^2*atanh(cos(c + d*x)))/(8*d) - (a^2*cos(c + d*x))/d + (2*a^2*cot(c + d*x))/d - (2*a^2*cot(c + d*x)^3)/(3*d) + (a^2*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 10, a^2*x - (3*a^2*atanh(cos(c + d*x)))/(4*d) + (a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)^5)/(5*d) + (3*a^2*cot(c + d*x)*csc(c + d*x))/(4*d) - (a^2*cot(c + d*x)^3*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 11, -((7*a^2*atanh(cos(c + d*x)))/(16*d)) - (2*a^2*cot(c + d*x)^5)/(5*d) + (5*a^2*cot(c + d*x)*csc(c + d*x))/(16*d) - (a^2*cot(c + d*x)^3*csc(c + d*x))/(4*d) + (a^2*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + a*sin(c + d*x))^2, x, 14, -((11*a^2*atanh(cos(c + d*x)))/(128*d)) - (2*a^2*cot(c + d*x)^5)/(5*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (11*a^2*cot(c + d*x)*csc(c + d*x))/(128*d) + (7*a^2*cot(c + d*x)*csc(c + d*x)^3)/(64*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d) + (a^2*cot(c + d*x)*csc(c + d*x)^5)/(16*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^10*(a + a*sin(c + d*x))^2, x, 13, -((3*a^2*atanh(cos(c + d*x)))/(64*d)) - (2*a^2*cot(c + d*x)^5)/(5*d) - (3*a^2*cot(c + d*x)^7)/(7*d) - (a^2*cot(c + d*x)^9)/(9*d) - (3*a^2*cot(c + d*x)*csc(c + d*x))/(64*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(32*d) + (a^2*cot(c + d*x)*csc(c + d*x)^5)/(8*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^11*(a + a*sin(c + d*x))^2, x, 16, -((9*a^2*atanh(cos(c + d*x)))/(256*d)) - (2*a^2*cot(c + d*x)^5)/(5*d) - (4*a^2*cot(c + d*x)^7)/(7*d) - (2*a^2*cot(c + d*x)^9)/(9*d) - (9*a^2*cot(c + d*x)*csc(c + d*x))/(256*d) - (3*a^2*cot(c + d*x)*csc(c + d*x)^3)/(128*d) + (9*a^2*cot(c + d*x)*csc(c + d*x)^5)/(160*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d) + (3*a^2*cot(c + d*x)*csc(c + d*x)^7)/(80*d) - (a^2*cot(c + d*x)^3*csc(c + d*x)^7)/(10*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 19, (15*a^3*x)/256 - (4*a^3*cos(c + d*x)^5)/(5*d) + (9*a^3*cos(c + d*x)^7)/(7*d) - (2*a^3*cos(c + d*x)^9)/(3*d) + (a^3*cos(c + d*x)^11)/(11*d) + (15*a^3*cos(c + d*x)*sin(c + d*x))/(256*d) + (5*a^3*cos(c + d*x)^3*sin(c + d*x))/(128*d) - (5*a^3*cos(c + d*x)^5*sin(c + d*x))/(32*d) - (5*a^3*cos(c + d*x)^5*sin(c + d*x)^3)/(16*d) - (3*a^3*cos(c + d*x)^5*sin(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 19, (21*a^3*x)/256 - (4*a^3*cos(c + d*x)^5)/(5*d) + (a^3*cos(c + d*x)^7)/d - (a^3*cos(c + d*x)^9)/(3*d) + (21*a^3*cos(c + d*x)*sin(c + d*x))/(256*d) + (7*a^3*cos(c + d*x)^3*sin(c + d*x))/(128*d) - (7*a^3*cos(c + d*x)^5*sin(c + d*x))/(32*d) - (7*a^3*cos(c + d*x)^5*sin(c + d*x)^3)/(16*d) - (a^3*cos(c + d*x)^5*sin(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 17, (17*a^3*x)/128 - (4*a^3*cos(c + d*x)^5)/(5*d) + (5*a^3*cos(c + d*x)^7)/(7*d) - (a^3*cos(c + d*x)^9)/(9*d) + (17*a^3*cos(c + d*x)*sin(c + d*x))/(128*d) + (17*a^3*cos(c + d*x)^3*sin(c + d*x))/(192*d) - (17*a^3*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (3*a^3*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 7, (27*a^3*x)/128 - (9*a^3*cos(c + d*x)^5)/(80*d) + (27*a^3*cos(c + d*x)*sin(c + d*x))/(128*d) + (9*a^3*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (3*a*cos(c + d*x)^5*(a + a*sin(c + d*x))^2)/(56*d) - (cos(c + d*x)^5*(a + a*sin(c + d*x))^3)/(8*d) - (9*cos(c + d*x)^5*(a^3 + a^3*sin(c + d*x)))/(112*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 15, (19*a^3*x)/16 - (a^3*atanh(cos(c + d*x)))/d + (a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/(3*d) - (3*a^3*cos(c + d*x)^5)/(5*d) + (19*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (19*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a^3*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 15, -((3*a^3*x)/8) - (3*a^3*atanh(cos(c + d*x)))/d + (3*a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/d - (a^3*cos(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x))/d + (11*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) - (3*a^3*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 15, -((33*a^3*x)/8) - (3*a^3*atanh(cos(c + d*x)))/(2*d) + (2*a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/d - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)*csc(c + d*x))/(2*d) - (7*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^3*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 14, -((7*a^3*x)/2) + (7*a^3*atanh(cos(c + d*x)))/(2*d) - (2*a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/(3*d) - (2*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)*csc(c + d*x))/(2*d) - (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 15, (3*a^3*x)/2 + (33*a^3*atanh(cos(c + d*x)))/(8*d) - (3*a^3*cos(c + d*x))/d + (2*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/d - (7*a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d) - (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 15, 3*a^3*x + (3*a^3*atanh(cos(c + d*x)))/(8*d) - (a^3*cos(c + d*x))/d + (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/d - (a^3*cot(c + d*x)^5)/(5*d) + (11*a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 14, a^3*x - (19*a^3*atanh(cos(c + d*x)))/(16*d) + (a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)^5)/(5*d) + (17*a^3*cot(c + d*x)*csc(c + d*x))/(16*d) - (3*a^3*cot(c + d*x)^3*csc(c + d*x))/(4*d) + (a^3*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + a*sin(c + d*x))^3, x, 14, -((9*a^3*atanh(cos(c + d*x)))/(16*d)) - (4*a^3*cot(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x)^7)/(7*d) + (3*a^3*cot(c + d*x)*csc(c + d*x))/(16*d) - (a^3*cot(c + d*x)^3*csc(c + d*x))/(4*d) + (3*a^3*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + a*sin(c + d*x))^3, x, 16, -((27*a^3*atanh(cos(c + d*x)))/(128*d)) - (4*a^3*cot(c + d*x)^5)/(5*d) - (3*a^3*cot(c + d*x)^7)/(7*d) - (27*a^3*cot(c + d*x)*csc(c + d*x))/(128*d) + (23*a^3*cot(c + d*x)*csc(c + d*x)^3)/(64*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(2*d) + (a^3*cot(c + d*x)*csc(c + d*x)^5)/(16*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^10*(a + a*sin(c + d*x))^3, x, 17, -((17*a^3*atanh(cos(c + d*x)))/(128*d)) - (4*a^3*cot(c + d*x)^5)/(5*d) - (5*a^3*cot(c + d*x)^7)/(7*d) - (a^3*cot(c + d*x)^9)/(9*d) - (17*a^3*cot(c + d*x)*csc(c + d*x))/(128*d) + (5*a^3*cot(c + d*x)*csc(c + d*x)^3)/(64*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d) + (3*a^3*cot(c + d*x)*csc(c + d*x)^5)/(16*d) - (3*a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^11*(a + a*sin(c + d*x))^3, x, 19, -((21*a^3*atanh(cos(c + d*x)))/(256*d)) - (4*a^3*cot(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x)^7)/d - (a^3*cot(c + d*x)^9)/(3*d) - (21*a^3*cot(c + d*x)*csc(c + d*x))/(256*d) - (7*a^3*cot(c + d*x)*csc(c + d*x)^3)/(128*d) + (29*a^3*cot(c + d*x)*csc(c + d*x)^5)/(160*d) - (3*a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d) + (3*a^3*cot(c + d*x)*csc(c + d*x)^7)/(80*d) - (a^3*cot(c + d*x)^3*csc(c + d*x)^7)/(10*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^4, x, 8, (55*a^4*x)/256 - (11*a^4*cos(c + d*x)^7)/(112*d) + (55*a^4*cos(c + d*x)*sin(c + d*x))/(256*d) + (55*a^4*cos(c + d*x)^3*sin(c + d*x))/(384*d) + (11*a^4*cos(c + d*x)^5*sin(c + d*x))/(96*d) - (cos(c + d*x)^5*(a + a*sin(c + d*x))^5)/(10*a*d) - (cos(c + d*x)^7*(a^2 + a^2*sin(c + d*x))^2)/(18*d) - (11*cos(c + d*x)^7*(a^4 + a^4*sin(c + d*x)))/(144*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x))^4, x, 17, -((61*a^4*x)/8) + (2*a^4*atanh(cos(c + d*x)))/d + (4*a^4*cos(c + d*x)^3)/(3*d) - (5*a^4*cot(c + d*x))/d - (a^4*cot(c + d*x)^3)/(3*d) - (2*a^4*cot(c + d*x)*csc(c + d*x))/d - (19*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^4*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 8, x/(16*a) + cos(c + d*x)^3/(3*a*d) - (2*cos(c + d*x)^5)/(5*a*d) + cos(c + d*x)^7/(7*a*d) + (cos(c + d*x)*sin(c + d*x))/(16*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(8*a*d) - (cos(c + d*x)^3*sin(c + d*x)^3)/(6*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 8, -(x/(16*a)) - cos(c + d*x)^3/(3*a*d) + cos(c + d*x)^5/(5*a*d) - (cos(c + d*x)*sin(c + d*x))/(16*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(8*a*d) + (cos(c + d*x)^3*sin(c + d*x)^3)/(6*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 7, x/(8*a) + cos(c + d*x)^3/(3*a*d) - cos(c + d*x)^5/(5*a*d) + (cos(c + d*x)*sin(c + d*x))/(8*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 6, -(x/(8*a)) - cos(c + d*x)^3/(3*a*d) - (cos(c + d*x)*sin(c + d*x))/(8*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 6, -(x/(2*a)) - atanh(cos(c + d*x))/(a*d) + cos(c + d*x)/(a*d) - (cos(c + d*x)*sin(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 6, -(x/a) + atanh(cos(c + d*x))/(a*d) - cos(c + d*x)/(a*d) - cot(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 5, x/a + atanh(cos(c + d*x))/(2*a*d) + cot(c + d*x)/(a*d) - (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 5, -(atanh(cos(c + d*x))/(2*a*d)) - cot(c + d*x)^3/(3*a*d) + (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 6, atanh(cos(c + d*x))/(8*a*d) + cot(c + d*x)^3/(3*a*d) + (cot(c + d*x)*csc(c + d*x))/(8*a*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 7, -(atanh(cos(c + d*x))/(8*a*d)) - cot(c + d*x)^3/(3*a*d) - cot(c + d*x)^5/(5*a*d) - (cot(c + d*x)*csc(c + d*x))/(8*a*d) + (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7/(a + a*sin(c + d*x)), x, 8, atanh(cos(c + d*x))/(16*a*d) + cot(c + d*x)^3/(3*a*d) + cot(c + d*x)^5/(5*a*d) + (cot(c + d*x)*csc(c + d*x))/(16*a*d) + (cot(c + d*x)*csc(c + d*x)^3)/(24*a*d) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 11, -((5*x)/(8*a^2)) - (2*cos(c + d*x))/(a^2*d) + (5*cos(c + d*x)^3)/(3*a^2*d) - (4*cos(c + d*x)^5)/(5*a^2*d) + cos(c + d*x)^7/(7*a^2*d) + (5*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) + (5*cos(c + d*x)*sin(c + d*x)^3)/(12*a^2*d) + (cos(c + d*x)*sin(c + d*x)^5)/(3*a^2*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 12, (11*x)/(16*a^2) + (2*cos(c + d*x))/(a^2*d) - (4*cos(c + d*x)^3)/(3*a^2*d) + (2*cos(c + d*x)^5)/(5*a^2*d) - (11*cos(c + d*x)*sin(c + d*x))/(16*a^2*d) - (11*cos(c + d*x)*sin(c + d*x)^3)/(24*a^2*d) - (cos(c + d*x)*sin(c + d*x)^5)/(6*a^2*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 10, -((3*x)/(4*a^2)) - (2*cos(c + d*x))/(a^2*d) + cos(c + d*x)^3/(a^2*d) - cos(c + d*x)^5/(5*a^2*d) + (3*cos(c + d*x)*sin(c + d*x))/(4*a^2*d) + (cos(c + d*x)*sin(c + d*x)^3)/(2*a^2*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 10, (7*x)/(8*a^2) + (2*cos(c + d*x))/(a^2*d) - (2*cos(c + d*x)^3)/(3*a^2*d) - (7*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) - (cos(c + d*x)*sin(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, -(x/a^2) - (2*cos(c + d*x)^3)/(3*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(a^2*d) - cos(c + d*x)^5/(d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, -((2*x)/a^2) - atanh(cos(c + d*x))/(a^2*d) - cos(c + d*x)/(a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 6, x/a^2 + (2*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 8, -((3*atanh(cos(c + d*x)))/(2*a^2*d)) + (2*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 9, atanh(cos(c + d*x))/(a^2*d) - (2*cot(c + d*x))/(a^2*d) - cot(c + d*x)^3/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 10, -((7*atanh(cos(c + d*x)))/(8*a^2*d)) + (2*cot(c + d*x))/(a^2*d) + (2*cot(c + d*x)^3)/(3*a^2*d) - (7*cot(c + d*x)*csc(c + d*x))/(8*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 10, (3*atanh(cos(c + d*x)))/(4*a^2*d) - (2*cot(c + d*x))/(a^2*d) - cot(c + d*x)^3/(a^2*d) - cot(c + d*x)^5/(5*a^2*d) + (3*cot(c + d*x)*csc(c + d*x))/(4*a^2*d) + (cot(c + d*x)*csc(c + d*x)^3)/(2*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 12, -((11*atanh(cos(c + d*x)))/(16*a^2*d)) + (2*cot(c + d*x))/(a^2*d) + (4*cot(c + d*x)^3)/(3*a^2*d) + (2*cot(c + d*x)^5)/(5*a^2*d) - (11*cot(c + d*x)*csc(c + d*x))/(16*a^2*d) - (11*cot(c + d*x)*csc(c + d*x)^3)/(24*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a^2*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 12, (51*x)/(8*a^3) + (7*cos(c + d*x))/(a^3*d) - cos(c + d*x)^3/(a^3*d) - (19*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) - (cos(c + d*x)*sin(c + d*x)^3)/(4*a^3*d) + (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 9, -((11*x)/(2*a^3)) - (5*cos(c + d*x))/(a^3*d) + cos(c + d*x)^3/(3*a^3*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 4, (9*x)/(2*a^3) + (9*cos(c + d*x))/(2*a^3*d) + cos(c + d*x)^5/(d*(a + a*sin(c + d*x))^3) + (3*cos(c + d*x)^3)/(2*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 5, x/a^3 - atanh(cos(c + d*x))/(a^3*d) + (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 7, (3*atanh(cos(c + d*x)))/(a^3*d) - cot(c + d*x)/(a^3*d) - (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 9, -((9*atanh(cos(c + d*x)))/(2*a^3*d)) + (3*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d) + (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + a*sin(c + d*x))^3, x, If(13<9, 12, 11), If(13<9, (11*atanh(cos(c + d*x)))/(2*a^3*d) - (13*cot(c + d*x))/(a^3*d) - (13*cot(c + d*x)^3)/(3*a^3*d) + (11*cot(c + d*x)*csc(c + d*x))/(2*a^3*d) + (4*cot(c + d*x)*csc(c + d*x)^2)/(a^3*d*(1 + sin(c + d*x))), (11*atanh(cos(c + d*x)))/(2*a^3*d) - (5*cot(c + d*x))/(a^3*d) - cot(c + d*x)^3/(3*a^3*d) + (3*cot(c + d*x)*csc(c + d*x))/(2*a^3*d) - (4*cot(c + d*x))/(a^3*d*(1 + csc(c + d*x))))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 14, -((51*atanh(cos(c + d*x)))/(8*a^3*d)) + (7*cot(c + d*x))/(a^3*d) + cot(c + d*x)^3/(a^3*d) - (19*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^3*d) + (4*cos(c + d*x))/(a^3*d*(1 + sin(c + d*x)))]

    @test_int [cos(e + f*x)^4*sin(e + f*x)^1/(a + a*sin(e + f*x))^6, x, 2, cos(e + f*x)^5/(7*f*(a + a*sin(e + f*x))^6) - 6*cos(e + f*x)^5/(35*a*f*(a + a*sin(e + f*x))^5)]

    @test_int [cos(e + f*x)^4*sin(e + f*x)^2/(a + a*sin(e + f*x))^7, x, 18, -a*cos(e + f*x)^7/(18*f*(a + a*sin(e + f*x))^8) + 25*cos(e + f*x)^5/(126*a*f*(a + a*sin(e + f*x))^6) - 47*cos(e + f*x)^5/(315*a^2*f*(a + a*sin(e + f*x))^5), -((4*cos(e + f*x))/(9*a^7*f*(1 + sin(e + f*x))^5)) + (92*cos(e + f*x))/(63*a^7*f*(1 + sin(e + f*x))^4) - (181*cos(e + f*x))/(105*a^7*f*(1 + sin(e + f*x))^3) + (268*cos(e + f*x))/(315*a^7*f*(1 + sin(e + f*x))^2) - (47*cos(e + f*x))/(315*a^7*f*(1 + sin(e + f*x)))]

    @test_int [cos(e + f*x)^4*sin(e + f*x)^3/(a + a*sin(e + f*x))^8, x, 24, (4*cos(e + f*x))/(11*a^8*f*(1 + sin(e + f*x))^6) - (52*cos(e + f*x))/(33*a^8*f*(1 + sin(e + f*x))^5) + (617*cos(e + f*x))/(231*a^8*f*(1 + sin(e + f*x))^4) - (846*cos(e + f*x))/(385*a^8*f*(1 + sin(e + f*x))^3) + (1003*cos(e + f*x))/(1155*a^8*f*(1 + sin(e + f*x))^2) - (152*cos(e + f*x))/(1155*a^8*f*(1 + sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 5, -((1472*a^3*cos(c + d*x)^5)/(45045*d*(a + a*sin(c + d*x))^(5/2))) - (368*a^2*cos(c + d*x)^5)/(9009*d*(a + a*sin(c + d*x))^(3/2)) - (46*a*cos(c + d*x)^5)/(1287*d*sqrt(a + a*sin(c + d*x))) + (20*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(143*d) - (2*cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(13*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 4, -((64*a^3*cos(c + d*x)^5)/(3465*d*(a + a*sin(c + d*x))^(5/2))) - (16*a^2*cos(c + d*x)^5)/(693*d*(a + a*sin(c + d*x))^(3/2)) - (2*a*cos(c + d*x)^5)/(99*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(11*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 9, -((2*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (8*a*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)*sin(c + d*x)^3)/(7*d*sqrt(a + a*sin(c + d*x))) + (164*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*d) - (12*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 8, -((sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (61*a*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x))) + (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*d) - (cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/d - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 7, (13*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) - (2*a*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d) - (cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 7, (11*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) - (2*a*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) + (11*a*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*sqrt(a + a*sin(c + d*x)), x, 9, -((67*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*d)) + (61*a*cot(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) + (61*a*cot(c + d*x)*csc(c + d*x))/(96*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^2)/(24*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*sqrt(a + a*sin(c + d*x)), x, 11, -((31*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(128*d)) - (31*a*cot(c + d*x))/(128*d*sqrt(a + a*sin(c + d*x))) + (97*a*cot(c + d*x)*csc(c + d*x))/(192*d*sqrt(a + a*sin(c + d*x))) + (97*a*cot(c + d*x)*csc(c + d*x)^2)/(240*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^3)/(40*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(5*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*sqrt(a + a*sin(c + d*x)), x, 13, -((55*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(512*d)) - (55*a*cot(c + d*x))/(512*d*sqrt(a + a*sin(c + d*x))) - (55*a*cot(c + d*x)*csc(c + d*x))/(768*d*sqrt(a + a*sin(c + d*x))) + (329*a*cot(c + d*x)*csc(c + d*x)^2)/(960*d*sqrt(a + a*sin(c + d*x))) + (47*a*cot(c + d*x)*csc(c + d*x)^3)/(160*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^4)/(60*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*sqrt(a + a*sin(c + d*x)), x, 15, -((61*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(1024*d)) - (61*a*cot(c + d*x))/(1024*d*sqrt(a + a*sin(c + d*x))) - (61*a*cot(c + d*x)*csc(c + d*x))/(1536*d*sqrt(a + a*sin(c + d*x))) - (61*a*cot(c + d*x)*csc(c + d*x)^2)/(1920*d*sqrt(a + a*sin(c + d*x))) + (579*a*cot(c + d*x)*csc(c + d*x)^3)/(2240*d*sqrt(a + a*sin(c + d*x))) + (193*a*cot(c + d*x)*csc(c + d*x)^4)/(840*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^5)/(84*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^6*sqrt(a + a*sin(c + d*x)))/(7*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 6, -((256*a^4*cos(c + d*x)^5)/(6435*d*(a + a*sin(c + d*x))^(5/2))) - (64*a^3*cos(c + d*x)^5)/(1287*d*(a + a*sin(c + d*x))^(3/2)) - (56*a^2*cos(c + d*x)^5)/(1287*d*sqrt(a + a*sin(c + d*x))) - (14*a*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(429*d) + (4*cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(39*d) - (2*cos(c + d*x)^5*(a + a*sin(c + d*x))^(5/2))/(15*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 5, -((256*a^4*cos(c + d*x)^5)/(5005*d*(a + a*sin(c + d*x))^(5/2))) - (64*a^3*cos(c + d*x)^5)/(1001*d*(a + a*sin(c + d*x))^(3/2)) - (8*a^2*cos(c + d*x)^5)/(143*d*sqrt(a + a*sin(c + d*x))) - (6*a*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(143*d) - (2*cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(13*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 12, -((2*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (14*a^2*cos(c + d*x))/(45*d*sqrt(a + a*sin(c + d*x))) - (34*a^2*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x)^4)/(9*d*sqrt(a + a*sin(c + d*x))) + (388*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*d) + (16*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 10, -((3*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (171*a^2*cos(c + d*x))/(35*d*sqrt(a + a*sin(c + d*x))) + (69*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(35*d) + (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*d) - (cot(c + d*x)*(a + a*sin(c + d*x))^(3/2))/d - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(7*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 9, (9*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) + (73*a^2*cos(c + d*x))/(20*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*d) - (3*a*cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*d) - (cot(c + d*x)*csc(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 8, (37*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) - (8*a^2*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) + (29*a^2*cot(c + d*x))/(24*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d) - (a*cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + a*sin(c + d*x))^(3/2), x, 11, (21*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*d) - (2*a^2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) + (149*a^2*cot(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) + (19*a^2*cot(c + d*x)*csc(c + d*x))/(32*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(8*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + a*sin(c + d*x))^(3/2), x, 12, -((165*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(128*d)) + (91*a^2*cot(c + d*x))/(128*d*sqrt(a + a*sin(c + d*x))) + (73*a^2*cot(c + d*x)*csc(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) + (31*a^2*cot(c + d*x)*csc(c + d*x)^2)/(80*d*sqrt(a + a*sin(c + d*x))) - (3*a*cot(c + d*x)*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(40*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + a*sin(c + d*x))^(3/2), x, 14, -((179*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(512*d)) - (179*a^2*cot(c + d*x))/(512*d*sqrt(a + a*sin(c + d*x))) + (111*a^2*cot(c + d*x)*csc(c + d*x))/(256*d*sqrt(a + a*sin(c + d*x))) + (239*a^2*cot(c + d*x)*csc(c + d*x)^2)/(320*d*sqrt(a + a*sin(c + d*x))) + (137*a^2*cot(c + d*x)*csc(c + d*x)^3)/(480*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(20*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + a*sin(c + d*x))^(3/2), x, 16, -((171*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(1024*d)) - (171*a^2*cot(c + d*x))/(1024*d*sqrt(a + a*sin(c + d*x))) - (57*a^2*cot(c + d*x)*csc(c + d*x))/(512*d*sqrt(a + a*sin(c + d*x))) + (199*a^2*cot(c + d*x)*csc(c + d*x)^2)/(640*d*sqrt(a + a*sin(c + d*x))) + (1237*a^2*cot(c + d*x)*csc(c + d*x)^3)/(2240*d*sqrt(a + a*sin(c + d*x))) + (9*a^2*cot(c + d*x)*csc(c + d*x)^4)/(40*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(28*d) - (cot(c + d*x)*csc(c + d*x)^6*(a + a*sin(c + d*x))^(3/2))/(7*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + a*sin(c + d*x))^(3/2), x, 18, -((1587*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(16384*d)) - (1587*a^2*cot(c + d*x))/(16384*d*sqrt(a + a*sin(c + d*x))) - (529*a^2*cot(c + d*x)*csc(c + d*x))/(8192*d*sqrt(a + a*sin(c + d*x))) - (529*a^2*cot(c + d*x)*csc(c + d*x)^2)/(10240*d*sqrt(a + a*sin(c + d*x))) + (8653*a^2*cot(c + d*x)*csc(c + d*x)^3)/(35840*d*sqrt(a + a*sin(c + d*x))) + (1957*a^2*cot(c + d*x)*csc(c + d*x)^4)/(4480*d*sqrt(a + a*sin(c + d*x))) + (83*a^2*cot(c + d*x)*csc(c + d*x)^5)/(448*d*sqrt(a + a*sin(c + d*x))) - (3*a*cot(c + d*x)*csc(c + d*x)^6*sqrt(a + a*sin(c + d*x)))/(112*d) - (cot(c + d*x)*csc(c + d*x)^7*(a + a*sin(c + d*x))^(3/2))/(8*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 5, -((152*a^2*cos(c + d*x)^5)/(3465*d*(a + a*sin(c + d*x))^(5/2))) - (38*a*cos(c + d*x)^5)/(693*d*(a + a*sin(c + d*x))^(3/2)) + (20*cos(c + d*x)^5)/(99*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(11*a*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 3, (8*a^2*cos(c + d*x)^5)/(315*d*(a + a*sin(c + d*x))^(5/2)) + (2*a*cos(c + d*x)^5)/(63*d*(a + a*sin(c + d*x))^(3/2)) - (2*cos(c + d*x)^5)/(9*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 13, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(sqrt(a)*d)) + (32*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sin(c + d*x)^2)/(5*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 11, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(sqrt(a)*d) + (4*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) - cot(c + d*x)/(d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 11, (9*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*sqrt(a)*d) - (2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) + cot(c + d*x)/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/sqrt(a + a*sin(c + d*x)), x, 11, -((7*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*sqrt(a)*d)) + (9*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) + (cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/sqrt(a + a*sin(c + d*x)), x, 15, -((11*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*sqrt(a)*d)) - (11*cot(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) + (53*cot(c + d*x)*csc(c + d*x))/(96*d*sqrt(a + a*sin(c + d*x))) + (cot(c + d*x)*csc(c + d*x)^2)/(24*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3)/(4*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6/sqrt(a + a*sin(c + d*x)), x, 17, -((9*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(128*sqrt(a)*d)) - (9*cot(c + d*x))/(128*d*sqrt(a + a*sin(c + d*x))) - (3*cot(c + d*x)*csc(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) + (29*cot(c + d*x)*csc(c + d*x)^2)/(80*d*sqrt(a + a*sin(c + d*x))) + (cot(c + d*x)*csc(c + d*x)^3)/(40*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^4)/(5*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 12, -((4*cos(c + d*x))/(165*a*d*sqrt(a + a*sin(c + d*x)))) - (2*cos(c + d*x)*sin(c + d*x)^3)/(231*a*d*sqrt(a + a*sin(c + d*x))) + (14*cos(c + d*x)*sin(c + d*x)^4)/(33*a*d*sqrt(a + a*sin(c + d*x))) + (8*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(1155*a^2*d) - (2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(11*a^2*d) - (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(385*a^3*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 4, -((46*a*cos(c + d*x)^5)/(315*d*(a + a*sin(c + d*x))^(5/2))) + (20*cos(c + d*x)^5)/(63*d*(a + a*sin(c + d*x))^(3/2)) - (2*cos(c + d*x)^5)/(9*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 2, (6*a*cos(c + d*x)^5)/(35*d*(a + a*sin(c + d*x))^(5/2)) - (2*cos(c + d*x)^5)/(7*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 6, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d)) + (10*cos(c + d*x))/(3*a*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 9, (3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d) - cos(c + d*x)/(a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/(a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 8, -((3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(3/2)*d)) + (7*cot(c + d*x))/(4*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 10, -(atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(8*a^(3/2)*d)) - cot(c + d*x)/(8*a*d*sqrt(a + a*sin(c + d*x))) + (11*cot(c + d*x)*csc(c + d*x))/(12*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(3*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + a*sin(c + d*x))^(3/2), x, 12, -((3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*a^(3/2)*d)) - (3*cot(c + d*x))/(64*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(32*a*d*sqrt(a + a*sin(c + d*x))) + (5*cot(c + d*x)*csc(c + d*x)^2)/(8*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(4*a^2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6/(a + a*sin(c + d*x))^(3/2), x, 14, -((3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(128*a^(3/2)*d)) - (3*cot(c + d*x))/(128*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(64*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(80*a*d*sqrt(a + a*sin(c + d*x))) + (19*cot(c + d*x)*csc(c + d*x)^3)/(40*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(5*a^2*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x))^(5/2), x, 18, -((4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d)) + (4496*cos(c + d*x))/(693*a^2*d*sqrt(a + a*sin(c + d*x))) + (200*cos(c + d*x)*sin(c + d*x)^2)/(231*a^2*d*sqrt(a + a*sin(c + d*x))) - (424*cos(c + d*x)*sin(c + d*x)^3)/(693*a^2*d*sqrt(a + a*sin(c + d*x))) + (46*cos(c + d*x)*sin(c + d*x)^4)/(99*a^2*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sin(c + d*x)^5)/(11*a^2*d*sqrt(a + a*sin(c + d*x))) - (1048*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(693*a^3*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 16, (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) - (2048*cos(c + d*x))/(315*a^2*d*sqrt(a + a*sin(c + d*x))) - (92*cos(c + d*x)*sin(c + d*x)^2)/(105*a^2*d*sqrt(a + a*sin(c + d*x))) + (38*cos(c + d*x)*sin(c + d*x)^3)/(63*a^2*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sin(c + d*x)^4)/(9*a^2*d*sqrt(a + a*sin(c + d*x))) + (472*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*a^3*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 6, -((4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d)) + (4*cos(c + d*x)^5)/(7*d*(a + a*sin(c + d*x))^(5/2)) + (2*cos(c + d*x)^3)/(3*a*d*(a + a*sin(c + d*x))^(3/2)) - (2*cos(c + d*x)^5)/(7*a*d*(a + a*sin(c + d*x))^(3/2)) + (4*cos(c + d*x))/(a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 5, (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) - (2*cos(c + d*x)^5)/(5*d*(a + a*sin(c + d*x))^(5/2)) - (2*cos(c + d*x)^3)/(3*a*d*(a + a*sin(c + d*x))^(3/2)) - (4*cos(c + d*x))/(a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 9, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(5/2)*d)) + (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) - (2*cos(c + d*x))/(a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 12, (5*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(5/2)*d) - (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) - cot(c + d*x)/(a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 14, -((23*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(5/2)*d)) + (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) + (9*cot(c + d*x))/(4*a^2*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + a*sin(c + d*x))^(5/2), x, 16, (45*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*a^(5/2)*d) - (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) - (19*cot(c + d*x))/(8*a^2*d*sqrt(a + a*sin(c + d*x))) + (13*cot(c + d*x)*csc(c + d*x))/(12*a^2*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + a*sin(c + d*x))^(5/2), x, 18, -((363*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*a^(5/2)*d)) + (4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) + (149*cot(c + d*x))/(64*a^2*d*sqrt(a + a*sin(c + d*x))) - (107*cot(c + d*x)*csc(c + d*x))/(96*a^2*d*sqrt(a + a*sin(c + d*x))) + (17*cot(c + d*x)*csc(c + d*x)^2)/(24*a^2*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d*sqrt(a + a*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+a*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x, 5, (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (2*a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2)) + (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (3 + n)/2, (5 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(3 + n))/(d*(3 + n)*sqrt(cos(c + d*x)^2))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + a*sin(c + d*x))^1, x, 3, (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^n/(a + a*sin(c + d*x))^1, x, 3, (cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(a*d*(1 + n)*sqrt(cos(c + d*x)^2)) - (cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(a*d*(2 + n)*sqrt(cos(c + d*x)^2))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^n/(a + a*sin(c + d*x))^2, x, 5, -((cos(c + d*x)*sin(c + d*x)^(1 + n))/(a^2*d*(2 + n))) + ((3 + 2*n)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(a^2*d*(1 + n)*(2 + n)*sqrt(cos(c + d*x)^2)) - (2*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(a^2*d*(2 + n)*sqrt(cos(c + d*x)^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^5*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^6)/(6*d) + (a*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^8)/(4*d) - (2*a*sin(c + d*x)^9)/(9*d) + (a*sin(c + d*x)^10)/(10*d) + (a*sin(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^5)/(5*d) + (a*sin(c + d*x)^6)/(6*d) - (2*a*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^8)/(4*d) + (a*sin(c + d*x)^9)/(9*d) + (a*sin(c + d*x)^10)/(10*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 7, -((a*cos(c + d*x)^6)/(6*d)) + (a*cos(c + d*x)^8)/(8*d) + (a*sin(c + d*x)^5)/(5*d) - (2*a*sin(c + d*x)^7)/(7*d) + (a*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 7, -((a*cos(c + d*x)^6)/(6*d)) + (a*cos(c + d*x)^8)/(8*d) + (a*sin(c + d*x)^3)/(3*d) - (2*a*sin(c + d*x)^5)/(5*d) + (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, -((a*cos(c + d*x)^6)/(6*d)) + (a*sin(c + d*x)^3)/(3*d) - (2*a*sin(c + d*x)^5)/(5*d) + (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 4, (a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/d - (2*a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^4)/(4*d) + (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (a*log(sin(c + d*x)))/d - (2*a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/d + (a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (2*a*log(sin(c + d*x)))/d - (2*a*sin(c + d*x))/d + (a*sin(c + d*x)^2)/(2*d) + (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, (2*a*csc(c + d*x))/d - (a*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/(3*d) - (2*a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d + (a*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 3, (2*a*csc(c + d*x))/d + (a*csc(c + d*x)^2)/d - (a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^4)/(4*d) + (a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (a*csc(c + d*x)^2)/d + (2*a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^4)/(4*d) - (a*csc(c + d*x)^5)/(5*d) + (a*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7*(a + a*sin(c + d*x)), x, 6, -((a*cot(c + d*x)^6)/(6*d)) - (a*csc(c + d*x))/d + (2*a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^8*(a + a*sin(c + d*x)), x, 6, -((a*cot(c + d*x)^6)/(6*d)) - (a*csc(c + d*x)^3)/(3*d) + (2*a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^9*(a + a*sin(c + d*x)), x, 7, -((a*cot(c + d*x)^6)/(6*d)) - (a*cot(c + d*x)^8)/(8*d) - (a*csc(c + d*x)^3)/(3*d) + (2*a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^10*(a + a*sin(c + d*x)), x, 7, -((a*cot(c + d*x)^6)/(6*d)) - (a*cot(c + d*x)^8)/(8*d) - (a*csc(c + d*x)^5)/(5*d) + (2*a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^11*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x)^5)/(5*d)) - (a*csc(c + d*x)^6)/(6*d) + (2*a*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^8)/(4*d) - (a*csc(c + d*x)^9)/(9*d) - (a*csc(c + d*x)^10)/(10*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^12*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x)^6)/(6*d)) - (a*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^8)/(4*d) + (2*a*csc(c + d*x)^9)/(9*d) - (a*csc(c + d*x)^10)/(10*d) - (a*csc(c + d*x)^11)/(11*d)]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 4, (a^2*sin(c + d*x)^4)/(4*d) + (2*a^2*sin(c + d*x)^5)/(5*d) - (a^2*sin(c + d*x)^6)/(6*d) - (4*a^2*sin(c + d*x)^7)/(7*d) - (a^2*sin(c + d*x)^8)/(8*d) + (2*a^2*sin(c + d*x)^9)/(9*d) + (a^2*sin(c + d*x)^10)/(10*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, (4*(a + a*sin(c + d*x))^5)/(5*a^3*d) - (2*(a + a*sin(c + d*x))^6)/(a^4*d) + (13*(a + a*sin(c + d*x))^7)/(7*a^5*d) - (3*(a + a*sin(c + d*x))^8)/(4*a^6*d) + (a + a*sin(c + d*x))^9/(9*a^7*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 4, -((4*(a + a*sin(c + d*x))^5)/(5*a^3*d)) + (4*(a + a*sin(c + d*x))^6)/(3*a^4*d) - (5*(a + a*sin(c + d*x))^7)/(7*a^5*d) + (a + a*sin(c + d*x))^8/(8*a^6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 4, (a^2*log(sin(c + d*x)))/d + (2*a^2*sin(c + d*x))/d - (a^2*sin(c + d*x)^2)/(2*d) - (4*a^2*sin(c + d*x)^3)/(3*d) - (a^2*sin(c + d*x)^4)/(4*d) + (2*a^2*sin(c + d*x)^5)/(5*d) + (a^2*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x))/d) + (2*a^2*log(sin(c + d*x)))/d - (a^2*sin(c + d*x))/d - (2*a^2*sin(c + d*x)^2)/d - (a^2*sin(c + d*x)^3)/(3*d) + (a^2*sin(c + d*x)^4)/(2*d) + (a^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 4, -((2*a^2*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/(2*d) - (a^2*log(sin(c + d*x)))/d - (4*a^2*sin(c + d*x))/d - (a^2*sin(c + d*x)^2)/(2*d) + (2*a^2*sin(c + d*x)^3)/(3*d) + (a^2*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 4, (a^2*csc(c + d*x))/d - (a^2*csc(c + d*x)^2)/d - (a^2*csc(c + d*x)^3)/(3*d) - (4*a^2*log(sin(c + d*x)))/d - (a^2*sin(c + d*x))/d + (a^2*sin(c + d*x)^2)/d + (a^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 3, (4*a^2*csc(c + d*x))/d + (a^2*csc(c + d*x)^2)/(2*d) - (2*a^2*csc(c + d*x)^3)/(3*d) - (a^2*csc(c + d*x)^4)/(4*d) - (a^2*log(sin(c + d*x)))/d + (2*a^2*sin(c + d*x))/d + (a^2*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 4, (a^2*csc(c + d*x))/d + (2*a^2*csc(c + d*x)^2)/d + (a^2*csc(c + d*x)^3)/(3*d) - (a^2*csc(c + d*x)^4)/(2*d) - (a^2*csc(c + d*x)^5)/(5*d) + (2*a^2*log(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 4, -((2*a^2*csc(c + d*x))/d) + (a^2*csc(c + d*x)^2)/(2*d) + (4*a^2*csc(c + d*x)^3)/(3*d) + (a^2*csc(c + d*x)^4)/(4*d) - (2*a^2*csc(c + d*x)^5)/(5*d) - (a^2*csc(c + d*x)^6)/(6*d) + (a^2*log(sin(c + d*x)))/d]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, (2*(a + a*sin(c + d*x))^6)/(3*a^3*d) - (12*(a + a*sin(c + d*x))^7)/(7*a^4*d) + (13*(a + a*sin(c + d*x))^8)/(8*a^5*d) - (2*(a + a*sin(c + d*x))^9)/(3*a^6*d) + (a + a*sin(c + d*x))^10/(10*a^7*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, -((2*(a + a*sin(c + d*x))^6)/(3*a^3*d)) + (8*(a + a*sin(c + d*x))^7)/(7*a^4*d) - (5*(a + a*sin(c + d*x))^8)/(8*a^5*d) + (a + a*sin(c + d*x))^9/(9*a^6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, (a^3*log(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (a^3*sin(c + d*x)^2)/(2*d) - (5*a^3*sin(c + d*x)^3)/(3*d) - (5*a^3*sin(c + d*x)^4)/(4*d) + (a^3*sin(c + d*x)^5)/(5*d) + (a^3*sin(c + d*x)^6)/(2*d) + (a^3*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x))/d) + (3*a^3*log(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d - (5*a^3*sin(c + d*x)^2)/(2*d) - (5*a^3*sin(c + d*x)^3)/(3*d) + (a^3*sin(c + d*x)^4)/(4*d) + (3*a^3*sin(c + d*x)^5)/(5*d) + (a^3*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 4, -((3*a^3*csc(c + d*x))/d) - (a^3*csc(c + d*x)^2)/(2*d) + (a^3*log(sin(c + d*x)))/d - (5*a^3*sin(c + d*x))/d - (5*a^3*sin(c + d*x)^2)/(2*d) + (a^3*sin(c + d*x)^3)/(3*d) + (3*a^3*sin(c + d*x)^4)/(4*d) + (a^3*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x))/d) - (3*a^3*csc(c + d*x)^2)/(2*d) - (a^3*csc(c + d*x)^3)/(3*d) - (5*a^3*log(sin(c + d*x)))/d - (5*a^3*sin(c + d*x))/d + (a^3*sin(c + d*x)^2)/(2*d) + (a^3*sin(c + d*x)^3)/d + (a^3*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 3, (5*a^3*csc(c + d*x))/d - (a^3*csc(c + d*x)^2)/(2*d) - (a^3*csc(c + d*x)^3)/d - (a^3*csc(c + d*x)^4)/(4*d) - (5*a^3*log(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d + (3*a^3*sin(c + d*x)^2)/(2*d) + (a^3*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 4, (5*a^3*csc(c + d*x))/d + (5*a^3*csc(c + d*x)^2)/(2*d) - (a^3*csc(c + d*x)^3)/(3*d) - (3*a^3*csc(c + d*x)^4)/(4*d) - (a^3*csc(c + d*x)^5)/(5*d) + (a^3*log(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (a^3*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x))/d) + (5*a^3*csc(c + d*x)^2)/(2*d) + (5*a^3*csc(c + d*x)^3)/(3*d) - (a^3*csc(c + d*x)^4)/(4*d) - (3*a^3*csc(c + d*x)^5)/(5*d) - (a^3*csc(c + d*x)^6)/(6*d) + (3*a^3*log(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d]

    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x))^4, x, 4, -((4*a^4*csc(c + d*x))/d) - (2*a^4*csc(c + d*x)^2)/d - (a^4*csc(c + d*x)^3)/(3*d) - (4*a^4*log(sin(c + d*x)))/d - (10*a^4*sin(c + d*x))/d - (2*a^4*sin(c + d*x)^2)/d + (4*a^4*sin(c + d*x)^3)/(3*d) + (a^4*sin(c + d*x)^4)/d + (a^4*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + a*sin(c + d*x))^4, x, 3, (4*a^4*csc(c + d*x))/d - (2*a^4*csc(c + d*x)^2)/d - (4*a^4*csc(c + d*x)^3)/(3*d) - (a^4*csc(c + d*x)^4)/(4*d) - (10*a^4*log(sin(c + d*x)))/d - (4*a^4*sin(c + d*x))/d + (2*a^4*sin(c + d*x)^2)/d + (4*a^4*sin(c + d*x)^3)/(3*d) + (a^4*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + a*sin(c + d*x))^4, x, 4, (10*a^4*csc(c + d*x))/d + (2*a^4*csc(c + d*x)^2)/d - (4*a^4*csc(c + d*x)^3)/(3*d) - (a^4*csc(c + d*x)^4)/d - (a^4*csc(c + d*x)^5)/(5*d) - (4*a^4*log(sin(c + d*x)))/d + (4*a^4*sin(c + d*x))/d + (2*a^4*sin(c + d*x)^2)/d + (a^4*sin(c + d*x)^3)/(3*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, sin(c + d*x)^4/(4*a*d) - sin(c + d*x)^5/(5*a*d) - sin(c + d*x)^6/(6*a*d) + sin(c + d*x)^7/(7*a*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 7, sin(c + d*x)^3/(3*a*d) - sin(c + d*x)^4/(4*a*d) - sin(c + d*x)^5/(5*a*d) + sin(c + d*x)^6/(6*a*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 6, -(cos(c + d*x)^4/(4*a*d)) - sin(c + d*x)^3/(3*a*d) + sin(c + d*x)^5/(5*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d) + sin(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - log(sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d) + sin(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d) - log(sin(c + d*x))/(a*d) + sin(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) + csc(c + d*x)^2/(2*a*d) - csc(c + d*x)^3/(3*a*d) + log(sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 5, -(cot(c + d*x)^4/(4*a*d)) - csc(c + d*x)/(a*d) + csc(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 6, cot(c + d*x)^4/(4*a*d) + csc(c + d*x)^3/(3*a*d) - csc(c + d*x)^5/(5*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)^3/(3*a*d)) + csc(c + d*x)^4/(4*a*d) + csc(c + d*x)^5/(5*a*d) - csc(c + d*x)^6/(6*a*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^8/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)^4/(4*a*d)) + csc(c + d*x)^5/(5*a*d) + csc(c + d*x)^6/(6*a*d) - csc(c + d*x)^7/(7*a*d)]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 4, sin(c + d*x)^4/(4*a^2*d) - (2*sin(c + d*x)^5)/(5*a^2*d) + sin(c + d*x)^6/(6*a^2*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 4, sin(c + d*x)^3/(3*a^2*d) - sin(c + d*x)^4/(2*a^2*d) + sin(c + d*x)^5/(5*a^2*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, sin(c + d*x)^2/(2*a^2*d) - (2*sin(c + d*x)^3)/(3*a^2*d) + sin(c + d*x)^4/(4*a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, log(sin(c + d*x))/(a^2*d) - (2*sin(c + d*x))/(a^2*d) + sin(c + d*x)^2/(2*a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 4, -(csc(c + d*x)/(a^2*d)) - (2*log(sin(c + d*x)))/(a^2*d) + sin(c + d*x)/(a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 4, (2*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a^2*d) + log(sin(c + d*x))/(a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 3, -((csc(c + d*x)^3*(a - a*sin(c + d*x))^3)/(3*a^5*d))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 3, -(csc(c + d*x)^2/(2*a^2*d)) + (2*csc(c + d*x)^3)/(3*a^2*d) - csc(c + d*x)^4/(4*a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 4, -(csc(c + d*x)^3/(3*a^2*d)) + csc(c + d*x)^4/(2*a^2*d) - csc(c + d*x)^5/(5*a^2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 4, -(csc(c + d*x)^4/(4*a^2*d)) + (2*csc(c + d*x)^5)/(5*a^2*d) - csc(c + d*x)^6/(6*a^2*d)]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 4, -((4*log(1 + sin(c + d*x)))/(a^3*d)) + (4*sin(c + d*x))/(a^3*d) - (2*sin(c + d*x)^2)/(a^3*d) + (4*sin(c + d*x)^3)/(3*a^3*d) - (3*sin(c + d*x)^4)/(4*a^3*d) + sin(c + d*x)^5/(5*a^3*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 4, (4*log(1 + sin(c + d*x)))/(a^3*d) - (4*sin(c + d*x))/(a^3*d) + (2*sin(c + d*x)^2)/(a^3*d) - sin(c + d*x)^3/(a^3*d) + sin(c + d*x)^4/(4*a^3*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 4, -((4*log(1 + sin(c + d*x)))/(a^3*d)) + (4*sin(c + d*x))/(a^3*d) - (3*sin(c + d*x)^2)/(2*a^3*d) + sin(c + d*x)^3/(3*a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 4, log(sin(c + d*x))/(a^3*d) - (4*log(1 + sin(c + d*x)))/(a^3*d) + sin(c + d*x)/(a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 4, -(csc(c + d*x)/(a^3*d)) - (3*log(sin(c + d*x)))/(a^3*d) + (4*log(1 + sin(c + d*x)))/(a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 4, (3*csc(c + d*x))/(a^3*d) - csc(c + d*x)^2/(2*a^3*d) + (4*log(sin(c + d*x)))/(a^3*d) - (4*log(1 + sin(c + d*x)))/(a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 4, -((4*csc(c + d*x))/(a^3*d)) + (3*csc(c + d*x)^2)/(2*a^3*d) - csc(c + d*x)^3/(3*a^3*d) - (4*log(sin(c + d*x)))/(a^3*d) + (4*log(1 + sin(c + d*x)))/(a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 3, (4*csc(c + d*x))/(a^3*d) - (2*csc(c + d*x)^2)/(a^3*d) + csc(c + d*x)^3/(a^3*d) - csc(c + d*x)^4/(4*a^3*d) + (4*log(sin(c + d*x)))/(a^3*d) - (4*log(1 + sin(c + d*x)))/(a^3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 4, -((4*csc(c + d*x))/(a^3*d)) + (2*csc(c + d*x)^2)/(a^3*d) - (4*csc(c + d*x)^3)/(3*a^3*d) + (3*csc(c + d*x)^4)/(4*a^3*d) - csc(c + d*x)^5/(5*a^3*d) - (4*log(sin(c + d*x)))/(a^3*d) + (4*log(1 + sin(c + d*x)))/(a^3*d)]

    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + a*sin(c + d*x))^4, x, 3, (12*csc(c + d*x))/(a^4*d) - (4*csc(c + d*x)^2)/(a^4*d) + (4*csc(c + d*x)^3)/(3*a^4*d) - csc(c + d*x)^4/(4*a^4*d) + (16*log(sin(c + d*x)))/(a^4*d) - (16*log(1 + sin(c + d*x)))/(a^4*d) + 4/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + a*sin(c + d*x))^4, x, 4, -((16*csc(c + d*x))/(a^4*d)) + (6*csc(c + d*x)^2)/(a^4*d) - (8*csc(c + d*x)^3)/(3*a^4*d) + csc(c + d*x)^4/(a^4*d) - csc(c + d*x)^5/(5*a^4*d) - (20*log(sin(c + d*x)))/(a^4*d) + (20*log(1 + sin(c + d*x)))/(a^4*d) - 4/(d*(a^4 + a^4*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+a*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^3, x, 3, (a^3*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (3*a^3*sin(c + d*x)^(2 + n))/(d*(2 + n)) + (a^3*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (5*a^3*sin(c + d*x)^(4 + n))/(d*(4 + n)) - (5*a^3*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (a^3*sin(c + d*x)^(6 + n))/(d*(6 + n)) + (3*a^3*sin(c + d*x)^(7 + n))/(d*(7 + n)) + (a^3*sin(c + d*x)^(8 + n))/(d*(8 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x, 3, (a^2*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (2*a^2*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (a^2*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (4*a^2*sin(c + d*x)^(4 + n))/(d*(4 + n)) - (a^2*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (2*a^2*sin(c + d*x)^(6 + n))/(d*(6 + n)) + (a^2*sin(c + d*x)^(7 + n))/(d*(7 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^1, x, 3, (a*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (a*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (2*a*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (2*a*sin(c + d*x)^(4 + n))/(d*(4 + n)) + (a*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (a*sin(c + d*x)^(6 + n))/(d*(6 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + a*sin(c + d*x))^1, x, 3, sin(c + d*x)^(1 + n)/(a*d*(1 + n)) - sin(c + d*x)^(2 + n)/(a*d*(2 + n)) - sin(c + d*x)^(3 + n)/(a*d*(3 + n)) + sin(c + d*x)^(4 + n)/(a*d*(4 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + a*sin(c + d*x))^2, x, 3, sin(c + d*x)^(1 + n)/(a^2*d*(1 + n)) - (2*sin(c + d*x)^(2 + n))/(a^2*d*(2 + n)) + sin(c + d*x)^(3 + n)/(a^2*d*(3 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + a*sin(c + d*x))^3, x, 4, -((3*sin(c + d*x)^(1 + n))/(a^3*d*(1 + n))) + (4*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^3*d*(1 + n)) + sin(c + d*x)^(2 + n)/(a^3*d*(2 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + a*sin(c + d*x))^4, x, 4, sin(c + d*x)^(1 + n)/(a^4*d*(1 + n)) - (4*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^4*d) + (4*sin(c + d*x)^(1 + n))/(d*(a^4 + a^4*sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^6*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 10, (3*a*x)/256 - (a*cos(c + d*x)^7)/(7*d) + (2*a*cos(c + d*x)^9)/(9*d) - (a*cos(c + d*x)^11)/(11*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(256*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(128*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(160*d) - (3*a*cos(c + d*x)^7*sin(c + d*x))/(80*d) - (a*cos(c + d*x)^7*sin(c + d*x)^3)/(10*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 10, (3*a*x)/256 - (a*cos(c + d*x)^7)/(7*d) + (a*cos(c + d*x)^9)/(9*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(256*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(128*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(160*d) - (3*a*cos(c + d*x)^7*sin(c + d*x))/(80*d) - (a*cos(c + d*x)^7*sin(c + d*x)^3)/(10*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 9, (5*a*x)/128 - (a*cos(c + d*x)^7)/(7*d) + (a*cos(c + d*x)^9)/(9*d) + (5*a*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (a*cos(c + d*x)^7*sin(c + d*x))/(8*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 8, (5*a*x)/128 - (a*cos(c + d*x)^7)/(7*d) + (5*a*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (a*cos(c + d*x)^7*sin(c + d*x))/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 9, (5*a*x)/16 - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)^5)/(5*d) + (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 10, -((15*a*x)/8) - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)^5)/(5*d) - (15*a*cot(c + d*x))/(8*d) + (5*a*cos(c + d*x)^2*cot(c + d*x))/(8*d) + (a*cos(c + d*x)^4*cot(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 11, -((15*a*x)/8) + (5*a*atanh(cos(c + d*x)))/(2*d) - (5*a*cos(c + d*x))/(2*d) - (5*a*cos(c + d*x)^3)/(6*d) - (15*a*cot(c + d*x))/(8*d) + (5*a*cos(c + d*x)^2*cot(c + d*x))/(8*d) + (a*cos(c + d*x)^4*cot(c + d*x))/(4*d) - (a*cos(c + d*x)^3*cot(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 11, (5*a*x)/2 + (5*a*atanh(cos(c + d*x)))/(2*d) - (5*a*cos(c + d*x))/(2*d) - (5*a*cos(c + d*x)^3)/(6*d) + (5*a*cot(c + d*x))/(2*d) - (a*cos(c + d*x)^3*cot(c + d*x)^2)/(2*d) - (5*a*cot(c + d*x)^3)/(6*d) + (a*cos(c + d*x)^2*cot(c + d*x)^3)/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 11, (5*a*x)/2 - (15*a*atanh(cos(c + d*x)))/(8*d) + (15*a*cos(c + d*x))/(8*d) + (5*a*cot(c + d*x))/(2*d) + (5*a*cos(c + d*x)*cot(c + d*x)^2)/(8*d) - (5*a*cot(c + d*x)^3)/(6*d) + (a*cos(c + d*x)^2*cot(c + d*x)^3)/(2*d) - (a*cos(c + d*x)*cot(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6*(a + a*sin(c + d*x)), x, 11, (-a)*x - (15*a*atanh(cos(c + d*x)))/(8*d) + (15*a*cos(c + d*x))/(8*d) - (a*cot(c + d*x))/d + (5*a*cos(c + d*x)*cot(c + d*x)^2)/(8*d) + (a*cot(c + d*x)^3)/(3*d) - (a*cos(c + d*x)*cot(c + d*x)^4)/(4*d) - (a*cot(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7*(a + a*sin(c + d*x)), x, 9, (-a)*x + (5*a*atanh(cos(c + d*x)))/(16*d) - (a*cot(c + d*x))/d + (a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^5)/(5*d) - (5*a*cot(c + d*x)*csc(c + d*x))/(16*d) + (5*a*cot(c + d*x)^3*csc(c + d*x))/(24*d) - (a*cot(c + d*x)^5*csc(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8*(a + a*sin(c + d*x)), x, 7, (5*a*atanh(cos(c + d*x)))/(16*d) - (a*cot(c + d*x)^7)/(7*d) - (5*a*cot(c + d*x)*csc(c + d*x))/(16*d) + (5*a*cot(c + d*x)^3*csc(c + d*x))/(24*d) - (a*cot(c + d*x)^5*csc(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^9*(a + a*sin(c + d*x)), x, 8, (5*a*atanh(cos(c + d*x)))/(128*d) - (a*cot(c + d*x)^7)/(7*d) + (5*a*cot(c + d*x)*csc(c + d*x))/(128*d) - (5*a*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (5*a*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^10*(a + a*sin(c + d*x)), x, 9, (5*a*atanh(cos(c + d*x)))/(128*d) - (a*cot(c + d*x)^7)/(7*d) - (a*cot(c + d*x)^9)/(9*d) + (5*a*cot(c + d*x)*csc(c + d*x))/(128*d) - (5*a*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (5*a*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^11*(a + a*sin(c + d*x)), x, 10, (3*a*atanh(cos(c + d*x)))/(256*d) - (a*cot(c + d*x)^7)/(7*d) - (a*cot(c + d*x)^9)/(9*d) + (3*a*cot(c + d*x)*csc(c + d*x))/(256*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(128*d) - (a*cot(c + d*x)*csc(c + d*x)^5)/(32*d) + (a*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^12*(a + a*sin(c + d*x)), x, 10, (3*a*atanh(cos(c + d*x)))/(256*d) - (a*cot(c + d*x)^7)/(7*d) - (2*a*cot(c + d*x)^9)/(9*d) - (a*cot(c + d*x)^11)/(11*d) + (3*a*cot(c + d*x)*csc(c + d*x))/(256*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(128*d) - (a*cot(c + d*x)*csc(c + d*x)^5)/(32*d) + (a*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d)]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 18, (17*a^2*x)/1024 - (2*a^2*cos(c + d*x)^7)/(7*d) + (4*a^2*cos(c + d*x)^9)/(9*d) - (2*a^2*cos(c + d*x)^11)/(11*d) + (17*a^2*cos(c + d*x)*sin(c + d*x))/(1024*d) + (17*a^2*cos(c + d*x)^3*sin(c + d*x))/(1536*d) + (17*a^2*cos(c + d*x)^5*sin(c + d*x))/(1920*d) - (17*a^2*cos(c + d*x)^7*sin(c + d*x))/(320*d) - (17*a^2*cos(c + d*x)^7*sin(c + d*x)^3)/(120*d) - (a^2*cos(c + d*x)^7*sin(c + d*x)^5)/(12*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 14, (3*a^2*x)/128 - (2*a^2*cos(c + d*x)^7)/(7*d) + (a^2*cos(c + d*x)^9)/(3*d) - (a^2*cos(c + d*x)^11)/(11*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(128*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(64*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(80*d) - (3*a^2*cos(c + d*x)^7*sin(c + d*x))/(40*d) - (a^2*cos(c + d*x)^7*sin(c + d*x)^3)/(5*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 16, (13*a^2*x)/256 - (2*a^2*cos(c + d*x)^7)/(7*d) + (2*a^2*cos(c + d*x)^9)/(9*d) + (13*a^2*cos(c + d*x)*sin(c + d*x))/(256*d) + (13*a^2*cos(c + d*x)^3*sin(c + d*x))/(384*d) + (13*a^2*cos(c + d*x)^5*sin(c + d*x))/(480*d) - (13*a^2*cos(c + d*x)^7*sin(c + d*x))/(80*d) - (a^2*cos(c + d*x)^7*sin(c + d*x)^3)/(10*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 7, (5*a^2*x)/64 - (a^2*cos(c + d*x)^7)/(28*d) + (5*a^2*cos(c + d*x)*sin(c + d*x))/(64*d) + (5*a^2*cos(c + d*x)^3*sin(c + d*x))/(96*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(24*d) - (cos(c + d*x)^7*(a + a*sin(c + d*x))^2)/(9*d) - (cos(c + d*x)^7*(a^2 + a^2*sin(c + d*x)))/(36*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 12, (5*a^2*x)/8 - (a^2*atanh(cos(c + d*x)))/d + (a^2*cos(c + d*x))/d + (a^2*cos(c + d*x)^3)/(3*d) + (a^2*cos(c + d*x)^5)/(5*d) - (a^2*cos(c + d*x)^7)/(7*d) + (5*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (5*a^2*cos(c + d*x)^3*sin(c + d*x))/(12*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 17, -((25*a^2*x)/16) - (2*a^2*atanh(cos(c + d*x)))/d + (2*a^2*cos(c + d*x))/d + (2*a^2*cos(c + d*x)^3)/(3*d) + (2*a^2*cos(c + d*x)^5)/(5*d) - (a^2*cot(c + d*x))/d - (7*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) - (7*a^2*cos(c + d*x)*sin(c + d*x)^3)/(24*d) + (a^2*cos(c + d*x)*sin(c + d*x)^5)/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 16, -((15*a^2*x)/4) + (3*a^2*atanh(cos(c + d*x)))/(2*d) - (a^2*cos(c + d*x))/d + (a^2*cos(c + d*x)^5)/(5*d) - (2*a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)*csc(c + d*x))/(2*d) - (9*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^2*cos(c + d*x)*sin(c + d*x)^3)/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 17, (5*a^2*x)/8 + (5*a^2*atanh(cos(c + d*x)))/d - (4*a^2*cos(c + d*x))/d - (2*a^2*cos(c + d*x)^3)/(3*d) + (a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)*csc(c + d*x))/d - (5*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 16, 5*a^2*x + (5*a^2*atanh(cos(c + d*x)))/(8*d) - (a^2*cos(c + d*x))/d - (a^2*cos(c + d*x)^3)/(3*d) + (4*a^2*cot(c + d*x))/d - (2*a^2*cot(c + d*x)^3)/(3*d) + (5*a^2*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(4*d) + (a^2*cos(c + d*x)*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 15, (3*a^2*x)/2 - (15*a^2*atanh(cos(c + d*x)))/(4*d) + (2*a^2*cos(c + d*x))/d + (a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)^5)/(5*d) + (9*a^2*cot(c + d*x)*csc(c + d*x))/(4*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(2*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 17, -2*a^2*x - (25*a^2*atanh(cos(c + d*x)))/(16*d) + (a^2*cos(c + d*x))/d - (2*a^2*cot(c + d*x))/d + (2*a^2*cot(c + d*x)^3)/(3*d) - (2*a^2*cot(c + d*x)^5)/(5*d) + (7*a^2*cot(c + d*x)*csc(c + d*x))/(16*d) + (7*a^2*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a^2*cot(c + d*x)*csc(c + d*x)^5)/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8*(a + a*sin(c + d*x))^2, x, 12, (-a^2)*x + (5*a^2*atanh(cos(c + d*x)))/(8*d) - (a^2*cot(c + d*x))/d + (a^2*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)^5)/(5*d) - (a^2*cot(c + d*x)^7)/(7*d) - (5*a^2*cot(c + d*x)*csc(c + d*x))/(8*d) + (5*a^2*cot(c + d*x)^3*csc(c + d*x))/(12*d) - (a^2*cot(c + d*x)^5*csc(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^9*(a + a*sin(c + d*x))^2, x, 13, (45*a^2*atanh(cos(c + d*x)))/(128*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (35*a^2*cot(c + d*x)*csc(c + d*x))/(128*d) + (5*a^2*cot(c + d*x)^3*csc(c + d*x))/(24*d) - (a^2*cot(c + d*x)^5*csc(c + d*x))/(6*d) - (5*a^2*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (5*a^2*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^10*(a + a*sin(c + d*x))^2, x, 12, (5*a^2*atanh(cos(c + d*x)))/(64*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (a^2*cot(c + d*x)^9)/(9*d) + (5*a^2*cot(c + d*x)*csc(c + d*x))/(64*d) - (5*a^2*cot(c + d*x)*csc(c + d*x)^3)/(32*d) + (5*a^2*cot(c + d*x)^3*csc(c + d*x)^3)/(24*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^11*(a + a*sin(c + d*x))^2, x, 16, (13*a^2*atanh(cos(c + d*x)))/(256*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (2*a^2*cot(c + d*x)^9)/(9*d) + (13*a^2*cot(c + d*x)*csc(c + d*x))/(256*d) - (9*a^2*cot(c + d*x)*csc(c + d*x)^3)/(128*d) + (5*a^2*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d) - (a^2*cot(c + d*x)*csc(c + d*x)^5)/(32*d) + (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^12*(a + a*sin(c + d*x))^2, x, 14, (3*a^2*atanh(cos(c + d*x)))/(128*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (a^2*cot(c + d*x)^9)/(3*d) - (a^2*cot(c + d*x)^11)/(11*d) + (3*a^2*cot(c + d*x)*csc(c + d*x))/(128*d) + (a^2*cot(c + d*x)*csc(c + d*x)^3)/(64*d) - (a^2*cot(c + d*x)*csc(c + d*x)^5)/(16*d) + (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^13*(a + a*sin(c + d*x))^2, x, 18, (17*a^2*atanh(cos(c + d*x)))/(1024*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (4*a^2*cot(c + d*x)^9)/(9*d) - (2*a^2*cot(c + d*x)^11)/(11*d) + (17*a^2*cot(c + d*x)*csc(c + d*x))/(1024*d) + (17*a^2*cot(c + d*x)*csc(c + d*x)^3)/(1536*d) - (11*a^2*cot(c + d*x)*csc(c + d*x)^5)/(384*d) + (a^2*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d) - (a^2*cot(c + d*x)*csc(c + d*x)^7)/(64*d) + (a^2*cot(c + d*x)^3*csc(c + d*x)^7)/(24*d) - (a^2*cot(c + d*x)^5*csc(c + d*x)^7)/(12*d)]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 21, (27*a^3*x)/1024 - (4*a^3*cos(c + d*x)^7)/(7*d) + (a^3*cos(c + d*x)^9)/d - (6*a^3*cos(c + d*x)^11)/(11*d) + (a^3*cos(c + d*x)^13)/(13*d) + (27*a^3*cos(c + d*x)*sin(c + d*x))/(1024*d) + (9*a^3*cos(c + d*x)^3*sin(c + d*x))/(512*d) + (9*a^3*cos(c + d*x)^5*sin(c + d*x))/(640*d) - (27*a^3*cos(c + d*x)^7*sin(c + d*x))/(320*d) - (9*a^3*cos(c + d*x)^7*sin(c + d*x)^3)/(40*d) - (a^3*cos(c + d*x)^7*sin(c + d*x)^5)/(4*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 21, (41*a^3*x)/1024 - (4*a^3*cos(c + d*x)^7)/(7*d) + (7*a^3*cos(c + d*x)^9)/(9*d) - (3*a^3*cos(c + d*x)^11)/(11*d) + (41*a^3*cos(c + d*x)*sin(c + d*x))/(1024*d) + (41*a^3*cos(c + d*x)^3*sin(c + d*x))/(1536*d) + (41*a^3*cos(c + d*x)^5*sin(c + d*x))/(1920*d) - (41*a^3*cos(c + d*x)^7*sin(c + d*x))/(320*d) - (41*a^3*cos(c + d*x)^7*sin(c + d*x)^3)/(120*d) - (a^3*cos(c + d*x)^7*sin(c + d*x)^5)/(12*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 19, (19*a^3*x)/256 - (4*a^3*cos(c + d*x)^7)/(7*d) + (5*a^3*cos(c + d*x)^9)/(9*d) - (a^3*cos(c + d*x)^11)/(11*d) + (19*a^3*cos(c + d*x)*sin(c + d*x))/(256*d) + (19*a^3*cos(c + d*x)^3*sin(c + d*x))/(384*d) + (19*a^3*cos(c + d*x)^5*sin(c + d*x))/(480*d) - (19*a^3*cos(c + d*x)^7*sin(c + d*x))/(80*d) - (3*a^3*cos(c + d*x)^7*sin(c + d*x)^3)/(10*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 8, (33*a^3*x)/256 - (33*a^3*cos(c + d*x)^7)/(560*d) + (33*a^3*cos(c + d*x)*sin(c + d*x))/(256*d) + (11*a^3*cos(c + d*x)^3*sin(c + d*x))/(128*d) + (11*a^3*cos(c + d*x)^5*sin(c + d*x))/(160*d) - (a*cos(c + d*x)^7*(a + a*sin(c + d*x))^2)/(30*d) - (cos(c + d*x)^7*(a + a*sin(c + d*x))^3)/(10*d) - (11*cos(c + d*x)^7*(a^3 + a^3*sin(c + d*x)))/(240*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 17, (125*a^3*x)/128 - (a^3*atanh(cos(c + d*x)))/d + (a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/(3*d) + (a^3*cos(c + d*x)^5)/(5*d) - (3*a^3*cos(c + d*x)^7)/(7*d) + (125*a^3*cos(c + d*x)*sin(c + d*x))/(128*d) + (125*a^3*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (25*a^3*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (a^3*cos(c + d*x)^7*sin(c + d*x))/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 19, -((15*a^3*x)/16) - (3*a^3*atanh(cos(c + d*x)))/d + (3*a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/d + (3*a^3*cos(c + d*x)^5)/(5*d) - (a^3*cos(c + d*x)^7)/(7*d) - (a^3*cot(c + d*x))/d + (15*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (11*a^3*cos(c + d*x)*sin(c + d*x)^3)/(8*d) + (a^3*cos(c + d*x)*sin(c + d*x)^5)/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 17, -((85*a^3*x)/16) - (a^3*atanh(cos(c + d*x)))/(2*d) + (a^3*cos(c + d*x))/d + (2*a^3*cos(c + d*x)^3)/(3*d) + (3*a^3*cos(c + d*x)^5)/(5*d) - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)*csc(c + d*x))/(2*d) - (43*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*a^3*cos(c + d*x)*sin(c + d*x)^3)/(24*d) + (a^3*cos(c + d*x)*sin(c + d*x)^5)/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 15, -((25*a^3*x)/8) + (13*a^3*atanh(cos(c + d*x)))/(2*d) - (5*a^3*cos(c + d*x))/d - (2*a^3*cos(c + d*x)^3)/(3*d) + (a^3*cos(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)*csc(c + d*x))/(2*d) - (23*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (3*a^3*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 16, (45*a^3*x)/8 + (45*a^3*atanh(cos(c + d*x)))/(8*d) - (5*a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/d + (5*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/d - (3*a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d) + (3*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 16, (13*a^3*x)/2 - (25*a^3*atanh(cos(c + d*x)))/(8*d) + (a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/(3*d) + (5*a^3*cot(c + d*x))/d - (2*a^3*cot(c + d*x)^3)/(3*d) - (a^3*cot(c + d*x)^5)/(5*d) + (23*a^3*cot(c + d*x)*csc(c + d*x))/(8*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^3)/(4*d) + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 18, -((a^3*x)/2) - (85*a^3*atanh(cos(c + d*x)))/(16*d) + (3*a^3*cos(c + d*x))/d - (a^3*cot(c + d*x))/d + (2*a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)^5)/(5*d) + (43*a^3*cot(c + d*x)*csc(c + d*x))/(16*d) - (5*a^3*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a^3*cot(c + d*x)*csc(c + d*x)^5)/(6*d) + (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8*(a + a*sin(c + d*x))^3, x, 18, -3*a^3*x - (15*a^3*atanh(cos(c + d*x)))/(16*d) + (a^3*cos(c + d*x))/d - (3*a^3*cot(c + d*x))/d + (a^3*cot(c + d*x)^3)/d - (3*a^3*cot(c + d*x)^5)/(5*d) - (a^3*cot(c + d*x)^7)/(7*d) - (15*a^3*cot(c + d*x)*csc(c + d*x))/(16*d) + (11*a^3*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a^3*cot(c + d*x)*csc(c + d*x)^5)/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^9*(a + a*sin(c + d*x))^3, x, 17, (-a^3)*x + (125*a^3*atanh(cos(c + d*x)))/(128*d) - (a^3*cot(c + d*x))/d + (a^3*cot(c + d*x)^3)/(3*d) - (a^3*cot(c + d*x)^5)/(5*d) - (3*a^3*cot(c + d*x)^7)/(7*d) - (115*a^3*cot(c + d*x)*csc(c + d*x))/(128*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x))/(8*d) - (a^3*cot(c + d*x)^5*csc(c + d*x))/(2*d) - (5*a^3*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^10*(a + a*sin(c + d*x))^3, x, 16, (55*a^3*atanh(cos(c + d*x)))/(128*d) - (4*a^3*cot(c + d*x)^7)/(7*d) - (a^3*cot(c + d*x)^9)/(9*d) - (25*a^3*cot(c + d*x)*csc(c + d*x))/(128*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x))/(24*d) - (a^3*cot(c + d*x)^5*csc(c + d*x))/(6*d) - (15*a^3*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(16*d) - (3*a^3*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^11*(a + a*sin(c + d*x))^3, x, 18, (33*a^3*atanh(cos(c + d*x)))/(256*d) - (4*a^3*cot(c + d*x)^7)/(7*d) - (a^3*cot(c + d*x)^9)/(3*d) + (33*a^3*cot(c + d*x)*csc(c + d*x))/(256*d) - (29*a^3*cot(c + d*x)*csc(c + d*x)^3)/(128*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(16*d) - (3*a^3*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d) - (a^3*cot(c + d*x)*csc(c + d*x)^5)/(32*d) + (a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^12*(a + a*sin(c + d*x))^3, x, 19, (19*a^3*atanh(cos(c + d*x)))/(256*d) - (4*a^3*cot(c + d*x)^7)/(7*d) - (5*a^3*cot(c + d*x)^9)/(9*d) - (a^3*cot(c + d*x)^11)/(11*d) + (19*a^3*cot(c + d*x)*csc(c + d*x))/(256*d) - (7*a^3*cot(c + d*x)*csc(c + d*x)^3)/(128*d) + (5*a^3*cot(c + d*x)^3*csc(c + d*x)^3)/(48*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^3)/(8*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^5)/(32*d) + (3*a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (3*a^3*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^13*(a + a*sin(c + d*x))^3, x, 21, (41*a^3*atanh(cos(c + d*x)))/(1024*d) - (4*a^3*cot(c + d*x)^7)/(7*d) - (7*a^3*cot(c + d*x)^9)/(9*d) - (3*a^3*cot(c + d*x)^11)/(11*d) + (41*a^3*cot(c + d*x)*csc(c + d*x))/(1024*d) + (41*a^3*cot(c + d*x)*csc(c + d*x)^3)/(1536*d) - (35*a^3*cot(c + d*x)*csc(c + d*x)^5)/(384*d) + (3*a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (3*a^3*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d) - (a^3*cot(c + d*x)*csc(c + d*x)^7)/(64*d) + (a^3*cot(c + d*x)^3*csc(c + d*x)^7)/(24*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^7)/(12*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^14*(a + a*sin(c + d*x))^3, x, 21, (27*a^3*atanh(cos(c + d*x)))/(1024*d) - (4*a^3*cot(c + d*x)^7)/(7*d) - (a^3*cot(c + d*x)^9)/d - (6*a^3*cot(c + d*x)^11)/(11*d) - (a^3*cot(c + d*x)^13)/(13*d) + (27*a^3*cot(c + d*x)*csc(c + d*x))/(1024*d) + (9*a^3*cot(c + d*x)*csc(c + d*x)^3)/(512*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^5)/(128*d) + (a^3*cot(c + d*x)^3*csc(c + d*x)^5)/(16*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^5)/(10*d) - (3*a^3*cot(c + d*x)*csc(c + d*x)^7)/(64*d) + (a^3*cot(c + d*x)^3*csc(c + d*x)^7)/(8*d) - (a^3*cot(c + d*x)^5*csc(c + d*x)^7)/(4*d)]

    @test_int [cos(c + d*x)^6*csc(c + d*x)^4*(a + a*sin(c + d*x))^4, x, 22, -((135*a^4*x)/16) + (6*a^4*atanh(cos(c + d*x)))/d - (4*a^4*cos(c + d*x))/d + (4*a^4*cos(c + d*x)^5)/(5*d) - (4*a^4*cot(c + d*x))/d - (a^4*cot(c + d*x)^3)/(3*d) - (2*a^4*cot(c + d*x)*csc(c + d*x))/d - (89*a^4*cos(c + d*x)*sin(c + d*x))/(16*d) + (23*a^4*cos(c + d*x)*sin(c + d*x)^3)/(24*d) + (a^4*cos(c + d*x)*sin(c + d*x)^5)/(6*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^6*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 9, (3*x)/(128*a) + cos(c + d*x)^5/(5*a*d) - (2*cos(c + d*x)^7)/(7*a*d) + cos(c + d*x)^9/(9*a*d) + (3*cos(c + d*x)*sin(c + d*x))/(128*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(64*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(16*a*d) - (cos(c + d*x)^5*sin(c + d*x)^3)/(8*a*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 9, -((3*x)/(128*a)) - cos(c + d*x)^5/(5*a*d) + cos(c + d*x)^7/(7*a*d) - (3*cos(c + d*x)*sin(c + d*x))/(128*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(64*a*d) + (cos(c + d*x)^5*sin(c + d*x))/(16*a*d) + (cos(c + d*x)^5*sin(c + d*x)^3)/(8*a*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 8, x/(16*a) + cos(c + d*x)^5/(5*a*d) - cos(c + d*x)^7/(7*a*d) + (cos(c + d*x)*sin(c + d*x))/(16*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(24*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(6*a*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 7, -(x/(16*a)) - cos(c + d*x)^5/(5*a*d) - (cos(c + d*x)*sin(c + d*x))/(16*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(24*a*d) + (cos(c + d*x)^5*sin(c + d*x))/(6*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 8, -((3*x)/(8*a)) - atanh(cos(c + d*x))/(a*d) + cos(c + d*x)/(a*d) + cos(c + d*x)^3/(3*a*d) - (3*cos(c + d*x)*sin(c + d*x))/(8*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 9, -((3*x)/(2*a)) + atanh(cos(c + d*x))/(a*d) - cos(c + d*x)/(a*d) - cos(c + d*x)^3/(3*a*d) - (3*cot(c + d*x))/(2*a*d) + (cos(c + d*x)^2*cot(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 9, (3*x)/(2*a) + (3*atanh(cos(c + d*x)))/(2*a*d) - (3*cos(c + d*x))/(2*a*d) + (3*cot(c + d*x))/(2*a*d) - (cos(c + d*x)^2*cot(c + d*x))/(2*a*d) - (cos(c + d*x)*cot(c + d*x)^2)/(2*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 8, x/a - (3*atanh(cos(c + d*x)))/(2*a*d) + (3*cos(c + d*x))/(2*a*d) + cot(c + d*x)/(a*d) + (cos(c + d*x)*cot(c + d*x)^2)/(2*a*d) - cot(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 7, -(x/a) - (3*atanh(cos(c + d*x)))/(8*a*d) - cot(c + d*x)/(a*d) + cot(c + d*x)^3/(3*a*d) + (3*cot(c + d*x)*csc(c + d*x))/(8*a*d) - (cot(c + d*x)^3*csc(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 6, (3*atanh(cos(c + d*x)))/(8*a*d) - cot(c + d*x)^5/(5*a*d) - (3*cot(c + d*x)*csc(c + d*x))/(8*a*d) + (cot(c + d*x)^3*csc(c + d*x))/(4*a*d)]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 13, -(x/(8*a^2)) - (2*cos(c + d*x)^3)/(3*a^2*d) + (3*cos(c + d*x)^5)/(5*a^2*d) - cos(c + d*x)^7/(7*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(8*a^2*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a^2*d) + (cos(c + d*x)^3*sin(c + d*x)^3)/(3*a^2*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 6, (3*x)/(16*a^2) + cos(c + d*x)^5/(10*a^2*d) + (3*cos(c + d*x)*sin(c + d*x))/(16*a^2*d) + (cos(c + d*x)^3*sin(c + d*x))/(8*a^2*d) + (cos(c + d*x)^3*(a - a*sin(c + d*x))^3)/(6*a^5*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 5, -(x/(4*a^2)) - (2*cos(c + d*x)^5)/(15*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(4*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(6*a^2*d) - cos(c + d*x)^7/(3*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 10, -(x/a^2) - atanh(cos(c + d*x))/(a^2*d) + cos(c + d*x)/(a^2*d) - cos(c + d*x)^3/(3*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 9, -(x/(2*a^2)) + (2*atanh(cos(c + d*x)))/(a^2*d) - (2*cos(c + d*x))/(a^2*d) - cot(c + d*x)/(a^2*d) + (cos(c + d*x)*sin(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 8, (2*x)/a^2 - atanh(cos(c + d*x))/(2*a^2*d) + cos(c + d*x)/(a^2*d) + (2*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 9, -(x/a^2) - atanh(cos(c + d*x))/(a^2*d) - cot(c + d*x)/(a^2*d) - cot(c + d*x)^3/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 10, (5*atanh(cos(c + d*x)))/(8*a^2*d) + (2*cot(c + d*x)^3)/(3*a^2*d) - (3*cot(c + d*x)*csc(c + d*x))/(8*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 11, -(atanh(cos(c + d*x))/(4*a^2*d)) - (2*cot(c + d*x)^3)/(3*a^2*d) - cot(c + d*x)^5/(5*a^2*d) - (cot(c + d*x)*csc(c + d*x))/(4*a^2*d) + (cot(c + d*x)*csc(c + d*x)^3)/(2*a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 13, (3*atanh(cos(c + d*x)))/(16*a^2*d) + (2*cot(c + d*x)^3)/(3*a^2*d) + (2*cot(c + d*x)^5)/(5*a^2*d) + (3*cot(c + d*x)*csc(c + d*x))/(16*a^2*d) - (5*cot(c + d*x)*csc(c + d*x)^3)/(24*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a^2*d)]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 14, -((23*x)/(16*a^3)) - (4*cos(c + d*x))/(a^3*d) + (7*cos(c + d*x)^3)/(3*a^3*d) - (3*cos(c + d*x)^5)/(5*a^3*d) + (23*cos(c + d*x)*sin(c + d*x))/(16*a^3*d) + (23*cos(c + d*x)*sin(c + d*x)^3)/(24*a^3*d) + (cos(c + d*x)*sin(c + d*x)^5)/(6*a^3*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 12, (13*x)/(8*a^3) + (4*cos(c + d*x))/(a^3*d) - (5*cos(c + d*x)^3)/(3*a^3*d) + cos(c + d*x)^5/(5*a^3*d) - (13*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) - (3*cos(c + d*x)*sin(c + d*x)^3)/(4*a^3*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 5, -((15*x)/(8*a^3)) - (4*cos(c + d*x))/(a^3*d) + cos(c + d*x)^3/(a^3*d) + (15*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) + (cos(c + d*x)*sin(c + d*x)^3)/(4*a^3*d), -((15*x)/(8*a^3)) - (5*cos(c + d*x)^3)/(4*a^3*d) - (15*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) - cos(c + d*x)^7/(d*(a + a*sin(c + d*x))^3) - (3*cos(c + d*x)^5)/(4*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 7, -((7*x)/(2*a^3)) - atanh(cos(c + d*x))/(a^3*d) - (3*cos(c + d*x))/(a^3*d) + (cos(c + d*x)*sin(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 7, (3*x)/a^3 + (3*atanh(cos(c + d*x)))/(a^3*d) + cos(c + d*x)/(a^3*d) - cot(c + d*x)/(a^3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 8, -(x/a^3) - (7*atanh(cos(c + d*x)))/(2*a^3*d) + (3*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 10, (5*atanh(cos(c + d*x)))/(2*a^3*d) - (4*cot(c + d*x))/(a^3*d) - cot(c + d*x)^3/(3*a^3*d) + (3*cot(c + d*x)*csc(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 12, -((15*atanh(cos(c + d*x)))/(8*a^3*d)) + (4*cot(c + d*x))/(a^3*d) + cot(c + d*x)^3/(a^3*d) - (15*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 12, (13*atanh(cos(c + d*x)))/(8*a^3*d) - (4*cot(c + d*x))/(a^3*d) - (5*cot(c + d*x)^3)/(3*a^3*d) - cot(c + d*x)^5/(5*a^3*d) + (13*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) + (3*cot(c + d*x)*csc(c + d*x)^3)/(4*a^3*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)*with*n*symbolic=#

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^3, x, 6, (a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (3*a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2)) + (3*a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (3 + n)/2, (5 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(3 + n))/(d*(3 + n)*sqrt(cos(c + d*x)^2)) + (a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (4 + n)/2, (6 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(4 + n))/(d*(4 + n)*sqrt(cos(c + d*x)^2))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x, 5, (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (2*a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2)) + (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (3 + n)/2, (5 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(3 + n))/(d*(3 + n)*sqrt(cos(c + d*x)^2))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x)), x, 3, (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^7*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^7*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^7*sin(c + d*x)^6*(a + a*sin(c + d*x)), x, 4, (a*sin(c + d*x)^7)/(7*d) + (a*sin(c + d*x)^8)/(8*d) - (a*sin(c + d*x)^9)/(3*d) - (3*a*sin(c + d*x)^10)/(10*d) + (3*a*sin(c + d*x)^11)/(11*d) + (a*sin(c + d*x)^12)/(4*d) - (a*sin(c + d*x)^13)/(13*d) - (a*sin(c + d*x)^14)/(14*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^5*(a + a*sin(c + d*x)), x, 8, -((a*cos(c + d*x)^8)/(8*d)) + (a*cos(c + d*x)^10)/(5*d) - (a*cos(c + d*x)^12)/(12*d) + (a*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^9)/(3*d) + (3*a*sin(c + d*x)^11)/(11*d) - (a*sin(c + d*x)^13)/(13*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 8, -((a*cos(c + d*x)^8)/(8*d)) + (a*cos(c + d*x)^10)/(5*d) - (a*cos(c + d*x)^12)/(12*d) + (a*sin(c + d*x)^5)/(5*d) - (3*a*sin(c + d*x)^7)/(7*d) + (a*sin(c + d*x)^9)/(3*d) - (a*sin(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 7, -((a*cos(c + d*x)^8)/(8*d)) + (a*cos(c + d*x)^10)/(10*d) + (a*sin(c + d*x)^5)/(5*d) - (3*a*sin(c + d*x)^7)/(7*d) + (a*sin(c + d*x)^9)/(3*d) - (a*sin(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 7, -((a*cos(c + d*x)^8)/(8*d)) + (a*cos(c + d*x)^10)/(10*d) + (a*sin(c + d*x)^3)/(3*d) - (3*a*sin(c + d*x)^5)/(5*d) + (3*a*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, -((a*cos(c + d*x)^8)/(8*d)) + (a*sin(c + d*x)^3)/(3*d) - (3*a*sin(c + d*x)^5)/(5*d) + (3*a*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 4, (a*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d - (3*a*sin(c + d*x)^2)/(2*d) - (a*sin(c + d*x)^3)/d + (3*a*sin(c + d*x)^4)/(4*d) + (3*a*sin(c + d*x)^5)/(5*d) - (a*sin(c + d*x)^6)/(6*d) - (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (a*log(sin(c + d*x)))/d - (3*a*sin(c + d*x))/d - (3*a*sin(c + d*x)^2)/(2*d) + (a*sin(c + d*x)^3)/d + (3*a*sin(c + d*x)^4)/(4*d) - (a*sin(c + d*x)^5)/(5*d) - (a*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (3*a*log(sin(c + d*x)))/d - (3*a*sin(c + d*x))/d + (3*a*sin(c + d*x)^2)/(2*d) + (a*sin(c + d*x)^3)/d - (a*sin(c + d*x)^4)/(4*d) - (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, (3*a*csc(c + d*x))/d - (a*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/(3*d) - (3*a*log(sin(c + d*x)))/d + (3*a*sin(c + d*x))/d + (3*a*sin(c + d*x)^2)/(2*d) - (a*sin(c + d*x)^3)/(3*d) - (a*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^5*(a + a*sin(c + d*x)), x, 4, (3*a*csc(c + d*x))/d + (3*a*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^4)/(4*d) + (3*a*log(sin(c + d*x)))/d + (3*a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/(2*d) - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^6*(a + a*sin(c + d*x)), x, 4, -((3*a*csc(c + d*x))/d) + (3*a*csc(c + d*x)^2)/(2*d) + (a*csc(c + d*x)^3)/d - (a*csc(c + d*x)^4)/(4*d) - (a*csc(c + d*x)^5)/(5*d) + (3*a*log(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^7*(a + a*sin(c + d*x)), x, 3, -((3*a*csc(c + d*x))/d) - (3*a*csc(c + d*x)^2)/(2*d) + (a*csc(c + d*x)^3)/d + (3*a*csc(c + d*x)^4)/(4*d) - (a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^6)/(6*d) - (a*log(sin(c + d*x)))/d - (a*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^8*(a + a*sin(c + d*x)), x, 4, (a*csc(c + d*x))/d - (3*a*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/d + (3*a*csc(c + d*x)^4)/(4*d) + (3*a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^6)/(6*d) - (a*csc(c + d*x)^7)/(7*d) - (a*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^9*(a + a*sin(c + d*x)), x, 6, -((a*cot(c + d*x)^8)/(8*d)) + (a*csc(c + d*x))/d - (a*csc(c + d*x)^3)/d + (3*a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^10*(a + a*sin(c + d*x)), x, 6, -((a*cot(c + d*x)^8)/(8*d)) + (a*csc(c + d*x)^3)/(3*d) - (3*a*csc(c + d*x)^5)/(5*d) + (3*a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^11*(a + a*sin(c + d*x)), x, 7, -((a*cot(c + d*x)^8)/(8*d)) - (a*cot(c + d*x)^10)/(10*d) + (a*csc(c + d*x)^3)/(3*d) - (3*a*csc(c + d*x)^5)/(5*d) + (3*a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^12*(a + a*sin(c + d*x)), x, 7, -((a*cot(c + d*x)^8)/(8*d)) - (a*cot(c + d*x)^10)/(10*d) + (a*csc(c + d*x)^5)/(5*d) - (3*a*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^9)/(3*d) - (a*csc(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^13*(a + a*sin(c + d*x)), x, 8, -((a*cot(c + d*x)^8)/(8*d)) - (a*cot(c + d*x)^10)/(5*d) - (a*cot(c + d*x)^12)/(12*d) + (a*csc(c + d*x)^5)/(5*d) - (3*a*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^9)/(3*d) - (a*csc(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^14*(a + a*sin(c + d*x)), x, 8, -((a*cot(c + d*x)^8)/(8*d)) - (a*cot(c + d*x)^10)/(5*d) - (a*cot(c + d*x)^12)/(12*d) + (a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(3*d) + (3*a*csc(c + d*x)^11)/(11*d) - (a*csc(c + d*x)^13)/(13*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^15*(a + a*sin(c + d*x)), x, 4, (a*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^8)/(8*d) - (a*csc(c + d*x)^9)/(3*d) - (3*a*csc(c + d*x)^10)/(10*d) + (3*a*csc(c + d*x)^11)/(11*d) + (a*csc(c + d*x)^12)/(4*d) - (a*csc(c + d*x)^13)/(13*d) - (a*csc(c + d*x)^14)/(14*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^7*sin(c + d*x)^6/(a + a*sin(c + d*x)), x, 4, sin(c + d*x)^7/(7*a*d) - sin(c + d*x)^8/(8*a*d) - (2*sin(c + d*x)^9)/(9*a*d) + sin(c + d*x)^10/(5*a*d) + sin(c + d*x)^11/(11*a*d) - sin(c + d*x)^12/(12*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^5/(a + a*sin(c + d*x)), x, 4, sin(c + d*x)^6/(6*a*d) - sin(c + d*x)^7/(7*a*d) - sin(c + d*x)^8/(4*a*d) + (2*sin(c + d*x)^9)/(9*a*d) + sin(c + d*x)^10/(10*a*d) - sin(c + d*x)^11/(11*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, sin(c + d*x)^5/(5*a*d) - sin(c + d*x)^6/(6*a*d) - (2*sin(c + d*x)^7)/(7*a*d) + sin(c + d*x)^8/(4*a*d) + sin(c + d*x)^9/(9*a*d) - sin(c + d*x)^10/(10*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 7, -(cos(c + d*x)^6/(6*a*d)) + cos(c + d*x)^8/(8*a*d) - sin(c + d*x)^5/(5*a*d) + (2*sin(c + d*x)^7)/(7*a*d) - sin(c + d*x)^9/(9*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 7, cos(c + d*x)^6/(6*a*d) - cos(c + d*x)^8/(8*a*d) + sin(c + d*x)^3/(3*a*d) - (2*sin(c + d*x)^5)/(5*a*d) + sin(c + d*x)^7/(7*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 6, -(cos(c + d*x)^6/(6*a*d)) - sin(c + d*x)^3/(3*a*d) + (2*sin(c + d*x)^5)/(5*a*d) - sin(c + d*x)^7/(7*a*d)]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^0/(a + a*sin(c + d*x)), x, 3, -((a - a*sin(c + d*x))^4/(a^5*d)) + (4*(a - a*sin(c + d*x))^5)/(5*a^6*d) - (a - a*sin(c + d*x))^6/(6*a^7*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d) - sin(c + d*x)^2/(a*d) + (2*sin(c + d*x)^3)/(3*a*d) + sin(c + d*x)^4/(4*a*d) - sin(c + d*x)^5/(5*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - log(sin(c + d*x))/(a*d) - (2*sin(c + d*x))/(a*d) + sin(c + d*x)^2/(a*d) + sin(c + d*x)^3/(3*a*d) - sin(c + d*x)^4/(4*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d) - (2*log(sin(c + d*x)))/(a*d) + (2*sin(c + d*x))/(a*d) + sin(c + d*x)^2/(2*a*d) - sin(c + d*x)^3/(3*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, (2*csc(c + d*x))/(a*d) + csc(c + d*x)^2/(2*a*d) - csc(c + d*x)^3/(3*a*d) + (2*log(sin(c + d*x)))/(a*d) + sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 4, -((2*csc(c + d*x))/(a*d)) + csc(c + d*x)^2/(a*d) + csc(c + d*x)^3/(3*a*d) - csc(c + d*x)^4/(4*a*d) + log(sin(c + d*x))/(a*d) - sin(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - csc(c + d*x)^2/(a*d) + (2*csc(c + d*x)^3)/(3*a*d) + csc(c + d*x)^4/(4*a*d) - csc(c + d*x)^5/(5*a*d) - log(sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^7/(a + a*sin(c + d*x)), x, 6, -(cot(c + d*x)^6/(6*a*d)) + csc(c + d*x)/(a*d) - (2*csc(c + d*x)^3)/(3*a*d) + csc(c + d*x)^5/(5*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^8/(a + a*sin(c + d*x)), x, 6, cot(c + d*x)^6/(6*a*d) - csc(c + d*x)^3/(3*a*d) + (2*csc(c + d*x)^5)/(5*a*d) - csc(c + d*x)^7/(7*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^9/(a + a*sin(c + d*x)), x, 7, -(cot(c + d*x)^6/(6*a*d)) - cot(c + d*x)^8/(8*a*d) + csc(c + d*x)^3/(3*a*d) - (2*csc(c + d*x)^5)/(5*a*d) + csc(c + d*x)^7/(7*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^10/(a + a*sin(c + d*x)), x, 7, cot(c + d*x)^6/(6*a*d) + cot(c + d*x)^8/(8*a*d) - csc(c + d*x)^5/(5*a*d) + (2*csc(c + d*x)^7)/(7*a*d) - csc(c + d*x)^9/(9*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^11/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)^5/(5*a*d) - csc(c + d*x)^6/(6*a*d) - (2*csc(c + d*x)^7)/(7*a*d) + csc(c + d*x)^8/(4*a*d) + csc(c + d*x)^9/(9*a*d) - csc(c + d*x)^10/(10*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^12/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)^6/(6*a*d) - csc(c + d*x)^7/(7*a*d) - csc(c + d*x)^8/(4*a*d) + (2*csc(c + d*x)^9)/(9*a*d) + csc(c + d*x)^10/(10*a*d) - csc(c + d*x)^11/(11*a*d)]
    @test_int [cos(c + d*x)^7*csc(c + d*x)^13/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)^7/(7*a*d) - csc(c + d*x)^8/(8*a*d) - (2*csc(c + d*x)^9)/(9*a*d) + csc(c + d*x)^10/(5*a*d) + csc(c + d*x)^11/(11*a*d) - csc(c + d*x)^12/(12*a*d)]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^7*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^7*sin(e+f*x)^n*(a+a*sin(e+f*x))^m*win*n*symbolic=#

    @test_int [cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^3, x, 3, (a^3*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (3*a^3*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (8*a^3*sin(c + d*x)^(4 + n))/(d*(4 + n)) - (6*a^3*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (6*a^3*sin(c + d*x)^(6 + n))/(d*(6 + n)) + (8*a^3*sin(c + d*x)^(7 + n))/(d*(7 + n)) - (3*a^3*sin(c + d*x)^(9 + n))/(d*(9 + n)) - (a^3*sin(c + d*x)^(10 + n))/(d*(10 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^2, x, 3, (a^2*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (2*a^2*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (2*a^2*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (6*a^2*sin(c + d*x)^(4 + n))/(d*(4 + n)) + (6*a^2*sin(c + d*x)^(6 + n))/(d*(6 + n)) + (2*a^2*sin(c + d*x)^(7 + n))/(d*(7 + n)) - (2*a^2*sin(c + d*x)^(8 + n))/(d*(8 + n)) - (a^2*sin(c + d*x)^(9 + n))/(d*(9 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^1, x, 3, (a*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (a*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (3*a*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (3*a*sin(c + d*x)^(4 + n))/(d*(4 + n)) + (3*a*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (3*a*sin(c + d*x)^(6 + n))/(d*(6 + n)) - (a*sin(c + d*x)^(7 + n))/(d*(7 + n)) - (a*sin(c + d*x)^(8 + n))/(d*(8 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n/(a + a*sin(c + d*x))^1, x, 3, sin(c + d*x)^(1 + n)/(a*d*(1 + n)) - sin(c + d*x)^(2 + n)/(a*d*(2 + n)) - (2*sin(c + d*x)^(3 + n))/(a*d*(3 + n)) + (2*sin(c + d*x)^(4 + n))/(a*d*(4 + n)) + sin(c + d*x)^(5 + n)/(a*d*(5 + n)) - sin(c + d*x)^(6 + n)/(a*d*(6 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n/(a + a*sin(c + d*x))^2, x, 3, sin(c + d*x)^(1 + n)/(a^2*d*(1 + n)) - (2*sin(c + d*x)^(2 + n))/(a^2*d*(2 + n)) + (2*sin(c + d*x)^(4 + n))/(a^2*d*(4 + n)) - sin(c + d*x)^(5 + n)/(a^2*d*(5 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n/(a + a*sin(c + d*x))^3, x, 3, sin(c + d*x)^(1 + n)/(a^3*d*(1 + n)) - (3*sin(c + d*x)^(2 + n))/(a^3*d*(2 + n)) + (3*sin(c + d*x)^(3 + n))/(a^3*d*(3 + n)) - sin(c + d*x)^(4 + n)/(a^3*d*(4 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n/(a + a*sin(c + d*x))^4, x, 8, -((7*sin(c + d*x)^(1 + n))/(a^4*d*(1 + n))) + (8*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^4*d*(1 + n)) + (4*sin(c + d*x)^(2 + n))/(a^4*d*(2 + n)) - sin(c + d*x)^(3 + n)/(a^4*d*(3 + n))]
    @test_int [cos(c + d*x)^7*sin(c + d*x)^n/(a + a*sin(c + d*x))^5, x, 4, -((4*(3 + 2*n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(a^5*d*(1 + n))) - (sin(c + d*x)^(1 + n)*(a - a*sin(c + d*x))^2)/(d*(2 + n)*(a^7 + a^7*sin(c + d*x))) + (sin(c + d*x)^(1 + n)*(a*(27 + 30*n + 8*n^2) + a*(7 + 2*n)*sin(c + d*x)))/(d*(2 + 3*n + n^2)*(a^6 + a^6*sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^8*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^8*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^8*sin(c + d*x)^5/(a + a*sin(c + d*x)), x, 11, (-5*x)/(1024*a) - cos(c + d*x)^7/(7*a*d) + (2*cos(c + d*x)^9)/(9*a*d) - cos(c + d*x)^11/(11*a*d) - (5*cos(c + d*x)*sin(c + d*x))/(1024*a*d) - (5*cos(c + d*x)^3*sin(c + d*x))/(1536*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(384*a*d) + (cos(c + d*x)^7*sin(c + d*x))/(64*a*d) + (cos(c + d*x)^7*sin(c + d*x)^3)/(24*a*d) + (cos(c + d*x)^7*sin(c + d*x)^5)/(12*a*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 10, (3*x)/(256*a) + cos(c + d*x)^7/(7*a*d) - (2*cos(c + d*x)^9)/(9*a*d) + cos(c + d*x)^11/(11*a*d) + (3*cos(c + d*x)*sin(c + d*x))/(256*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(128*a*d) + (cos(c + d*x)^5*sin(c + d*x))/(160*a*d) - (3*cos(c + d*x)^7*sin(c + d*x))/(80*a*d) - (cos(c + d*x)^7*sin(c + d*x)^3)/(10*a*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 10, (-3*x)/(256*a) - cos(c + d*x)^7/(7*a*d) + cos(c + d*x)^9/(9*a*d) - (3*cos(c + d*x)*sin(c + d*x))/(256*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(128*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(160*a*d) + (3*cos(c + d*x)^7*sin(c + d*x))/(80*a*d) + (cos(c + d*x)^7*sin(c + d*x)^3)/(10*a*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 9, (5*x)/(128*a) + cos(c + d*x)^7/(7*a*d) - cos(c + d*x)^9/(9*a*d) + (5*cos(c + d*x)*sin(c + d*x))/(128*a*d) + (5*cos(c + d*x)^3*sin(c + d*x))/(192*a*d) + (cos(c + d*x)^5*sin(c + d*x))/(48*a*d) - (cos(c + d*x)^7*sin(c + d*x))/(8*a*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 8, (-5*x)/(128*a) - cos(c + d*x)^7/(7*a*d) - (5*cos(c + d*x)*sin(c + d*x))/(128*a*d) - (5*cos(c + d*x)^3*sin(c + d*x))/(192*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(48*a*d) + (cos(c + d*x)^7*sin(c + d*x))/(8*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 9, (-5*x)/(16*a) - atanh(cos(c + d*x))/(a*d) + cos(c + d*x)/(a*d) + cos(c + d*x)^3/(3*a*d) + cos(c + d*x)^5/(5*a*d) - (5*cos(c + d*x)*sin(c + d*x))/(16*a*d) - (5*cos(c + d*x)^3*sin(c + d*x))/(24*a*d) - (cos(c + d*x)^5*sin(c + d*x))/(6*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 10, (-15*x)/(8*a) + atanh(cos(c + d*x))/(a*d) - cos(c + d*x)/(a*d) - cos(c + d*x)^3/(3*a*d) - cos(c + d*x)^5/(5*a*d) - (15*cot(c + d*x))/(8*a*d) + (5*cos(c + d*x)^2*cot(c + d*x))/(8*a*d) + (cos(c + d*x)^4*cot(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 11, (15*x)/(8*a) + (5*atanh(cos(c + d*x)))/(2*a*d) - (5*cos(c + d*x))/(2*a*d) - (5*cos(c + d*x)^3)/(6*a*d) + (15*cot(c + d*x))/(8*a*d) - (5*cos(c + d*x)^2*cot(c + d*x))/(8*a*d) - (cos(c + d*x)^4*cot(c + d*x))/(4*a*d) - (cos(c + d*x)^3*cot(c + d*x)^2)/(2*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 11, (5*x)/(2*a) - (5*atanh(cos(c + d*x)))/(2*a*d) + (5*cos(c + d*x))/(2*a*d) + (5*cos(c + d*x)^3)/(6*a*d) + (5*cot(c + d*x))/(2*a*d) + (cos(c + d*x)^3*cot(c + d*x)^2)/(2*a*d) - (5*cot(c + d*x)^3)/(6*a*d) + (cos(c + d*x)^2*cot(c + d*x)^3)/(2*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^5/(a + a*sin(c + d*x)), x, 11, (-5*x)/(2*a) - (15*atanh(cos(c + d*x)))/(8*a*d) + (15*cos(c + d*x))/(8*a*d) - (5*cot(c + d*x))/(2*a*d) + (5*cos(c + d*x)*cot(c + d*x)^2)/(8*a*d) + (5*cot(c + d*x)^3)/(6*a*d) - (cos(c + d*x)^2*cot(c + d*x)^3)/(2*a*d) - (cos(c + d*x)*cot(c + d*x)^4)/(4*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^6/(a + a*sin(c + d*x)), x, 10, -(x/a) + (15*atanh(cos(c + d*x)))/(8*a*d) - (15*cos(c + d*x))/(8*a*d) - cot(c + d*x)/(a*d) - (5*cos(c + d*x)*cot(c + d*x)^2)/(8*a*d) + cot(c + d*x)^3/(3*a*d) + (cos(c + d*x)*cot(c + d*x)^4)/(4*a*d) - cot(c + d*x)^5/(5*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^7/(a + a*sin(c + d*x)), x, 9, x/a + (5*atanh(cos(c + d*x)))/(16*a*d) + cot(c + d*x)/(a*d) - cot(c + d*x)^3/(3*a*d) + cot(c + d*x)^5/(5*a*d) - (5*cot(c + d*x)*csc(c + d*x))/(16*a*d) + (5*cot(c + d*x)^3*csc(c + d*x))/(24*a*d) - (cot(c + d*x)^5*csc(c + d*x))/(6*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^8/(a + a*sin(c + d*x)), x, 7, (-5*atanh(cos(c + d*x)))/(16*a*d) - cot(c + d*x)^7/(7*a*d) + (5*cot(c + d*x)*csc(c + d*x))/(16*a*d) - (5*cot(c + d*x)^3*csc(c + d*x))/(24*a*d) + (cot(c + d*x)^5*csc(c + d*x))/(6*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^9/(a + a*sin(c + d*x)), x, 8, (5*atanh(cos(c + d*x)))/(128*a*d) + cot(c + d*x)^7/(7*a*d) + (5*cot(c + d*x)*csc(c + d*x))/(128*a*d) - (5*cot(c + d*x)*csc(c + d*x)^3)/(64*a*d) + (5*cot(c + d*x)^3*csc(c + d*x)^3)/(48*a*d) - (cot(c + d*x)^5*csc(c + d*x)^3)/(8*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^10/(a + a*sin(c + d*x)), x, 9, -((5*atanh(cos(c + d*x)))/(128*a*d)) - cot(c + d*x)^7/(7*a*d) - cot(c + d*x)^9/(9*a*d) - (5*cot(c + d*x)*csc(c + d*x))/(128*a*d) + (5*cot(c + d*x)*csc(c + d*x)^3)/(64*a*d) - (5*cot(c + d*x)^3*csc(c + d*x)^3)/(48*a*d) + (cot(c + d*x)^5*csc(c + d*x)^3)/(8*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^11/(a + a*sin(c + d*x)), x, 10, (3*atanh(cos(c + d*x)))/(256*a*d) + cot(c + d*x)^7/(7*a*d) + cot(c + d*x)^9/(9*a*d) + (3*cot(c + d*x)*csc(c + d*x))/(256*a*d) + (cot(c + d*x)*csc(c + d*x)^3)/(128*a*d) - (cot(c + d*x)*csc(c + d*x)^5)/(32*a*d) + (cot(c + d*x)^3*csc(c + d*x)^5)/(16*a*d) - (cot(c + d*x)^5*csc(c + d*x)^5)/(10*a*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^12/(a + a*sin(c + d*x)), x, 10, -((3*atanh(cos(c + d*x)))/(256*a*d)) - cot(c + d*x)^7/(7*a*d) - (2*cot(c + d*x)^9)/(9*a*d) - cot(c + d*x)^11/(11*a*d) - (3*cot(c + d*x)*csc(c + d*x))/(256*a*d) - (cot(c + d*x)*csc(c + d*x)^3)/(128*a*d) + (cot(c + d*x)*csc(c + d*x)^5)/(32*a*d) - (cot(c + d*x)^3*csc(c + d*x)^5)/(16*a*d) + (cot(c + d*x)^5*csc(c + d*x)^5)/(10*a*d)]

    @test_int [cos(c + d*x)^8*sin(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 15, (-3*x)/(128*a^2) - (2*cos(c + d*x)^5)/(5*a^2*d) + (5*cos(c + d*x)^7)/(7*a^2*d) - (4*cos(c + d*x)^9)/(9*a^2*d) + cos(c + d*x)^11/(11*a^2*d) - (3*cos(c + d*x)*sin(c + d*x))/(128*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(64*a^2*d) + (cos(c + d*x)^5*sin(c + d*x))/(16*a^2*d) + (cos(c + d*x)^5*sin(c + d*x)^3)/(8*a^2*d) + (cos(c + d*x)^5*sin(c + d*x)^5)/(5*a^2*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 17, (9*x)/(256*a^2) + (2*cos(c + d*x)^5)/(5*a^2*d) - (4*cos(c + d*x)^7)/(7*a^2*d) + (2*cos(c + d*x)^9)/(9*a^2*d) + (9*cos(c + d*x)*sin(c + d*x))/(256*a^2*d) + (3*cos(c + d*x)^3*sin(c + d*x))/(128*a^2*d) - (3*cos(c + d*x)^5*sin(c + d*x))/(32*a^2*d) - (3*cos(c + d*x)^5*sin(c + d*x)^3)/(16*a^2*d) - (cos(c + d*x)^5*sin(c + d*x)^5)/(10*a^2*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 14, (-3*x)/(64*a^2) - (2*cos(c + d*x)^5)/(5*a^2*d) + (3*cos(c + d*x)^7)/(7*a^2*d) - cos(c + d*x)^9/(9*a^2*d) - (3*cos(c + d*x)*sin(c + d*x))/(64*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(32*a^2*d) + (cos(c + d*x)^5*sin(c + d*x))/(8*a^2*d) + (cos(c + d*x)^5*sin(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 15, (11*x)/(128*a^2) + (2*cos(c + d*x)^5)/(5*a^2*d) - (2*cos(c + d*x)^7)/(7*a^2*d) + (11*cos(c + d*x)*sin(c + d*x))/(128*a^2*d) + (11*cos(c + d*x)^3*sin(c + d*x))/(192*a^2*d) - (11*cos(c + d*x)^5*sin(c + d*x))/(48*a^2*d) - (cos(c + d*x)^5*sin(c + d*x)^3)/(8*a^2*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 6, -x/(8*a^2) - (2*cos(c + d*x)^7)/(35*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(8*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(12*a^2*d) - (cos(c + d*x)^5*sin(c + d*x))/(15*a^2*d) - cos(c + d*x)^9/(5*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 12, (-3*x)/(4*a^2) - atanh(cos(c + d*x))/(a^2*d) + cos(c + d*x)/(a^2*d) + cos(c + d*x)^3/(3*a^2*d) - cos(c + d*x)^5/(5*a^2*d) - (3*cos(c + d*x)*sin(c + d*x))/(4*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 14, (-9*x)/(8*a^2) + (2*atanh(cos(c + d*x)))/(a^2*d) - (2*cos(c + d*x))/(a^2*d) - (2*cos(c + d*x)^3)/(3*a^2*d) - cot(c + d*x)/(a^2*d) + (cos(c + d*x)*sin(c + d*x))/(8*a^2*d) - (cos(c + d*x)*sin(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 13, (3*x)/a^2 + atanh(cos(c + d*x))/(2*a^2*d) + cos(c + d*x)^3/(3*a^2*d) + (2*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d) + (cos(c + d*x)*sin(c + d*x))/(a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 13, -x/(2*a^2) - (3*atanh(cos(c + d*x)))/(a^2*d) + (2*cos(c + d*x))/(a^2*d) - cot(c + d*x)^3/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(a^2*d) - (cos(c + d*x)*sin(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 14, (-2*x)/a^2 + (9*atanh(cos(c + d*x)))/(8*a^2*d) - cos(c + d*x)/(a^2*d) - (2*cot(c + d*x))/(a^2*d) + (2*cot(c + d*x)^3)/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(8*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 11, x/a^2 + (3*atanh(cos(c + d*x)))/(4*a^2*d) + cot(c + d*x)/(a^2*d) - cot(c + d*x)^3/(3*a^2*d) - cot(c + d*x)^5/(5*a^2*d) - (3*cot(c + d*x)*csc(c + d*x))/(4*a^2*d) + (cot(c + d*x)^3*csc(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 12, -((7*atanh(cos(c + d*x)))/(16*a^2*d)) + (2*cot(c + d*x)^5)/(5*a^2*d) + (5*cot(c + d*x)*csc(c + d*x))/(16*a^2*d) - (cot(c + d*x)^3*csc(c + d*x))/(4*a^2*d) + (cot(c + d*x)*csc(c + d*x)^3)/(8*a^2*d) - (cot(c + d*x)^3*csc(c + d*x)^3)/(6*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^8/(a + a*sin(c + d*x))^2, x, 19, atanh(cos(c + d*x))/(8*a^2*d) - (2*cot(c + d*x)^5)/(5*a^2*d) - cot(c + d*x)^7/(7*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(8*a^2*d) - (7*cot(c + d*x)*csc(c + d*x)^3)/(12*a^2*d) + (cot(c + d*x)*csc(c + d*x)^5)/(3*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^9/(a + a*sin(c + d*x))^2, x, 15, -((11*atanh(cos(c + d*x)))/(128*a^2*d)) + (2*cot(c + d*x)^5)/(5*a^2*d) + (2*cot(c + d*x)^7)/(7*a^2*d) - (11*cot(c + d*x)*csc(c + d*x))/(128*a^2*d) + (7*cot(c + d*x)*csc(c + d*x)^3)/(64*a^2*d) - (cot(c + d*x)^3*csc(c + d*x)^3)/(6*a^2*d) + (cot(c + d*x)*csc(c + d*x)^5)/(16*a^2*d) - (cot(c + d*x)^3*csc(c + d*x)^5)/(8*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^10/(a + a*sin(c + d*x))^2, x, 14, (3*atanh(cos(c + d*x)))/(64*a^2*d) - (2*cot(c + d*x)^5)/(5*a^2*d) - (3*cot(c + d*x)^7)/(7*a^2*d) - cot(c + d*x)^9/(9*a^2*d) + (3*cot(c + d*x)*csc(c + d*x))/(64*a^2*d) + (cot(c + d*x)*csc(c + d*x)^3)/(32*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5)/(8*a^2*d) + (cot(c + d*x)^3*csc(c + d*x)^5)/(4*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^11/(a + a*sin(c + d*x))^2, x, 17, -((9*atanh(cos(c + d*x)))/(256*a^2*d)) + (2*cot(c + d*x)^5)/(5*a^2*d) + (4*cot(c + d*x)^7)/(7*a^2*d) + (2*cot(c + d*x)^9)/(9*a^2*d) - (9*cot(c + d*x)*csc(c + d*x))/(256*a^2*d) - (3*cot(c + d*x)*csc(c + d*x)^3)/(128*a^2*d) + (9*cot(c + d*x)*csc(c + d*x)^5)/(160*a^2*d) - (cot(c + d*x)^3*csc(c + d*x)^5)/(8*a^2*d) + (3*cot(c + d*x)*csc(c + d*x)^7)/(80*a^2*d) - (cot(c + d*x)^3*csc(c + d*x)^7)/(10*a^2*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^12/(a + a*sin(c + d*x))^2, x, 15, (3*atanh(cos(c + d*x)))/(128*a^2*d) - (2*cot(c + d*x)^5)/(5*a^2*d) - (5*cot(c + d*x)^7)/(7*a^2*d) - (4*cot(c + d*x)^9)/(9*a^2*d) - cot(c + d*x)^11/(11*a^2*d) + (3*cot(c + d*x)*csc(c + d*x))/(128*a^2*d) + (cot(c + d*x)*csc(c + d*x)^3)/(64*a^2*d) + (cot(c + d*x)*csc(c + d*x)^5)/(80*a^2*d) - (3*cot(c + d*x)*csc(c + d*x)^7)/(40*a^2*d) + (cot(c + d*x)^3*csc(c + d*x)^7)/(5*a^2*d)]

    @test_int [cos(c + d*x)^8*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 18, -((29*x)/(128*a^3)) - (4*cos(c + d*x)^3)/(3*a^3*d) + (7*cos(c + d*x)^5)/(5*a^3*d) - (3*cos(c + d*x)^7)/(7*a^3*d) - (29*cos(c + d*x)*sin(c + d*x))/(128*a^3*d) + (29*cos(c + d*x)^3*sin(c + d*x))/(64*a^3*d) + (29*cos(c + d*x)^3*sin(c + d*x)^3)/(48*a^3*d) + (cos(c + d*x)^3*sin(c + d*x)^5)/(8*a^3*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 16, (5*x)/(16*a^3) + (4*cos(c + d*x)^3)/(3*a^3*d) - cos(c + d*x)^5/(a^3*d) + cos(c + d*x)^7/(7*a^3*d) + (5*cos(c + d*x)*sin(c + d*x))/(16*a^3*d) - (5*cos(c + d*x)^3*sin(c + d*x))/(8*a^3*d) - (cos(c + d*x)^3*sin(c + d*x)^3)/(2*a^3*d)]
    @test_int [cos(c + d*x)^8*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 6, -((7*x)/(16*a^3)) - (7*cos(c + d*x)^5)/(30*a^3*d) - (7*cos(c + d*x)*sin(c + d*x))/(16*a^3*d) - (7*cos(c + d*x)^3*sin(c + d*x))/(24*a^3*d) - cos(c + d*x)^9/(3*d*(a + a*sin(c + d*x))^3) - cos(c + d*x)^7/(6*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 13, -((13*x)/(8*a^3)) - atanh(cos(c + d*x))/(a^3*d) + cos(c + d*x)/(a^3*d) - cos(c + d*x)^3/(a^3*d) - (13*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 11, x/(2*a^3) + (3*atanh(cos(c + d*x)))/(a^3*d) - (3*cos(c + d*x))/(a^3*d) + cos(c + d*x)^3/(3*a^3*d) - cot(c + d*x)/(a^3*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 11, (5*x)/(2*a^3) - (5*atanh(cos(c + d*x)))/(2*a^3*d) + (3*cos(c + d*x))/(a^3*d) + (3*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d) - (cos(c + d*x)*sin(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 11, -((3*x)/a^3) - atanh(cos(c + d*x))/(2*a^3*d) - cos(c + d*x)/(a^3*d) - (3*cot(c + d*x))/(a^3*d) - cot(c + d*x)^3/(3*a^3*d) + (3*cot(c + d*x)*csc(c + d*x))/(2*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 12, x/a^3 + (13*atanh(cos(c + d*x)))/(8*a^3*d) + cot(c + d*x)/(a^3*d) + cot(c + d*x)^3/(a^3*d) - (11*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 13, -((7*atanh(cos(c + d*x)))/(8*a^3*d)) - (4*cot(c + d*x)^3)/(3*a^3*d) - cot(c + d*x)^5/(5*a^3*d) + (cot(c + d*x)*csc(c + d*x))/(8*a^3*d) + (3*cot(c + d*x)*csc(c + d*x)^3)/(4*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^7/(a + a*sin(c + d*x))^3, x, 15, (7*atanh(cos(c + d*x)))/(16*a^3*d) + (4*cot(c + d*x)^3)/(3*a^3*d) + (3*cot(c + d*x)^5)/(5*a^3*d) + (7*cot(c + d*x)*csc(c + d*x))/(16*a^3*d) - (17*cot(c + d*x)*csc(c + d*x)^3)/(24*a^3*d) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^8/(a + a*sin(c + d*x))^3, x, 17, -((5*atanh(cos(c + d*x)))/(16*a^3*d)) - (4*cot(c + d*x)^3)/(3*a^3*d) - cot(c + d*x)^5/(a^3*d) - cot(c + d*x)^7/(7*a^3*d) - (5*cot(c + d*x)*csc(c + d*x))/(16*a^3*d) + (cot(c + d*x)*csc(c + d*x)^3)/(8*a^3*d) + (cot(c + d*x)*csc(c + d*x)^5)/(2*a^3*d)]
    @test_int [cos(c + d*x)^8*csc(c + d*x)^9/(a + a*sin(c + d*x))^3, x, 18, (29*atanh(cos(c + d*x)))/(128*a^3*d) + (4*cot(c + d*x)^3)/(3*a^3*d) + (7*cot(c + d*x)^5)/(5*a^3*d) + (3*cot(c + d*x)^7)/(7*a^3*d) + (29*cot(c + d*x)*csc(c + d*x))/(128*a^3*d) + (29*cot(c + d*x)*csc(c + d*x)^3)/(192*a^3*d) - (23*cot(c + d*x)*csc(c + d*x)^5)/(48*a^3*d) - (cot(c + d*x)*csc(c + d*x)^7)/(8*a^3*d)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Section:: =#
    #=Integrands*of*the*form*sec(e+f*x)^1*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^2*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 8, -((3*a*x)/2) + (2*a*cos(c + d*x))/d - (a*cos(c + d*x)^3)/(3*d) + (a*sec(c + d*x))/d + (3*a*tan(c + d*x))/(2*d) - (a*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 8, -((3*a*x)/2) + (a*cos(c + d*x))/d + (a*sec(c + d*x))/d + (3*a*tan(c + d*x))/(2*d) - (a*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 5, (-a)*x + (a*cos(c + d*x))/d + (a*cos(c + d*x))/(d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 5, (-a)*x + (a*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 6, -((a*atanh(cos(c + d*x)))/d) + (a*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 7, -((a*atanh(cos(c + d*x)))/d) - (a*cot(c + d*x))/d + (a*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 8, -((3*a*atanh(cos(c + d*x)))/(2*d)) - (a*cot(c + d*x))/d + (3*a*sec(c + d*x))/(2*d) - (a*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 8, -((3*a*atanh(cos(c + d*x)))/(2*d)) - (2*a*cot(c + d*x))/d - (a*cot(c + d*x)^3)/(3*d) + (3*a*sec(c + d*x))/(2*d) - (a*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (a*tan(c + d*x))/d]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 12, -3*a^2*x + (3*a^2*cos(c + d*x))/d - (a^2*cos(c + d*x)^3)/(3*d) + (2*a^2*sec(c + d*x))/d + (3*a^2*tan(c + d*x))/d - (a^2*sin(c + d*x)^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 6, -((5*a^2*x)/2) + (2*a^2*cos(c + d*x))/d + (2*a^2*cos(c + d*x))/(d*(1 - sin(c + d*x))) + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 3, -2*a^2*x + (2*a^2*cos(c + d*x))/d + (sec(c + d*x)*(a + a*sin(c + d*x))^2)/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 9, -((a^2*atanh(cos(c + d*x)))/d) + (2*a^2*sec(c + d*x))/d + (2*a^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 10, -((2*a^2*atanh(cos(c + d*x)))/d) - (a^2*cot(c + d*x))/d + (2*a^2*sec(c + d*x))/d + (2*a^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 12, -((5*a^2*atanh(cos(c + d*x)))/(2*d)) - (2*a^2*cot(c + d*x))/d + (5*a^2*sec(c + d*x))/(2*d) - (a^2*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (2*a^2*tan(c + d*x))/d]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 11, -((51*a^3*x)/8) + (7*a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/d + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x))) + (19*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 8, -((11*a^3*x)/2) + (5*a^3*cos(c + d*x))/d - (a^3*cos(c + d*x)^3)/(3*d) + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x))) + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 2, -((9*a^3*x)/2) + (6*a^3*cos(c + d*x))/d + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d) + (sec(c + d*x)*(a + a*sin(c + d*x))^3)/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, (-a^3)*x - (a^3*atanh(cos(c + d*x)))/d + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 6, -((3*a^3*atanh(cos(c + d*x)))/d) - (a^3*cot(c + d*x))/d + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 8, -((9*a^3*atanh(cos(c + d*x)))/(2*d)) - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)*csc(c + d*x))/(2*d) + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 10, -((11*a^3*atanh(cos(c + d*x)))/(2*d)) - (5*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)*csc(c + d*x))/(2*d) + (4*a^3*cos(c + d*x))/(d*(1 - sin(c + d*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 7, x/a + cos(c + d*x)/(a*d) + (2*sec(c + d*x))/(a*d) - sec(c + d*x)^3/(3*a*d) - tan(c + d*x)/(a*d) + tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 6, -(x/a) - sec(c + d*x)/(a*d) + sec(c + d*x)^3/(3*a*d) + tan(c + d*x)/(a*d) - tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, sec(c + d*x)/(a*d) - sec(c + d*x)^3/(3*a*d) + tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 5, sec(c + d*x)^3/(3*a*d) - tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 7, -(atanh(cos(c + d*x))/(a*d)) + sec(c + d*x)/(a*d) + sec(c + d*x)^3/(3*a*d) - tan(c + d*x)/(a*d) - tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 8, atanh(cos(c + d*x))/(a*d) - cot(c + d*x)/(a*d) - sec(c + d*x)/(a*d) - sec(c + d*x)^3/(3*a*d) + (2*tan(c + d*x))/(a*d) + tan(c + d*x)^3/(3*a*d)]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 15, -((9*x)/(2*a^2)) - (2*cos(c + d*x))/(a^2*d) - (6*sec(c + d*x))/(a^2*d) + (2*sec(c + d*x)^3)/(a^2*d) - (2*sec(c + d*x)^5)/(5*a^2*d) + (9*tan(c + d*x))/(2*a^2*d) - (3*tan(c + d*x)^3)/(2*a^2*d) + (9*tan(c + d*x)^5)/(10*a^2*d) - (sin(c + d*x)^2*tan(c + d*x)^5)/(2*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 13, (2*x)/a^2 + cos(c + d*x)/(a^2*d) + (4*sec(c + d*x))/(a^2*d) - (5*sec(c + d*x)^3)/(3*a^2*d) + (2*sec(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x))/(a^2*d) + (2*tan(c + d*x)^3)/(3*a^2*d) - (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 12, -(x/a^2) - (2*sec(c + d*x))/(a^2*d) + (4*sec(c + d*x)^3)/(3*a^2*d) - (2*sec(c + d*x)^5)/(5*a^2*d) + tan(c + d*x)/(a^2*d) - tan(c + d*x)^3/(3*a^2*d) + (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 11, sec(c + d*x)/(a^2*d) - sec(c + d*x)^3/(a^2*d) + (2*sec(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 10, (2*sec(c + d*x)^3)/(3*a^2*d) - (2*sec(c + d*x)^5)/(5*a^2*d) + tan(c + d*x)^3/(3*a^2*d) + (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, sec(c + d*x)/(5*d*(a + a*sin(c + d*x))^2) - (2*sec(c + d*x))/(15*d*(a^2 + a^2*sin(c + d*x))) + (4*tan(c + d*x))/(15*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 11, -(atanh(cos(c + d*x))/(a^2*d)) + sec(c + d*x)/(a^2*d) + sec(c + d*x)^3/(3*a^2*d) + (2*sec(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x))/(a^2*d) - (4*tan(c + d*x)^3)/(3*a^2*d) - (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 12, (2*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a^2*d) - (2*sec(c + d*x))/(a^2*d) - (2*sec(c + d*x)^3)/(3*a^2*d) - (2*sec(c + d*x)^5)/(5*a^2*d) + (4*tan(c + d*x))/(a^2*d) + (5*tan(c + d*x)^3)/(3*a^2*d) + (2*tan(c + d*x)^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 15, -((9*atanh(cos(c + d*x)))/(2*a^2*d)) + (2*cot(c + d*x))/(a^2*d) + (9*sec(c + d*x))/(2*a^2*d) + (3*sec(c + d*x)^3)/(2*a^2*d) + (9*sec(c + d*x)^5)/(10*a^2*d) - (csc(c + d*x)^2*sec(c + d*x)^5)/(2*a^2*d) - (6*tan(c + d*x))/(a^2*d) - (2*tan(c + d*x)^3)/(a^2*d) - (2*tan(c + d*x)^5)/(5*a^2*d)]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 16, (3*x)/a^3 + cos(c + d*x)/(a^3*d) + (7*sec(c + d*x))/(a^3*d) - (5*sec(c + d*x)^3)/(a^3*d) + (13*sec(c + d*x)^5)/(5*a^3*d) - (4*sec(c + d*x)^7)/(7*a^3*d) - (3*tan(c + d*x))/(a^3*d) + tan(c + d*x)^3/(a^3*d) - (3*tan(c + d*x)^5)/(5*a^3*d) + (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 16, -(x/a^3) - (3*sec(c + d*x))/(a^3*d) + (10*sec(c + d*x)^3)/(3*a^3*d) - (11*sec(c + d*x)^5)/(5*a^3*d) + (4*sec(c + d*x)^7)/(7*a^3*d) + tan(c + d*x)/(a^3*d) - tan(c + d*x)^3/(3*a^3*d) + tan(c + d*x)^5/(5*a^3*d) - (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 14, sec(c + d*x)/(a^3*d) - (2*sec(c + d*x)^3)/(a^3*d) + (9*sec(c + d*x)^5)/(5*a^3*d) - (4*sec(c + d*x)^7)/(7*a^3*d) + tan(c + d*x)^5/(5*a^3*d) + (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 14, sec(c + d*x)^3/(a^3*d) - (7*sec(c + d*x)^5)/(5*a^3*d) + (4*sec(c + d*x)^7)/(7*a^3*d) - (3*tan(c + d*x)^5)/(5*a^3*d) - (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 14, -(sec(c + d*x)^3/(3*a^3*d)) + sec(c + d*x)^5/(a^3*d) - (4*sec(c + d*x)^7)/(7*a^3*d) + tan(c + d*x)^3/(3*a^3*d) + tan(c + d*x)^5/(a^3*d) + (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 5, sec(c + d*x)/(7*d*(a + a*sin(c + d*x))^3) - (3*sec(c + d*x))/(35*a*d*(a + a*sin(c + d*x))^2) - (3*sec(c + d*x))/(35*d*(a^3 + a^3*sin(c + d*x))) + (6*tan(c + d*x))/(35*a^3*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 14, -(atanh(cos(c + d*x))/(a^3*d)) + sec(c + d*x)/(a^3*d) + sec(c + d*x)^3/(3*a^3*d) + sec(c + d*x)^5/(5*a^3*d) + (4*sec(c + d*x)^7)/(7*a^3*d) - (3*tan(c + d*x))/(a^3*d) - (10*tan(c + d*x)^3)/(3*a^3*d) - (11*tan(c + d*x)^5)/(5*a^3*d) - (4*tan(c + d*x)^7)/(7*a^3*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 14, (3*atanh(cos(c + d*x)))/(a^3*d) - cot(c + d*x)/(a^3*d) - (3*sec(c + d*x))/(a^3*d) - sec(c + d*x)^3/(a^3*d) - (3*sec(c + d*x)^5)/(5*a^3*d) - (4*sec(c + d*x)^7)/(7*a^3*d) + (7*tan(c + d*x))/(a^3*d) + (5*tan(c + d*x)^3)/(a^3*d) + (13*tan(c + d*x)^5)/(5*a^3*d) + (4*tan(c + d*x)^7)/(7*a^3*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    #= [sec(c + d*x)^2*sin(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 7, -((76*a*cos(c + d*x))/(495*d*sqrt(a + a*sin(c + d*x)))) - (38*a*cos(c + d*x)*sin(c + d*x)^3)/(693*d*sqrt(a + a*sin(c + d*x))) + (2*a*cos(c + d*x)*sin(c + d*x)^4)/(99*d*sqrt(a + a*sin(c + d*x))) + (152*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3465*d) + (2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(11*d) - (76*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(1155*a*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 6, -((2*a*cos(c + d*x))/(9*d*sqrt(a + a*sin(c + d*x)))) + (2*a*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) + (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(63*d) + (2*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(9*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(21*a*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 5, -((16*a*cos(c + d*x))/(105*d*sqrt(a + a*sin(c + d*x)))) - (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*d) - (18*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*a*d) + (2*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(7*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 4, -((2*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (2*a*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 4, -((sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (3*a*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 4, (5*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) - (a*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 5, (3*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) + (3*a*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(3*d)] =#

    #= [sec(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 8, -((1724*a^2*cos(c + d*x))/(6435*d*sqrt(a + a*sin(c + d*x)))) - (862*a^2*cos(c + d*x)*sin(c + d*x)^3)/(9009*d*sqrt(a + a*sin(c + d*x))) - (38*a^2*cos(c + d*x)*sin(c + d*x)^4)/(1287*d*sqrt(a + a*sin(c + d*x))) + (3448*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(45045*d) + (6*a*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(143*d) - (1724*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(15015*d) + (2*cos(c + d*x)*sin(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(13*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 7, -((62*a^2*cos(c + d*x))/(165*d*sqrt(a + a*sin(c + d*x)))) - (10*a^2*cos(c + d*x)*sin(c + d*x)^3)/(231*d*sqrt(a + a*sin(c + d*x))) + (124*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(1155*d) + (2*a*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(33*d) - (62*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(385*d) + (2*cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(11*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 6, -((128*a^2*cos(c + d*x))/(315*d*sqrt(a + a*sin(c + d*x)))) - (32*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*d) - (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*d) - (22*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(63*a*d) + (2*cos(c + d*x)*(a + a*sin(c + d*x))^(7/2))/(9*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 5, -((2*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (2*a^2*cos(c + d*x))/(5*d*sqrt(a + a*sin(c + d*x))) + (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*d) + (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 5, -((3*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) + (11*a^2*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) + (5*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d) - (cot(c + d*x)*(a + a*sin(c + d*x))^(3/2))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 5, (a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d) + (13*a^2*cos(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (3*a*cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (cot(c + d*x)*csc(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 5, (13*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d) + (5*a^2*cot(c + d*x))/(24*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(3*d)] =#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    #= [sec(c + d*x)^2*sin(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 6, -((4*cos(c + d*x))/(45*d*sqrt(a + a*sin(c + d*x)))) - (2*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sin(c + d*x)^4)/(9*d*sqrt(a + a*sin(c + d*x))) + (8*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*a*d) - (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 5, -((2*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x)))) + (2*cos(c + d*x)*sin(c + d*x)^3)/(7*d*sqrt(a + a*sin(c + d*x))) + (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*a*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 4, (4*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x))) - (14*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*a*d) + (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 3, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(sqrt(a)*d)) + (2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 3, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(sqrt(a)*d) - cot(c + d*x)/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 4, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(4*sqrt(a)*d) + cot(c + d*x)/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4/sqrt(a + a*sin(c + d*x)), x, 5, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(8*sqrt(a)*d) + cot(c + d*x)/(8*d*sqrt(a + a*sin(c + d*x))) + (cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*d*sqrt(a + a*sin(c + d*x)))] =#

    #= [sec(c + d*x)^2*sin(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 6, (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (344*cos(c + d*x))/(105*a*d*sqrt(a + a*sin(c + d*x))) - (16*cos(c + d*x)*sin(c + d*x)^2)/(35*a*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sin(c + d*x)^3)/(7*a*d*sqrt(a + a*sin(c + d*x))) + (76*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 5, -((2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d)) + (16*cos(c + d*x))/(5*a*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sin(c + d*x)^2)/(5*a*d*sqrt(a + a*sin(c + d*x))) - (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*a^2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 4, (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (10*cos(c + d*x))/(3*a*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*a^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 4, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d)) + (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 5, (3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d) - (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - cot(c + d*x)/(a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 6, -((11*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(3/2)*d)) + (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) + (5*cot(c + d*x))/(4*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 7, (23*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*a^(3/2)*d) - (2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) - (9*cot(c + d*x))/(8*a*d*sqrt(a + a*sin(c + d*x))) + (7*cot(c + d*x)*csc(c + d*x))/(12*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*sqrt(a + a*sin(c + d*x)))] =#

    #= ::Section:: =#
    #=Integrands*of*the*form*sec(e+f*x)^3*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^4*sin(c + d*x)^6*(a + a*sin(c + d*x)), x, 9, (5*a*x)/2 - (3*a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) - (3*a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) - (5*a*tan(c + d*x))/(2*d) + (5*a*tan(c + d*x)^3)/(6*d) - (a*sin(c + d*x)^2*tan(c + d*x)^3)/(2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^5*(a + a*sin(c + d*x)), x, 9, (5*a*x)/2 - (a*cos(c + d*x))/d - (2*a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) - (5*a*tan(c + d*x))/(2*d) + (5*a*tan(c + d*x)^3)/(6*d) - (a*sin(c + d*x)^2*tan(c + d*x)^3)/(2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 8, a*x - (a*cos(c + d*x))/d - (2*a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) - (a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 6, a*x - (a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) - (a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 5, -((a*sec(c + d*x))/d) + (a*sec(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 5, (a*sec(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 7, -((a*atanh(cos(c + d*x)))/d) + (a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) + (a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 8, -((a*atanh(cos(c + d*x)))/d) - (a*cot(c + d*x))/d + (a*sec(c + d*x))/d + (a*sec(c + d*x)^3)/(3*d) + (2*a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 9, -((5*a*atanh(cos(c + d*x)))/(2*d)) - (a*cot(c + d*x))/d + (5*a*sec(c + d*x))/(2*d) + (5*a*sec(c + d*x)^3)/(6*d) - (a*csc(c + d*x)^2*sec(c + d*x)^3)/(2*d) + (2*a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 4, (7*a^2*x)/2 - (2*a^2*cos(c + d*x))/d + (a^2*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) - (11*a^2*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) - (a^2*cos(c + d*x)*sin(c + d*x))/(2*d), (7*a^2*x)/2 - (16*a^2*cos(c + d*x))/(3*d) - (7*a^2*cos(c + d*x)*sin(c + d*x))/(2*d) - (8*a^2*cos(c + d*x)*sin(c + d*x)^2)/(3*d*(1 - sin(c + d*x))) + (a^4*cos(c + d*x)*sin(c + d*x)^3)/(3*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 7, 2*a^2*x - (4*a^2*cos(c + d*x))/(3*d) - (2*a^2*cos(c + d*x))/(d*(1 - sin(c + d*x))) + (a^4*cos(c + d*x)*sin(c + d*x)^2)/(3*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, a^2*x - (5*a^2*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) + (a^4*cos(c + d*x))/(3*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 4, -((2*a^2*sec(c + d*x))/(3*d)) + (sec(c + d*x)^3*(a + a*sin(c + d*x))^2)/(3*d) - (2*a^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 5, -((a^2*atanh(cos(c + d*x)))/d) + (4*a^2*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) + (a^4*cos(c + d*x))/(3*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 7, -((2*a^2*atanh(cos(c + d*x)))/d) - (10*a^2*cot(c + d*x))/(3*d) + (2*a^2*cot(c + d*x))/(d*(1 - sin(c + d*x))) + (a^4*cot(c + d*x))/(3*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 8, -((7*a^2*atanh(cos(c + d*x)))/(2*d)) - (16*a^2*cot(c + d*x))/(3*d) - (7*a^2*cot(c + d*x)*csc(c + d*x))/(2*d) + (8*a^2*cot(c + d*x)*csc(c + d*x))/(3*d*(1 - sin(c + d*x))) + (a^4*cot(c + d*x)*csc(c + d*x))/(3*d*(a - a*sin(c + d*x))^2)]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 10, (17*a^3*x)/2 - (6*a^3*cos(c + d*x))/d + (a^3*cos(c + d*x)^3)/(3*d) + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) - (25*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) - (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 8, (11*a^3*x)/2 - (3*a^3*cos(c + d*x))/d + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) - (19*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) - (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 5, 3*a^3*x - (3*a^3*cos(c + d*x))/d - (2*a^5*cos(c + d*x)^3)/(d*(a - a*sin(c + d*x))^2) + (sec(c + d*x)^3*(a + a*sin(c + d*x))^3)/(3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, a^3*x + (sec(c + d*x)^3*(a + a*sin(c + d*x))^3)/(3*d) - (2*a^5*cos(c + d*x))/(d*(a^2 - a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 6, -((a^3*atanh(cos(c + d*x)))/d) + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) + (5*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 8, -((3*a^3*atanh(cos(c + d*x)))/d) - (a^3*cot(c + d*x))/d + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) + (11*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 10, -((11*a^3*atanh(cos(c + d*x)))/(2*d)) - (3*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)*csc(c + d*x))/(2*d) + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) + (17*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 12, -((17*a^3*atanh(cos(c + d*x)))/(2*d)) - (6*a^3*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (3*a^3*cot(c + d*x)*csc(c + d*x))/(2*d) + (2*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) + (23*a^3*cos(c + d*x))/(3*d*(1 - sin(c + d*x)))]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^4, x, 13, (163*a^4*x)/8 - (16*a^4*cos(c + d*x))/d + (4*a^4*cos(c + d*x)^3)/(3*d) + (4*a^4*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) - (56*a^4*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) - (35*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) - (a^4*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^4, x, 8, (17*a^4*x)/2 - (4*a^4*cos(c + d*x))/d + (4*a^4*cos(c + d*x))/(3*d*(1 - sin(c + d*x))^2) - (32*a^4*cos(c + d*x))/(3*d*(1 - sin(c + d*x))) - (a^4*cos(c + d*x)*sin(c + d*x))/(2*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(c + d*x)^4*sin(c + d*x)^6/(a + a*sin(c + d*x)), x, 8, -(x/a) - cos(c + d*x)/(a*d) - (3*sec(c + d*x))/(a*d) + sec(c + d*x)^3/(a*d) - sec(c + d*x)^5/(5*a*d) + tan(c + d*x)/(a*d) - tan(c + d*x)^3/(3*a*d) + tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^5/(a + a*sin(c + d*x)), x, 8, x/a + sec(c + d*x)/(a*d) - (2*sec(c + d*x)^3)/(3*a*d) + sec(c + d*x)^5/(5*a*d) - tan(c + d*x)/(a*d) + tan(c + d*x)^3/(3*a*d) - tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 6, -(sec(c + d*x)/(a*d)) + (2*sec(c + d*x)^3)/(3*a*d) - sec(c + d*x)^5/(5*a*d) + tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 6, -(sec(c + d*x)^3/(3*a*d)) + sec(c + d*x)^5/(5*a*d) - tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 7, sec(c + d*x)^3/(3*a*d) - sec(c + d*x)^5/(5*a*d) + tan(c + d*x)^3/(3*a*d) + tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 6, sec(c + d*x)^5/(5*a*d) - tan(c + d*x)^3/(3*a*d) - tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 7, -(atanh(cos(c + d*x))/(a*d)) + sec(c + d*x)/(a*d) + sec(c + d*x)^3/(3*a*d) + sec(c + d*x)^5/(5*a*d) - tan(c + d*x)/(a*d) - (2*tan(c + d*x)^3)/(3*a*d) - tan(c + d*x)^5/(5*a*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 8, atanh(cos(c + d*x))/(a*d) - cot(c + d*x)/(a*d) - sec(c + d*x)/(a*d) - sec(c + d*x)^3/(3*a*d) - sec(c + d*x)^5/(5*a*d) + (3*tan(c + d*x))/(a*d) + tan(c + d*x)^3/(a*d) + tan(c + d*x)^5/(5*a*d)]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 14, -((2*x)/a^2) - cos(c + d*x)/(a^2*d) - (5*sec(c + d*x))/(a^2*d) + (3*sec(c + d*x)^3)/(a^2*d) - (7*sec(c + d*x)^5)/(5*a^2*d) + (2*sec(c + d*x)^7)/(7*a^2*d) + (2*tan(c + d*x))/(a^2*d) - (2*tan(c + d*x)^3)/(3*a^2*d) + (2*tan(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 13, x/a^2 + (2*sec(c + d*x))/(a^2*d) - (2*sec(c + d*x)^3)/(a^2*d) + (6*sec(c + d*x)^5)/(5*a^2*d) - (2*sec(c + d*x)^7)/(7*a^2*d) - tan(c + d*x)/(a^2*d) + tan(c + d*x)^3/(3*a^2*d) - tan(c + d*x)^5/(5*a^2*d) + (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 11, -(sec(c + d*x)/(a^2*d)) + (4*sec(c + d*x)^3)/(3*a^2*d) - sec(c + d*x)^5/(a^2*d) + (2*sec(c + d*x)^7)/(7*a^2*d) - (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 10, -((2*sec(c + d*x)^3)/(3*a^2*d)) + (4*sec(c + d*x)^5)/(5*a^2*d) - (2*sec(c + d*x)^7)/(7*a^2*d) + tan(c + d*x)^5/(5*a^2*d) + (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 12, sec(c + d*x)^3/(3*a^2*d) - (3*sec(c + d*x)^5)/(5*a^2*d) + (2*sec(c + d*x)^7)/(7*a^2*d) - (2*tan(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 12, (2*sec(c + d*x)^5)/(5*a^2*d) - (2*sec(c + d*x)^7)/(7*a^2*d) + tan(c + d*x)^3/(3*a^2*d) + (3*tan(c + d*x)^5)/(5*a^2*d) + (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, sec(c + d*x)^3/(7*d*(a + a*sin(c + d*x))^2) - (2*sec(c + d*x)^3)/(35*d*(a^2 + a^2*sin(c + d*x))) + (8*tan(c + d*x))/(35*a^2*d) + (8*tan(c + d*x)^3)/(105*a^2*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 11, -(atanh(cos(c + d*x))/(a^2*d)) + sec(c + d*x)/(a^2*d) + sec(c + d*x)^3/(3*a^2*d) + sec(c + d*x)^5/(5*a^2*d) + (2*sec(c + d*x)^7)/(7*a^2*d) - (2*tan(c + d*x))/(a^2*d) - (2*tan(c + d*x)^3)/(a^2*d) - (6*tan(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 12, (2*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a^2*d) - (2*sec(c + d*x))/(a^2*d) - (2*sec(c + d*x)^3)/(3*a^2*d) - (2*sec(c + d*x)^5)/(5*a^2*d) - (2*sec(c + d*x)^7)/(7*a^2*d) + (5*tan(c + d*x))/(a^2*d) + (3*tan(c + d*x)^3)/(a^2*d) + (7*tan(c + d*x)^5)/(5*a^2*d) + (2*tan(c + d*x)^7)/(7*a^2*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 15, -((11*atanh(cos(c + d*x)))/(2*a^2*d)) + (2*cot(c + d*x))/(a^2*d) + (11*sec(c + d*x))/(2*a^2*d) + (11*sec(c + d*x)^3)/(6*a^2*d) + (11*sec(c + d*x)^5)/(10*a^2*d) + (11*sec(c + d*x)^7)/(14*a^2*d) - (csc(c + d*x)^2*sec(c + d*x)^7)/(2*a^2*d) - (8*tan(c + d*x))/(a^2*d) - (4*tan(c + d*x)^3)/(a^2*d) - (8*tan(c + d*x)^5)/(5*a^2*d) - (2*tan(c + d*x)^7)/(7*a^2*d)]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^7/(a + a*sin(c + d*x))^3, x, 17, x/a^3 + (3*sec(c + d*x))/(a^3*d) - (13*sec(c + d*x)^3)/(3*a^3*d) + (21*sec(c + d*x)^5)/(5*a^3*d) - (15*sec(c + d*x)^7)/(7*a^3*d) + (4*sec(c + d*x)^9)/(9*a^3*d) - tan(c + d*x)/(a^3*d) + tan(c + d*x)^3/(3*a^3*d) - tan(c + d*x)^5/(5*a^3*d) + tan(c + d*x)^7/(7*a^3*d) - (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 14, -(sec(c + d*x)/(a^3*d)) + (7*sec(c + d*x)^3)/(3*a^3*d) - (3*sec(c + d*x)^5)/(a^3*d) + (13*sec(c + d*x)^7)/(7*a^3*d) - (4*sec(c + d*x)^9)/(9*a^3*d) + tan(c + d*x)^7/(7*a^3*d) + (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 14, -(sec(c + d*x)^3/(a^3*d)) + (2*sec(c + d*x)^5)/(a^3*d) - (11*sec(c + d*x)^7)/(7*a^3*d) + (4*sec(c + d*x)^9)/(9*a^3*d) - (3*tan(c + d*x)^7)/(7*a^3*d) - (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 14, sec(c + d*x)^3/(3*a^3*d) - (6*sec(c + d*x)^5)/(5*a^3*d) + (9*sec(c + d*x)^7)/(7*a^3*d) - (4*sec(c + d*x)^9)/(9*a^3*d) + tan(c + d*x)^5/(5*a^3*d) + (5*tan(c + d*x)^7)/(7*a^3*d) + (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 15, (3*sec(c + d*x)^5)/(5*a^3*d) - sec(c + d*x)^7/(a^3*d) + (4*sec(c + d*x)^9)/(9*a^3*d) - (3*tan(c + d*x)^5)/(5*a^3*d) - tan(c + d*x)^7/(a^3*d) - (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 15, -(sec(c + d*x)^5/(5*a^3*d)) + (5*sec(c + d*x)^7)/(7*a^3*d) - (4*sec(c + d*x)^9)/(9*a^3*d) + tan(c + d*x)^3/(3*a^3*d) + (6*tan(c + d*x)^5)/(5*a^3*d) + (9*tan(c + d*x)^7)/(7*a^3*d) + (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 5, sec(c + d*x)^3/(9*d*(a + a*sin(c + d*x))^3) - sec(c + d*x)^3/(21*a*d*(a + a*sin(c + d*x))^2) - sec(c + d*x)^3/(21*d*(a^3 + a^3*sin(c + d*x))) + (4*tan(c + d*x))/(21*a^3*d) + (4*tan(c + d*x)^3)/(63*a^3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 14, -(atanh(cos(c + d*x))/(a^3*d)) + sec(c + d*x)/(a^3*d) + sec(c + d*x)^3/(3*a^3*d) + sec(c + d*x)^5/(5*a^3*d) + sec(c + d*x)^7/(7*a^3*d) + (4*sec(c + d*x)^9)/(9*a^3*d) - (3*tan(c + d*x))/(a^3*d) - (13*tan(c + d*x)^3)/(3*a^3*d) - (21*tan(c + d*x)^5)/(5*a^3*d) - (15*tan(c + d*x)^7)/(7*a^3*d) - (4*tan(c + d*x)^9)/(9*a^3*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 14, (3*atanh(cos(c + d*x)))/(a^3*d) - cot(c + d*x)/(a^3*d) - (3*sec(c + d*x))/(a^3*d) - sec(c + d*x)^3/(a^3*d) - (3*sec(c + d*x)^5)/(5*a^3*d) - (3*sec(c + d*x)^7)/(7*a^3*d) - (4*sec(c + d*x)^9)/(9*a^3*d) + (8*tan(c + d*x))/(a^3*d) + (22*tan(c + d*x)^3)/(3*a^3*d) + (28*tan(c + d*x)^5)/(5*a^3*d) + (17*tan(c + d*x)^7)/(7*a^3*d) + (4*tan(c + d*x)^9)/(9*a^3*d)]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^4/(a + a*sin(c + d*x))^4, x, 17, (4*sec(c + d*x)^5)/(5*a^4*d) - (16*sec(c + d*x)^7)/(7*a^4*d) + (20*sec(c + d*x)^9)/(9*a^4*d) - (8*sec(c + d*x)^11)/(11*a^4*d) + tan(c + d*x)^5/(5*a^4*d) + (9*tan(c + d*x)^7)/(7*a^4*d) + (16*tan(c + d*x)^9)/(9*a^4*d) + (8*tan(c + d*x)^11)/(11*a^4*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3/(a + a*sin(c + d*x))^4, x, 18, -(sec(c + d*x)^5/(5*a^4*d)) + (9*sec(c + d*x)^7)/(7*a^4*d) - (16*sec(c + d*x)^9)/(9*a^4*d) + (8*sec(c + d*x)^11)/(11*a^4*d) - (4*tan(c + d*x)^5)/(5*a^4*d) - (16*tan(c + d*x)^7)/(7*a^4*d) - (20*tan(c + d*x)^9)/(9*a^4*d) - (8*tan(c + d*x)^11)/(11*a^4*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2/(a + a*sin(c + d*x))^4, x, 8, -((4*sec(c + d*x)^7)/(7*a^4*d)) + (4*sec(c + d*x)^9)/(3*a^4*d) - (8*sec(c + d*x)^11)/(11*a^4*d) + tan(c + d*x)^3/(3*a^4*d) + (2*tan(c + d*x)^5)/(a^4*d) + (25*tan(c + d*x)^7)/(7*a^4*d) + (8*tan(c + d*x)^9)/(3*a^4*d) + (8*tan(c + d*x)^11)/(11*a^4*d), -((a*sec(c + d*x))/(22*d*(a + a*sin(c + d*x))^5)) - sec(c + d*x)/(33*d*(a + a*sin(c + d*x))^4) - (5*sec(c + d*x))/(231*a*d*(a + a*sin(c + d*x))^3) + sec(c + d*x)^3/(6*a*d*(a + a*sin(c + d*x))^3) - (4*sec(c + d*x))/(231*d*(a^2 + a^2*sin(c + d*x))^2) - (4*sec(c + d*x))/(231*d*(a^4 + a^4*sin(c + d*x))) + (8*tan(c + d*x))/(231*a^4*d)]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^5*sin(c + d*x)^6*(a + a*sin(c + d*x)), x, 4, -((39*a*log(1 - sin(c + d*x)))/(16*d)) - (9*a*log(1 + sin(c + d*x)))/(16*d) - (a*sin(c + d*x))/d - (a*sin(c + d*x)^2)/(2*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) - (5*a^2)/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + a*sin(c + d*x)), x, 3, -((23*a*log(1 - sin(c + d*x)))/(16*d)) + (7*a*log(1 + sin(c + d*x)))/(16*d) - (a*sin(c + d*x))/d + a^3/(8*d*(a - a*sin(c + d*x))^2) - a^2/(d*(a - a*sin(c + d*x))) + a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, -((11*a*log(1 - sin(c + d*x)))/(16*d)) - (5*a*log(1 + sin(c + d*x)))/(16*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) - (3*a^2)/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + a*sin(c + d*x)), x, 5, (3*a*atanh(sin(c + d*x)))/(8*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) - a^2/(2*d*(a - a*sin(c + d*x))) + a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x)), x, 5, -((a*atanh(sin(c + d*x)))/(8*d)) + a^3/(8*d*(a - a*sin(c + d*x))^2) - a^2/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x)), x, 5, -((a*atanh(sin(c + d*x)))/(8*d)) + a^3/(8*d*(a - a*sin(c + d*x))^2) + a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x)), x, 4, -((11*a*log(1 - sin(c + d*x)))/(16*d)) + (a*log(sin(c + d*x)))/d - (5*a*log(1 + sin(c + d*x)))/(16*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) + a^2/(2*d*(a - a*sin(c + d*x))) + a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) - (23*a*log(1 - sin(c + d*x)))/(16*d) + (a*log(sin(c + d*x)))/d + (7*a*log(1 + sin(c + d*x)))/(16*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) + (3*a^2)/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (39*a*log(1 - sin(c + d*x)))/(16*d) + (3*a*log(sin(c + d*x)))/d - (9*a*log(1 + sin(c + d*x)))/(16*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) + a^2/(d*(a - a*sin(c + d*x))) + a^2/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, -((3*a*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/(3*d) - (59*a*log(1 - sin(c + d*x)))/(16*d) + (3*a*log(sin(c + d*x)))/d + (11*a*log(1 + sin(c + d*x)))/(16*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) + (5*a^2)/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]

    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 3, -((31*a^2*log(1 - sin(c + d*x)))/(8*d)) - (a^2*log(1 + sin(c + d*x)))/(8*d) - (2*a^2*sin(c + d*x))/d - (a^2*sin(c + d*x)^2)/(2*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) - (9*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 4, -((17*a^2*log(1 - sin(c + d*x)))/(8*d)) + (a^2*log(1 + sin(c + d*x)))/(8*d) - (a^2*sin(c + d*x))/d + a^4/(4*d*(a - a*sin(c + d*x))^2) - (7*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 4, -((7*a^2*log(1 - sin(c + d*x)))/(8*d)) - (a^2*log(1 + sin(c + d*x)))/(8*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) - (5*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 5, (a^2*atanh(sin(c + d*x)))/(4*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) - (3*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 5, -((a^2*atanh(sin(c + d*x)))/(4*d)) + a^4/(4*d*(a - a*sin(c + d*x))^2) - a^3/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 4, -((7*a^2*log(1 - sin(c + d*x)))/(8*d)) + (a^2*log(sin(c + d*x)))/d - (a^2*log(1 + sin(c + d*x)))/(8*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) + (3*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x))/d) - (17*a^2*log(1 - sin(c + d*x)))/(8*d) + (2*a^2*log(sin(c + d*x)))/d + (a^2*log(1 + sin(c + d*x)))/(8*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) + (5*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 4, -((2*a^2*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/(2*d) - (31*a^2*log(1 - sin(c + d*x)))/(8*d) + (4*a^2*log(sin(c + d*x)))/d - (a^2*log(1 + sin(c + d*x)))/(8*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) + (7*a^3)/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 4, -((4*a^2*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/d - (a^2*csc(c + d*x)^3)/(3*d) - (49*a^2*log(1 - sin(c + d*x)))/(8*d) + (6*a^2*log(sin(c + d*x)))/d + (a^2*log(1 + sin(c + d*x)))/(8*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) + (9*a^3)/(4*d*(a - a*sin(c + d*x)))]

    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 3, -((10*a^3*log(1 - sin(c + d*x)))/d) - (6*a^3*sin(c + d*x))/d - (3*a^3*sin(c + d*x)^2)/(2*d) - (a^3*sin(c + d*x)^3)/(3*d) + a^5/(2*d*(a - a*sin(c + d*x))^2) - (5*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 4, -((6*a^3*log(1 - sin(c + d*x)))/d) - (3*a^3*sin(c + d*x))/d - (a^3*sin(c + d*x)^2)/(2*d) + a^5/(2*d*(a - a*sin(c + d*x))^2) - (4*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 4, -((3*a^3*log(1 - sin(c + d*x)))/d) - (a^3*sin(c + d*x))/d + a^5/(2*d*(a - a*sin(c + d*x))^2) - (3*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, -((a^3*log(1 - sin(c + d*x)))/d) + a^5/(2*d*(a - a*sin(c + d*x))^2) - (2*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 3, (a^5*sin(c + d*x)^2)/(2*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 4, -((a^3*log(1 - sin(c + d*x)))/d) + (a^3*log(sin(c + d*x)))/d + a^5/(2*d*(a - a*sin(c + d*x))^2) + a^4/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, -((a^3*csc(c + d*x))/d) - (3*a^3*log(1 - sin(c + d*x)))/d + (3*a^3*log(sin(c + d*x)))/d + a^5/(2*d*(a - a*sin(c + d*x))^2) + (2*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 4, -((3*a^3*csc(c + d*x))/d) - (a^3*csc(c + d*x)^2)/(2*d) - (6*a^3*log(1 - sin(c + d*x)))/d + (6*a^3*log(sin(c + d*x)))/d + a^5/(2*d*(a - a*sin(c + d*x))^2) + (3*a^4)/(d*(a - a*sin(c + d*x)))]

    #= ::Subsubsection:: =#
    #=m<0=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Section:: =#
    #=Integrands*of*the*form*sec(e+f*x)^6*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^7*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^7*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(c + d*x)^7*sin(c + d*x)^11/(a + a*sin(c + d*x)), x, 4, (515*log(1 - sin(c + d*x)))/(256*a*d) - (1795*log(1 + sin(c + d*x)))/(256*a*d) + (5*sin(c + d*x))/(a*d) - sin(c + d*x)^2/(2*a*d) + sin(c + d*x)^3/(3*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) - (17*a)/(128*d*(a - a*sin(c + d*x))^2) + 125/(128*d*(a - a*sin(c + d*x))) + a^3/(64*d*(a + a*sin(c + d*x))^4) - (3*a^2)/(16*d*(a + a*sin(c + d*x))^3) + (71*a)/(64*d*(a + a*sin(c + d*x))^2) - 5/(d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^10/(a + a*sin(c + d*x)), x, 4, (325*log(1 - sin(c + d*x)))/(256*a*d) + (955*log(1 + sin(c + d*x)))/(256*a*d) - sin(c + d*x)/(a*d) + sin(c + d*x)^2/(2*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) - (15*a)/(128*d*(a - a*sin(c + d*x))^2) + 95/(128*d*(a - a*sin(c + d*x))) - a^3/(64*d*(a + a*sin(c + d*x))^4) + a^2/(6*d*(a + a*sin(c + d*x))^3) - (55*a)/(64*d*(a + a*sin(c + d*x))^2) + 105/(32*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^9/(a + a*sin(c + d*x)), x, 4, (187*log(1 - sin(c + d*x)))/(256*a*d) - (443*log(1 + sin(c + d*x)))/(256*a*d) + sin(c + d*x)/(a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) - (13*a)/(128*d*(a - a*sin(c + d*x))^2) + 69/(128*d*(a - a*sin(c + d*x))) + a^3/(64*d*(a + a*sin(c + d*x))^4) - (7*a^2)/(48*d*(a + a*sin(c + d*x))^3) + (41*a)/(64*d*(a + a*sin(c + d*x))^2) - 2/(d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^8/(a + a*sin(c + d*x)), x, 4, (93*log(1 - sin(c + d*x)))/(256*a*d) + (163*log(1 + sin(c + d*x)))/(256*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) - (11*a)/(128*d*(a - a*sin(c + d*x))^2) + 47/(128*d*(a - a*sin(c + d*x))) - a^3/(64*d*(a + a*sin(c + d*x))^4) + a^2/(8*d*(a + a*sin(c + d*x))^3) - (29*a)/(64*d*(a + a*sin(c + d*x))^2) + 35/(32*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^7/(a + a*sin(c + d*x)), x, 8, -((35*atanh(sin(c + d*x)))/(128*a*d)) + (35*sec(c + d*x)*tan(c + d*x))/(128*a*d) - (35*sec(c + d*x)*tan(c + d*x)^3)/(192*a*d) + (7*sec(c + d*x)*tan(c + d*x)^5)/(48*a*d) - (sec(c + d*x)*tan(c + d*x)^7)/(8*a*d) + tan(c + d*x)^8/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^6/(a + a*sin(c + d*x)), x, 8, -((5*atanh(sin(c + d*x)))/(128*a*d)) - (5*sec(c + d*x)*tan(c + d*x))/(128*a*d) + (5*sec(c + d*x)^3*tan(c + d*x))/(64*a*d) - (5*sec(c + d*x)^3*tan(c + d*x)^3)/(48*a*d) + (sec(c + d*x)^3*tan(c + d*x)^5)/(8*a*d) - tan(c + d*x)^8/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^5/(a + a*sin(c + d*x)), x, 9, (5*atanh(sin(c + d*x)))/(128*a*d) + (5*sec(c + d*x)*tan(c + d*x))/(128*a*d) - (5*sec(c + d*x)^3*tan(c + d*x))/(64*a*d) + (5*sec(c + d*x)^3*tan(c + d*x)^3)/(48*a*d) - (sec(c + d*x)^3*tan(c + d*x)^5)/(8*a*d) + tan(c + d*x)^6/(6*a*d) + tan(c + d*x)^8/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 9, (3*atanh(sin(c + d*x)))/(128*a*d) + (3*sec(c + d*x)*tan(c + d*x))/(128*a*d) + (sec(c + d*x)^3*tan(c + d*x))/(64*a*d) - (sec(c + d*x)^5*tan(c + d*x))/(16*a*d) + (sec(c + d*x)^5*tan(c + d*x)^3)/(8*a*d) - tan(c + d*x)^6/(6*a*d) - tan(c + d*x)^8/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 9, -((3*atanh(sin(c + d*x)))/(128*a*d)) - sec(c + d*x)^6/(6*a*d) + sec(c + d*x)^8/(8*a*d) - (3*sec(c + d*x)*tan(c + d*x))/(128*a*d) - (sec(c + d*x)^3*tan(c + d*x))/(64*a*d) + (sec(c + d*x)^5*tan(c + d*x))/(16*a*d) - (sec(c + d*x)^5*tan(c + d*x)^3)/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 9, -((5*atanh(sin(c + d*x)))/(128*a*d)) + sec(c + d*x)^6/(6*a*d) - sec(c + d*x)^8/(8*a*d) - (5*sec(c + d*x)*tan(c + d*x))/(128*a*d) - (5*sec(c + d*x)^3*tan(c + d*x))/(192*a*d) - (sec(c + d*x)^5*tan(c + d*x))/(48*a*d) + (sec(c + d*x)^7*tan(c + d*x))/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 8, (5*atanh(sin(c + d*x)))/(128*a*d) + sec(c + d*x)^8/(8*a*d) + (5*sec(c + d*x)*tan(c + d*x))/(128*a*d) + (5*sec(c + d*x)^3*tan(c + d*x))/(192*a*d) + (sec(c + d*x)^5*tan(c + d*x))/(48*a*d) - (sec(c + d*x)^7*tan(c + d*x))/(8*a*d)]
    @test_int [sec(c + d*x)^7*sin(c + d*x)^0/(a + a*sin(c + d*x)), x, 4, (35*atanh(sin(c + d*x)))/(128*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) + (5*a)/(128*d*(a - a*sin(c + d*x))^2) + 15/(128*d*(a - a*sin(c + d*x))) - a^3/(64*d*(a + a*sin(c + d*x))^4) - a^2/(24*d*(a + a*sin(c + d*x))^3) - (5*a)/(64*d*(a + a*sin(c + d*x))^2) - 5/(32*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, -((93*log(1 - sin(c + d*x)))/(256*a*d)) + log(sin(c + d*x))/(a*d) - (163*log(1 + sin(c + d*x)))/(256*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) + (7*a)/(128*d*(a - a*sin(c + d*x))^2) + 29/(128*d*(a - a*sin(c + d*x))) + a^3/(64*d*(a + a*sin(c + d*x))^4) + a^2/(16*d*(a + a*sin(c + d*x))^3) + (11*a)/(64*d*(a + a*sin(c + d*x))^2) + 1/(2*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (187*log(1 - sin(c + d*x)))/(256*a*d) - log(sin(c + d*x))/(a*d) + (443*log(1 + sin(c + d*x)))/(256*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) + (9*a)/(128*d*(a - a*sin(c + d*x))^2) + 47/(128*d*(a - a*sin(c + d*x))) - a^3/(64*d*(a + a*sin(c + d*x))^4) - a^2/(12*d*(a + a*sin(c + d*x))^3) - (19*a)/(64*d*(a + a*sin(c + d*x))^2) - 35/(32*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d) - (325*log(1 - sin(c + d*x)))/(256*a*d) + (5*log(sin(c + d*x)))/(a*d) - (955*log(1 + sin(c + d*x)))/(256*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) + (11*a)/(128*d*(a - a*sin(c + d*x))^2) + 69/(128*d*(a - a*sin(c + d*x))) + a^3/(64*d*(a + a*sin(c + d*x))^4) + (5*a^2)/(48*d*(a + a*sin(c + d*x))^3) + (29*a)/(64*d*(a + a*sin(c + d*x))^2) + 2/(d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*csc(c + d*x)^4/(a + a*sin(c + d*x)), x, 4, -((5*csc(c + d*x))/(a*d)) + csc(c + d*x)^2/(2*a*d) - csc(c + d*x)^3/(3*a*d) - (515*log(1 - sin(c + d*x)))/(256*a*d) - (5*log(sin(c + d*x)))/(a*d) + (1795*log(1 + sin(c + d*x)))/(256*a*d) + a^2/(96*d*(a - a*sin(c + d*x))^3) + (13*a)/(128*d*(a - a*sin(c + d*x))^2) + 95/(128*d*(a - a*sin(c + d*x))) - a^3/(64*d*(a + a*sin(c + d*x))^4) - a^2/(8*d*(a + a*sin(c + d*x))^3) - (41*a)/(64*d*(a + a*sin(c + d*x))^2) - 105/(32*d*(a + a*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^3*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^8*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^8*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^8*sin(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 11, (a^2*sec(c + d*x)^3)/(3*d) - (3*a^2*sec(c + d*x)^5)/(5*d) + (2*a^2*sec(c + d*x)^7)/(7*d) + (2*a^2*tan(c + d*x)^5)/(5*d) + (2*a^2*tan(c + d*x)^7)/(7*d)]

    #= ::Subsubsection:: =#
    #=m<0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^9*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^9*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(c + d*x)^9*sin(c + d*x)^12/(a + a*sin(c + d*x)), x, 4, -((843*log(1 - sin(c + d*x)))/(512*a*d)) - (2229*log(1 + sin(c + d*x)))/(512*a*d) + sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) - (3*a^2)/(64*d*(a - a*sin(c + d*x))^3) + (141*a)/(512*d*(a - a*sin(c + d*x))^2) - 39/(32*d*(a - a*sin(c + d*x))) - a^4/(160*d*(a + a*sin(c + d*x))^5) + (19*a^3)/(256*d*(a + a*sin(c + d*x))^4) - (53*a^2)/(128*d*(a + a*sin(c + d*x))^3) + (765*a)/(512*d*(a + a*sin(c + d*x))^2) - 1155/(256*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^11/(a + a*sin(c + d*x)), x, 4, -((437*log(1 - sin(c + d*x)))/(512*a*d)) + (949*log(1 + sin(c + d*x)))/(512*a*d) - sin(c + d*x)/(a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) - a^2/(24*d*(a - a*sin(c + d*x))^3) + (109*a)/(512*d*(a - a*sin(c + d*x))^2) - 203/(256*d*(a - a*sin(c + d*x))) + a^4/(160*d*(a + a*sin(c + d*x))^5) - (17*a^3)/(256*d*(a + a*sin(c + d*x))^4) + (125*a^2)/(384*d*(a + a*sin(c + d*x))^3) - (515*a)/(512*d*(a + a*sin(c + d*x))^2) + 5/(2*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^10/(a + a*sin(c + d*x)), x, 4, -((193*log(1 - sin(c + d*x)))/(512*a*d)) - (319*log(1 + sin(c + d*x)))/(512*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) - (7*a^2)/(192*d*(a - a*sin(c + d*x))^3) + (81*a)/(512*d*(a - a*sin(c + d*x))^2) - 61/(128*d*(a - a*sin(c + d*x))) - a^4/(160*d*(a + a*sin(c + d*x))^5) + (15*a^3)/(256*d*(a + a*sin(c + d*x))^4) - (95*a^2)/(384*d*(a + a*sin(c + d*x))^3) + (325*a)/(512*d*(a + a*sin(c + d*x))^2) - 315/(256*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^9/(a + a*sin(c + d*x)), x, 9, (63*atanh(sin(c + d*x)))/(256*a*d) - (63*sec(c + d*x)*tan(c + d*x))/(256*a*d) + (21*sec(c + d*x)*tan(c + d*x)^3)/(128*a*d) - (21*sec(c + d*x)*tan(c + d*x)^5)/(160*a*d) + (9*sec(c + d*x)*tan(c + d*x)^7)/(80*a*d) - (sec(c + d*x)*tan(c + d*x)^9)/(10*a*d) + tan(c + d*x)^10/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^8/(a + a*sin(c + d*x)), x, 9, (7*atanh(sin(c + d*x)))/(256*a*d) + (7*sec(c + d*x)*tan(c + d*x))/(256*a*d) - (7*sec(c + d*x)^3*tan(c + d*x))/(128*a*d) + (7*sec(c + d*x)^3*tan(c + d*x)^3)/(96*a*d) - (7*sec(c + d*x)^3*tan(c + d*x)^5)/(80*a*d) + (sec(c + d*x)^3*tan(c + d*x)^7)/(10*a*d) - tan(c + d*x)^10/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^7/(a + a*sin(c + d*x)), x, 10, -((7*atanh(sin(c + d*x)))/(256*a*d)) - (7*sec(c + d*x)*tan(c + d*x))/(256*a*d) + (7*sec(c + d*x)^3*tan(c + d*x))/(128*a*d) - (7*sec(c + d*x)^3*tan(c + d*x)^3)/(96*a*d) + (7*sec(c + d*x)^3*tan(c + d*x)^5)/(80*a*d) - (sec(c + d*x)^3*tan(c + d*x)^7)/(10*a*d) + tan(c + d*x)^8/(8*a*d) + tan(c + d*x)^10/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^6/(a + a*sin(c + d*x)), x, 10, -((3*atanh(sin(c + d*x)))/(256*a*d)) - (3*sec(c + d*x)*tan(c + d*x))/(256*a*d) - (sec(c + d*x)^3*tan(c + d*x))/(128*a*d) + (sec(c + d*x)^5*tan(c + d*x))/(32*a*d) - (sec(c + d*x)^5*tan(c + d*x)^3)/(16*a*d) + (sec(c + d*x)^5*tan(c + d*x)^5)/(10*a*d) - tan(c + d*x)^8/(8*a*d) - tan(c + d*x)^10/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^5/(a + a*sin(c + d*x)), x, 11, (3*atanh(sin(c + d*x)))/(256*a*d) + sec(c + d*x)^6/(6*a*d) - sec(c + d*x)^8/(4*a*d) + sec(c + d*x)^10/(10*a*d) + (3*sec(c + d*x)*tan(c + d*x))/(256*a*d) + (sec(c + d*x)^3*tan(c + d*x))/(128*a*d) - (sec(c + d*x)^5*tan(c + d*x))/(32*a*d) + (sec(c + d*x)^5*tan(c + d*x)^3)/(16*a*d) - (sec(c + d*x)^5*tan(c + d*x)^5)/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^4/(a + a*sin(c + d*x)), x, 11, (3*atanh(sin(c + d*x)))/(256*a*d) - sec(c + d*x)^6/(6*a*d) + sec(c + d*x)^8/(4*a*d) - sec(c + d*x)^10/(10*a*d) + (3*sec(c + d*x)*tan(c + d*x))/(256*a*d) + (sec(c + d*x)^3*tan(c + d*x))/(128*a*d) + (sec(c + d*x)^5*tan(c + d*x))/(160*a*d) - (3*sec(c + d*x)^7*tan(c + d*x))/(80*a*d) + (sec(c + d*x)^7*tan(c + d*x)^3)/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 10, -((3*atanh(sin(c + d*x)))/(256*a*d)) - sec(c + d*x)^8/(8*a*d) + sec(c + d*x)^10/(10*a*d) - (3*sec(c + d*x)*tan(c + d*x))/(256*a*d) - (sec(c + d*x)^3*tan(c + d*x))/(128*a*d) - (sec(c + d*x)^5*tan(c + d*x))/(160*a*d) + (3*sec(c + d*x)^7*tan(c + d*x))/(80*a*d) - (sec(c + d*x)^7*tan(c + d*x)^3)/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 10, -((7*atanh(sin(c + d*x)))/(256*a*d)) + sec(c + d*x)^8/(8*a*d) - sec(c + d*x)^10/(10*a*d) - (7*sec(c + d*x)*tan(c + d*x))/(256*a*d) - (7*sec(c + d*x)^3*tan(c + d*x))/(384*a*d) - (7*sec(c + d*x)^5*tan(c + d*x))/(480*a*d) - (sec(c + d*x)^7*tan(c + d*x))/(80*a*d) + (sec(c + d*x)^9*tan(c + d*x))/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^1/(a + a*sin(c + d*x)), x, 9, (7*atanh(sin(c + d*x)))/(256*a*d) + sec(c + d*x)^10/(10*a*d) + (7*sec(c + d*x)*tan(c + d*x))/(256*a*d) + (7*sec(c + d*x)^3*tan(c + d*x))/(384*a*d) + (7*sec(c + d*x)^5*tan(c + d*x))/(480*a*d) + (sec(c + d*x)^7*tan(c + d*x))/(80*a*d) - (sec(c + d*x)^9*tan(c + d*x))/(10*a*d)]
    @test_int [sec(c + d*x)^9*sin(c + d*x)^0/(a + a*sin(c + d*x)), x, 4, (63*atanh(sin(c + d*x)))/(256*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) + a^2/(64*d*(a - a*sin(c + d*x))^3) + (21*a)/(512*d*(a - a*sin(c + d*x))^2) + 7/(64*d*(a - a*sin(c + d*x))) - a^4/(160*d*(a + a*sin(c + d*x))^5) - (5*a^3)/(256*d*(a + a*sin(c + d*x))^4) - (5*a^2)/(128*d*(a + a*sin(c + d*x))^3) - (35*a)/(512*d*(a + a*sin(c + d*x))^2) - 35/(256*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*csc(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, -((193*log(1 - sin(c + d*x)))/(512*a*d)) + log(sin(c + d*x))/(a*d) - (319*log(1 + sin(c + d*x)))/(512*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) + a^2/(48*d*(a - a*sin(c + d*x))^3) + (37*a)/(512*d*(a - a*sin(c + d*x))^2) + 65/(256*d*(a - a*sin(c + d*x))) + a^4/(160*d*(a + a*sin(c + d*x))^5) + (7*a^3)/(256*d*(a + a*sin(c + d*x))^4) + (29*a^2)/(384*d*(a + a*sin(c + d*x))^3) + (93*a)/(512*d*(a + a*sin(c + d*x))^2) + 1/(2*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (437*log(1 - sin(c + d*x)))/(512*a*d) - log(sin(c + d*x))/(a*d) + (949*log(1 + sin(c + d*x)))/(512*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) + (5*a^2)/(192*d*(a - a*sin(c + d*x))^3) + (57*a)/(512*d*(a - a*sin(c + d*x))^2) + 61/(128*d*(a - a*sin(c + d*x))) - a^4/(160*d*(a + a*sin(c + d*x))^5) - (9*a^3)/(256*d*(a + a*sin(c + d*x))^4) - (47*a^2)/(384*d*(a + a*sin(c + d*x))^3) - (187*a)/(512*d*(a + a*sin(c + d*x))^2) - 315/(256*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, csc(c + d*x)/(a*d) - csc(c + d*x)^2/(2*a*d) - (843*log(1 - sin(c + d*x)))/(512*a*d) + (6*log(sin(c + d*x)))/(a*d) - (2229*log(1 + sin(c + d*x)))/(512*a*d) + a^3/(256*d*(a - a*sin(c + d*x))^4) + a^2/(32*d*(a - a*sin(c + d*x))^3) + (81*a)/(512*d*(a - a*sin(c + d*x))^2) + 203/(256*d*(a - a*sin(c + d*x))) + a^4/(160*d*(a + a*sin(c + d*x))^5) + (11*a^3)/(256*d*(a + a*sin(c + d*x))^4) + (23*a^2)/(128*d*(a + a*sin(c + d*x))^3) + (325*a)/(512*d*(a + a*sin(c + d*x))^2) + 5/(2*d*(a + a*sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m*with*m, n*and*p*symbolic=#

    @test_int [(g*sec(e + f*x))^p*(d*sin(e + f*x))^n*(a + a*sin(e + f*x))^m, x, 5, (AppellF1(1 + n, (1 + p)/2, (1/2)*(1 - 2*m + p), 2 + n, sin(e + f*x), -sin(e + f*x))*sec(e + f*x)*(g*sec(e + f*x))^p*(1 - sin(e + f*x))^((1 + p)/2)*(d*sin(e + f*x))^(1 + n)*(1 + sin(e + f*x))^((1/2)*(1 - 2*m + p))*(a + a*sin(e + f*x))^m)/(d*f*(1 + n))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^1*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 3, (HypergeometricFunctions._₂F₁(1 + m, -n, 2 + m, -((d*(1 + sin(e + f*x)))/(c - d)))*(a + a*sin(e + f*x))^(1 + m)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c - d))^n*(a*f*(1 + m)))]

    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^4*(c + d*sin(e + f*x))^n, x, 3, (a^4*(c - d)^4*(c + d*sin(e + f*x))^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*sin(e + f*x))^(2 + n))/(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*sin(e + f*x))^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*sin(e + f*x))^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*sin(e + f*x))^(5 + n))/(d^5*f*(5 + n))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((a^3*(c - d)^3*(c + d*sin(e + f*x))^(1 + n))/(d^4*f*(1 + n))) + (3*a^3*(c - d)^2*(c + d*sin(e + f*x))^(2 + n))/(d^4*f*(2 + n)) - (3*a^3*(c - d)*(c + d*sin(e + f*x))^(3 + n))/(d^4*f*(3 + n)) + (a^3*(c + d*sin(e + f*x))^(4 + n))/(d^4*f*(4 + n))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, (a^2*(c - d)^2*(c + d*sin(e + f*x))^(1 + n))/(d^3*f*(1 + n)) - (2*a^2*(c - d)*(c + d*sin(e + f*x))^(2 + n))/(d^3*f*(2 + n)) + (a^2*(c + d*sin(e + f*x))^(3 + n))/(d^3*f*(3 + n))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((a*(c - d)*(c + d*sin(e + f*x))^(1 + n))/(d^2*f*(1 + n))) + (a*(c + d*sin(e + f*x))^(2 + n))/(d^2*f*(2 + n))]
    @test_int [cos(e + f*x)/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 2, -((HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*sin(e + f*x))/(c - d))*(c + d*sin(e + f*x))^(1 + n))/(a*(c - d)*f*(1 + n)))]
    @test_int [cos(e + f*x)/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 2, (d*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, (c + d*sin(e + f*x))/(c - d))*(c + d*sin(e + f*x))^(1 + n))/(a^2*(c - d)^2*f*(1 + n))]
    @test_int [cos(e + f*x)/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 2, -((d^2*HypergeometricFunctions._₂F₁(3, 1 + n, 2 + n, (c + d*sin(e + f*x))/(c - d))*(c + d*sin(e + f*x))^(1 + n))/(a^3*(c - d)^3*f*(1 + n)))]

    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^4, x, 3, ((c - d)^4*(a + a*sin(e + f*x))^(1 + m))/(a*f*(1 + m)) + (4*(c - d)^3*d*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m)) + (6*(c - d)^2*d^2*(a + a*sin(e + f*x))^(3 + m))/(a^3*f*(3 + m)) + (4*(c - d)*d^3*(a + a*sin(e + f*x))^(4 + m))/(a^4*f*(4 + m)) + (d^4*(a + a*sin(e + f*x))^(5 + m))/(a^5*f*(5 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^3, x, 3, ((c - d)^3*(a + a*sin(e + f*x))^(1 + m))/(a*f*(1 + m)) + (3*(c - d)^2*d*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m)) + (3*(c - d)*d^2*(a + a*sin(e + f*x))^(3 + m))/(a^3*f*(3 + m)) + (d^3*(a + a*sin(e + f*x))^(4 + m))/(a^4*f*(4 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^2, x, 3, ((c - d)^2*(a + a*sin(e + f*x))^(1 + m))/(a*f*(1 + m)) + (2*(c - d)*d*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m)) + (d^2*(a + a*sin(e + f*x))^(3 + m))/(a^3*f*(3 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^1, x, 3, ((c - d)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(1 + m)) + (d*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^1, x, 2, (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((d*(1 + sin(e + f*x)))/(c - d)))*(a + a*sin(e + f*x))^(1 + m))/(a*(c - d)*f*(1 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^2, x, 2, (HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, -((d*(1 + sin(e + f*x)))/(c - d)))*(a + a*sin(e + f*x))^(1 + m))/(a*(c - d)^2*f*(1 + m))]
    @test_int [cos(e + f*x)*(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^3, x, 2, (HypergeometricFunctions._₂F₁(3, 1 + m, 2 + m, -((d*(1 + sin(e + f*x)))/(c - d)))*(a + a*sin(e + f*x))^(1 + m))/(a*(c - d)^3*f*(1 + m))]

    @test_int [cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^m, x, 3, -((HypergeometricFunctions._₂F₁(1, 2 + n + m, 2 + m, 1 + sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + a*sin(c + d*x))^(1 + m))/(a*d*(1 + m))), (HypergeometricFunctions._₂F₁(1 + n, -m, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + a*sin(c + d*x))^m)/((1 + sin(c + d*x))^m*(d*(1 + n)))]

    @test_int [cos(c + d*x)*sin(c + d*x)^4*(a + a*sin(c + d*x))^m, x, 4, (a + a*sin(c + d*x))^(1 + m)/(a*d*(1 + m)) - (4*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*(2 + m)) + (6*(a + a*sin(c + d*x))^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*sin(c + d*x))^(4 + m))/(a^4*d*(4 + m)) + (a + a*sin(c + d*x))^(5 + m)/(a^5*d*(5 + m))]
    @test_int [cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^m, x, 4, -((a + a*sin(c + d*x))^(1 + m)/(a*d*(1 + m))) + (3*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*(2 + m)) - (3*(a + a*sin(c + d*x))^(3 + m))/(a^3*d*(3 + m)) + (a + a*sin(c + d*x))^(4 + m)/(a^4*d*(4 + m))]
    @test_int [cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^m, x, 4, (a + a*sin(c + d*x))^(1 + m)/(a*d*(1 + m)) - (2*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*(2 + m)) + (a + a*sin(c + d*x))^(3 + m)/(a^3*d*(3 + m))]
    @test_int [cos(c + d*x)*sin(c + d*x)^1*(a + a*sin(c + d*x))^m, x, 4, -((a + a*sin(c + d*x))^(1 + m)/(a*d*(1 + m))) + (a + a*sin(c + d*x))^(2 + m)/(a^2*d*(2 + m))]
    @test_int [cos(c + d*x)*csc(c + d*x)^1*(a + a*sin(c + d*x))^m, x, 2, -((HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, 1 + sin(c + d*x))*(a + a*sin(c + d*x))^(1 + m))/(a*d*(1 + m)))]
    @test_int [cos(c + d*x)*csc(c + d*x)^2*(a + a*sin(c + d*x))^m, x, 3, (HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, 1 + sin(c + d*x))*(a + a*sin(c + d*x))^(1 + m))/(a*d*(1 + m))]
    @test_int [cos(c + d*x)*csc(c + d*x)^3*(a + a*sin(c + d*x))^m, x, 3, -((HypergeometricFunctions._₂F₁(3, 1 + m, 2 + m, 1 + sin(c + d*x))*(a + a*sin(c + d*x))^(1 + m))/(a*d*(1 + m)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))*(c + d*sin(e + f*x)), x, 4, (1/8)*a*(4*c + d)*x - (a*(c + d)*cos(e + f*x)^3)/(3*f) + (a*(4*c + d)*cos(e + f*x)*sin(e + f*x))/(8*f) - (a*d*cos(e + f*x)^3*sin(e + f*x))/(4*f), (1/8)*a*(4*c + d)*x - (a*(4*c + d)*cos(e + f*x)^3)/(12*f) + (a*(4*c + d)*cos(e + f*x)*sin(e + f*x))/(8*f) - (d*cos(e + f*x)^3*(a + a*sin(e + f*x)))/(4*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^(m/2)*(c+d*sin(e+f*x))^n=#

    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))), x, 6, -((2*sqrt(2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(a^(3/2)*(c - d)*f)) + (2*sqrt(c + d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(a^(3/2)*(c - d)*sqrt(d)*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^(m/2)*(c+d*sin(e+f*x))^(n/2)=#

    @test_int [cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x))), x, 6, (2*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(a^(3/2)*sqrt(d)*f) - (2*sqrt(2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(a^(3/2)*sqrt(c - d)*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 4, (2*sqrt(2)*AppellF1(3/2 + m, -(1/2), -n, 5/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c - d))^n*(a*f*(3 + 2*m)*sqrt(1 - sin(e + f*x))))]

    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((16*sqrt(2)*a^3*AppellF1(3/2, -(7/2), -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(3*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((8*sqrt(2)*a^2*AppellF1(3/2, -(5/2), -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(3*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((4*sqrt(2)*a*AppellF1(3/2, -(3/2), -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(3*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 4, -((sqrt(2)*AppellF1(3/2, 1/2, -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(3*a*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(3/2, 3/2, -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(3*sqrt(2)*a^2*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(3/2, 5/2, -n, 5/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(6*sqrt(2)*a^3*f*sqrt(1 + sin(e + f*x)))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [cos(e + f*x)^4*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 4, (4*sqrt(2)*AppellF1(5/2 + m, -(3/2), -n, 7/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^(2 + m)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c - d))^n*(a^2*f*(5 + 2*m)*sqrt(1 - sin(e + f*x))))]

    @test_int [cos(e + f*x)^4*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((16*sqrt(2)*a^2*AppellF1(5/2, -(7/2), -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(5*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^4*(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((8*sqrt(2)*a*AppellF1(5/2, -(5/2), -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)^3*(1 - sin(e + f*x))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(5*f*(1 + sin(e + f*x))^(3/2))))]
    @test_int [cos(e + f*x)^4/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((2*sqrt(2)*AppellF1(5/2, -(1/2), -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(5*a*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^4/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 4, -((sqrt(2)*AppellF1(5/2, 1/2, -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(5*a^2*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^4/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(5/2, 3/2, -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(5*sqrt(2)*a^3*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^4/(a + a*sin(e + f*x))^4*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(5/2, 5/2, -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(10*sqrt(2)*a^4*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^4/(a + a*sin(e + f*x))^5*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(5/2, 7/2, -n, 7/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(1 - sin(e + f*x))^2*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(20*sqrt(2)*a^5*f*sqrt(1 + sin(e + f*x)))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*cos(e+f*x))^m*(a+a*sin(e+f*x))^n*(A+B*sin(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (8*(A - B)*(a + a*sin(c + d*x))^5)/(5*a^4*d) - (2*(3*A - 5*B)*(a + a*sin(c + d*x))^6)/(3*a^5*d) + (6*(A - 3*B)*(a + a*sin(c + d*x))^7)/(7*a^6*d) - ((A - 7*B)*(a + a*sin(c + d*x))^8)/(8*a^7*d) - (B*(a + a*sin(c + d*x))^9)/(9*a^8*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((A - B)*(a + a*sin(c + d*x))^4)/(a^3*d) - (4*(A - 2*B)*(a + a*sin(c + d*x))^5)/(5*a^4*d) + ((A - 5*B)*(a + a*sin(c + d*x))^6)/(6*a^5*d) + (B*(a + a*sin(c + d*x))^7)/(7*a^6*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (2*(A - B)*(a + a*sin(c + d*x))^3)/(3*a^2*d) - ((A - 3*B)*(a + a*sin(c + d*x))^4)/(4*a^3*d) - (B*(a + a*sin(c + d*x))^5)/(5*a^4*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (a*A*sin(c + d*x))/d + (a*(A + B)*sin(c + d*x)^2)/(2*d) + (a*B*sin(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, -((a*(A + B)*log(1 - sin(c + d*x)))/d) - (a*B*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 4, (a*(A - B)*atanh(sin(c + d*x)))/(2*d) + (a^2*(A + B))/(2*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 4, (a*(3*A - B)*atanh(sin(c + d*x)))/(8*d) + (a^3*(A + B))/(8*d*(a - a*sin(c + d*x))^2) + (a^2*A)/(4*d*(a - a*sin(c + d*x))) - (a^2*(A - B))/(8*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 4, (a*(5*A - B)*atanh(sin(c + d*x)))/(16*d) + (a^4*(A + B))/(24*d*(a - a*sin(c + d*x))^3) + (a^3*(3*A + B))/(32*d*(a - a*sin(c + d*x))^2) + (3*a^2*A)/(16*d*(a - a*sin(c + d*x))) - (a^3*(A - B))/(32*d*(a + a*sin(c + d*x))^2) - (a^2*(2*A - B))/(16*d*(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 6, (5/128)*a*(8*A + B)*x - (a*(8*A + B)*cos(c + d*x)^7)/(56*d) + (5*a*(8*A + B)*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*a*(8*A + B)*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (a*(8*A + B)*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (B*cos(c + d*x)^7*(a + a*sin(c + d*x)))/(8*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 5, (1/16)*a*(6*A + B)*x - (a*(6*A + B)*cos(c + d*x)^5)/(30*d) + (a*(6*A + B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(6*A + B)*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (B*cos(c + d*x)^5*(a + a*sin(c + d*x)))/(6*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 4, (1/8)*a*(4*A + B)*x - (a*(4*A + B)*cos(c + d*x)^3)/(12*d) + (a*(4*A + B)*cos(c + d*x)*sin(c + d*x))/(8*d) - (B*cos(c + d*x)^3*(a + a*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 2, (-a)*B*x + ((A + B)*sec(c + d*x)*(a + a*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((A + B)*sec(c + d*x)^3*(a + a*sin(c + d*x)))/(3*d) + (a*(2*A - B)*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((A + B)*sec(c + d*x)^5*(a + a*sin(c + d*x)))/(5*d) + (a*(4*A - B)*tan(c + d*x))/(5*d) + (a*(4*A - B)*tan(c + d*x)^3)/(15*d)]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((A + B)*sec(c + d*x)^7*(a + a*sin(c + d*x)))/(7*d) + (a*(6*A - B)*tan(c + d*x))/(7*d) + (2*a*(6*A - B)*tan(c + d*x)^3)/(21*d) + (a*(6*A - B)*tan(c + d*x)^5)/(35*d)]
    @test_int [sec(c + d*x)^10*(a + a*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((A + B)*sec(c + d*x)^9*(a + a*sin(c + d*x)))/(9*d) + (a*(8*A - B)*tan(c + d*x))/(9*d) + (a*(8*A - B)*tan(c + d*x)^3)/(9*d) + (a*(8*A - B)*tan(c + d*x)^5)/(15*d) + (a*(8*A - B)*tan(c + d*x)^7)/(63*d)]

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, (4*(A - B)*(a + a*sin(c + d*x))^6)/(3*a^4*d) - (4*(3*A - 5*B)*(a + a*sin(c + d*x))^7)/(7*a^5*d) + (3*(A - 3*B)*(a + a*sin(c + d*x))^8)/(4*a^6*d) - ((A - 7*B)*(a + a*sin(c + d*x))^9)/(9*a^7*d) - (B*(a + a*sin(c + d*x))^10)/(10*a^8*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, (4*(A - B)*(a + a*sin(c + d*x))^5)/(5*a^3*d) - (2*(A - 2*B)*(a + a*sin(c + d*x))^6)/(3*a^4*d) + ((A - 5*B)*(a + a*sin(c + d*x))^7)/(7*a^5*d) + (B*(a + a*sin(c + d*x))^8)/(8*a^6*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, ((A - B)*(a + a*sin(c + d*x))^4)/(2*a^2*d) - ((A - 3*B)*(a + a*sin(c + d*x))^5)/(5*a^3*d) - (B*(a + a*sin(c + d*x))^6)/(6*a^4*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, ((A - B)*(a + a*sin(c + d*x))^3)/(3*a*d) + (B*(a + a*sin(c + d*x))^4)/(4*a^2*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, (-2*a^2*(A + B)*log(1 - sin(c + d*x)))/d - (a^2*(A + B)*sin(c + d*x))/d - (B*(a + a*sin(c + d*x))^2)/(2*d)]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, (a^2*B*log(1 - sin(c + d*x)))/d + (a^3*(A + B))/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(A - B)*atanh(sin(c + d*x)))/(4*d) + (a^4*(A + B))/(4*d*(a - a*sin(c + d*x))^2) + (a^3*(A - B))/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(2*A - B)*atanh(sin(c + d*x)))/(8*d) + (a^5*(A + B))/(12*d*(a - a*sin(c + d*x))^3) + (a^4*A)/(8*d*(a - a*sin(c + d*x))^2) + (a^3*(3*A - B))/(16*d*(a - a*sin(c + d*x))) - (a^3*(A - B))/(16*d*(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 7, (5/128)*a^2*(9*A + 2*B)*x - (a^2*(9*A + 2*B)*cos(c + d*x)^7)/(56*d) + (5*a^2*(9*A + 2*B)*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*a^2*(9*A + 2*B)*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (a^2*(9*A + 2*B)*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (B*cos(c + d*x)^7*(a + a*sin(c + d*x))^2)/(9*d) - ((9*A + 2*B)*cos(c + d*x)^7*(a^2 + a^2*sin(c + d*x)))/(72*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 6, (1/16)*a^2*(7*A + 2*B)*x - (a^2*(7*A + 2*B)*cos(c + d*x)^5)/(30*d) + (a^2*(7*A + 2*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*(7*A + 2*B)*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (B*cos(c + d*x)^5*(a + a*sin(c + d*x))^2)/(7*d) - ((7*A + 2*B)*cos(c + d*x)^5*(a^2 + a^2*sin(c + d*x)))/(42*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 5, (1/8)*a^2*(5*A + 2*B)*x - (a^2*(5*A + 2*B)*cos(c + d*x)^3)/(12*d) + (a^2*(5*A + 2*B)*cos(c + d*x)*sin(c + d*x))/(8*d) - (B*cos(c + d*x)^3*(a + a*sin(c + d*x))^2)/(5*d) - ((5*A + 2*B)*cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x)))/(20*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, (-a^2)*(A + 2*B)*x + (a^2*(A + 2*B)*cos(c + d*x))/d + ((A + B)*sec(c + d*x)*(a + a*sin(c + d*x))^2)/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(A - 2*B)*sec(c + d*x))/(3*d) + ((A + B)*sec(c + d*x)^3*(a + a*sin(c + d*x))^2)/(3*d) + (a^2*(A - 2*B)*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(3*A - 2*B)*sec(c + d*x)^3)/(15*d) + ((A + B)*sec(c + d*x)^5*(a + a*sin(c + d*x))^2)/(5*d) + (a^2*(3*A - 2*B)*tan(c + d*x))/(5*d) + (a^2*(3*A - 2*B)*tan(c + d*x)^3)/(15*d)]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(5*A - 2*B)*sec(c + d*x)^5)/(35*d) + ((A + B)*sec(c + d*x)^7*(a + a*sin(c + d*x))^2)/(7*d) + (a^2*(5*A - 2*B)*tan(c + d*x))/(7*d) + (2*a^2*(5*A - 2*B)*tan(c + d*x)^3)/(21*d) + (a^2*(5*A - 2*B)*tan(c + d*x)^5)/(35*d)]
    @test_int [sec(c + d*x)^10*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(7*A - 2*B)*sec(c + d*x)^7)/(63*d) + ((A + B)*sec(c + d*x)^9*(a + a*sin(c + d*x))^2)/(9*d) + (a^2*(7*A - 2*B)*tan(c + d*x))/(9*d) + (a^2*(7*A - 2*B)*tan(c + d*x)^3)/(9*d) + (a^2*(7*A - 2*B)*tan(c + d*x)^5)/(15*d) + (a^2*(7*A - 2*B)*tan(c + d*x)^7)/(63*d)]
    @test_int [sec(c + d*x)^12*(a + a*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, (a^2*(9*A - 2*B)*sec(c + d*x)^9)/(99*d) + ((A + B)*sec(c + d*x)^11*(a + a*sin(c + d*x))^2)/(11*d) + (a^2*(9*A - 2*B)*tan(c + d*x))/(11*d) + (4*a^2*(9*A - 2*B)*tan(c + d*x)^3)/(33*d) + (6*a^2*(9*A - 2*B)*tan(c + d*x)^5)/(55*d) + (4*a^2*(9*A - 2*B)*tan(c + d*x)^7)/(77*d) + (a^2*(9*A - 2*B)*tan(c + d*x)^9)/(99*d)]

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, (8*(A - B)*(a + a*sin(c + d*x))^7)/(7*a^4*d) - ((3*A - 5*B)*(a + a*sin(c + d*x))^8)/(2*a^5*d) + (2*(A - 3*B)*(a + a*sin(c + d*x))^9)/(3*a^6*d) - ((A - 7*B)*(a + a*sin(c + d*x))^10)/(10*a^7*d) - (B*(a + a*sin(c + d*x))^11)/(11*a^8*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, (2*(A - B)*(a + a*sin(c + d*x))^6)/(3*a^3*d) - (4*(A - 2*B)*(a + a*sin(c + d*x))^7)/(7*a^4*d) + ((A - 5*B)*(a + a*sin(c + d*x))^8)/(8*a^5*d) + (B*(a + a*sin(c + d*x))^9)/(9*a^6*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, (2*(A - B)*(a + a*sin(c + d*x))^5)/(5*a^2*d) - ((A - 3*B)*(a + a*sin(c + d*x))^6)/(6*a^3*d) - (B*(a + a*sin(c + d*x))^7)/(7*a^4*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, ((A - B)*(a + a*sin(c + d*x))^4)/(4*a*d) + (B*(a + a*sin(c + d*x))^5)/(5*a^2*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, -((4*a^3*(A + B)*log(1 - sin(c + d*x)))/d) - (3*a^3*(A + B)*sin(c + d*x))/d - (a^3*(A + B)*sin(c + d*x)^2)/(2*d) - (B*(a + a*sin(c + d*x))^3)/(3*d)]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 3, (a^3*(A + 3*B)*log(1 - sin(c + d*x)))/d + (a^3*B*sin(c + d*x))/d + (2*a^4*(A + B))/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 2, (a^3*(a*A + a*B*sin(c + d*x))^2)/(2*(A + B)*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, (a^3*(A - B)*atanh(sin(c + d*x)))/(8*d) + (a^6*(A + B))/(6*d*(a - a*sin(c + d*x))^3) + (a^5*(A - B))/(8*d*(a - a*sin(c + d*x))^2) + (a^4*(A - B))/(8*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^9*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, (a^3*(5*A - 3*B)*atanh(sin(c + d*x)))/(32*d) + (a^7*(A + B))/(16*d*(a - a*sin(c + d*x))^4) + (a^6*A)/(12*d*(a - a*sin(c + d*x))^3) + (a^5*(3*A - B))/(32*d*(a - a*sin(c + d*x))^2) + (a^4*(2*A - B))/(16*d*(a - a*sin(c + d*x))) - (a^4*(A - B))/(32*d*(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 8, (11/256)*a^3*(10*A + 3*B)*x - (11*a^3*(10*A + 3*B)*cos(c + d*x)^7)/(560*d) + (11*a^3*(10*A + 3*B)*cos(c + d*x)*sin(c + d*x))/(256*d) + (11*a^3*(10*A + 3*B)*cos(c + d*x)^3*sin(c + d*x))/(384*d) + (11*a^3*(10*A + 3*B)*cos(c + d*x)^5*sin(c + d*x))/(480*d) - (a*(10*A + 3*B)*cos(c + d*x)^7*(a + a*sin(c + d*x))^2)/(90*d) - (B*cos(c + d*x)^7*(a + a*sin(c + d*x))^3)/(10*d) - (11*(10*A + 3*B)*cos(c + d*x)^7*(a^3 + a^3*sin(c + d*x)))/(720*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 7, (9/128)*a^3*(8*A + 3*B)*x - (3*a^3*(8*A + 3*B)*cos(c + d*x)^5)/(80*d) + (9*a^3*(8*A + 3*B)*cos(c + d*x)*sin(c + d*x))/(128*d) + (3*a^3*(8*A + 3*B)*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (a*(8*A + 3*B)*cos(c + d*x)^5*(a + a*sin(c + d*x))^2)/(56*d) - (B*cos(c + d*x)^5*(a + a*sin(c + d*x))^3)/(8*d) - (3*(8*A + 3*B)*cos(c + d*x)^5*(a^3 + a^3*sin(c + d*x)))/(112*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 6, (7/16)*a^3*(2*A + B)*x - (7*a^3*(2*A + B)*cos(c + d*x)^3)/(24*d) + (7*a^3*(2*A + B)*cos(c + d*x)*sin(c + d*x))/(16*d) - (a*(2*A + B)*cos(c + d*x)^3*(a + a*sin(c + d*x))^2)/(10*d) - (B*cos(c + d*x)^3*(a + a*sin(c + d*x))^3)/(6*d) - (7*(2*A + B)*cos(c + d*x)^3*(a^3 + a^3*sin(c + d*x)))/(40*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 2, (-(3/2))*a^3*(2*A + 3*B)*x + (2*a^3*(2*A + 3*B)*cos(c + d*x))/d + (a^3*(2*A + 3*B)*cos(c + d*x)*sin(c + d*x))/(2*d) + ((A + B)*sec(c + d*x)*(a + a*sin(c + d*x))^3)/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, a^3*B*x + ((A + B)*sec(c + d*x)^3*(a + a*sin(c + d*x))^3)/(3*d) - (2*a^5*B*cos(c + d*x))/(d*(a^2 - a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, (a^5*(2*A - 3*B)*cos(c + d*x))/(15*d*(a - a*sin(c + d*x))^2) + ((A + B)*sec(c + d*x)^5*(a + a*sin(c + d*x))^3)/(5*d) + (a^5*(2*A - 3*B)*cos(c + d*x))/(15*d*(a^2 - a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, ((A + B)*sec(c + d*x)^7*(a + a*sin(c + d*x))^3)/(7*d) + (2*(4*A - 3*B)*sec(c + d*x)^5*(a^3 + a^3*sin(c + d*x)))/(35*d) + (3*a^3*(4*A - 3*B)*tan(c + d*x))/(35*d) + (a^3*(4*A - 3*B)*tan(c + d*x)^3)/(35*d)]
    @test_int [sec(c + d*x)^10*(a + a*sin(c + d*x))^3*(A + B*sin(c + d*x)), x, 4, ((A + B)*sec(c + d*x)^9*(a + a*sin(c + d*x))^3)/(9*d) + (2*(2*A - B)*sec(c + d*x)^7*(a^3 + a^3*sin(c + d*x)))/(21*d) + (5*a^3*(2*A - B)*tan(c + d*x))/(21*d) + (10*a^3*(2*A - B)*tan(c + d*x)^3)/(63*d) + (a^3*(2*A - B)*tan(c + d*x)^5)/(21*d)]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [(cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 3, -(((A + B)*(a - a*sin(c + d*x))^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*sin(c + d*x))^5)/(5*a^6*d) - ((A + 5*B)*(a - a*sin(c + d*x))^6)/(6*a^7*d) + (B*(a - a*sin(c + d*x))^7)/(7*a^8*d)]
    @test_int [(cos(c + d*x)^5*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 3, (-2*(A + B)*(a - a*sin(c + d*x))^3)/(3*a^4*d) + ((A + 3*B)*(a - a*sin(c + d*x))^4)/(4*a^5*d) - (B*(a - a*sin(c + d*x))^5)/(5*a^6*d)]
    @test_int [(cos(c + d*x)^3*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 3, (A*sin(c + d*x))/(a*d) - ((A - B)*sin(c + d*x)^2)/(2*a*d) - (B*sin(c + d*x)^3)/(3*a*d)]
    @test_int [(cos(c + d*x)^1*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 3, ((A - B)*log(1 + sin(c + d*x)))/(a*d) + (B*sin(c + d*x))/(a*d)]
    @test_int [(sec(c + d*x)^1*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 4, ((A + B)*atanh(sin(c + d*x)))/(2*a*d) - (A - B)/(2*d*(a + a*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^3*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 4, ((3*A + B)*atanh(sin(c + d*x)))/(8*a*d) + (A + B)/(8*d*(a - a*sin(c + d*x))) - (a*(A - B))/(8*d*(a + a*sin(c + d*x))^2) - A/(4*d*(a + a*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^5*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 4, ((5*A + B)*atanh(sin(c + d*x)))/(16*a*d) + (a*(A + B))/(32*d*(a - a*sin(c + d*x))^2) + (2*A + B)/(16*d*(a - a*sin(c + d*x))) - (a^2*(A - B))/(24*d*(a + a*sin(c + d*x))^3) - (a*(3*A - B))/(32*d*(a + a*sin(c + d*x))^2) - (3*A)/(16*d*(a + a*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x)), x, 4, (5*(7*A + B)*atanh(sin(c + d*x)))/(128*a*d) + (a^2*(A + B))/(96*d*(a - a*sin(c + d*x))^3) + (a*(5*A + 3*B))/(128*d*(a - a*sin(c + d*x))^2) + (5*(3*A + B))/(128*d*(a - a*sin(c + d*x))) - (a^3*(A - B))/(64*d*(a + a*sin(c + d*x))^4) - (a^2*(2*A - B))/(48*d*(a + a*sin(c + d*x))^3) - (a*(5*A - B))/(64*d*(a + a*sin(c + d*x))^2) - (5*A)/(32*d*(a + a*sin(c + d*x)))]

    @test_int [(cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 3, -((A + B)*(a - a*sin(c + d*x))^4)/(2*a^6*d) + ((A + 3*B)*(a - a*sin(c + d*x))^5)/(5*a^7*d) - (B*(a - a*sin(c + d*x))^6)/(6*a^8*d)]
    @test_int [(cos(c + d*x)^5*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 3, -((A + B)*(a - a*sin(c + d*x))^3)/(3*a^5*d) + (B*(a - a*sin(c + d*x))^4)/(4*a^6*d)]
    @test_int [(cos(c + d*x)^3*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 3, (2*(A - B)*log(1 + sin(c + d*x)))/(a^2*d) - ((A - B)*sin(c + d*x))/(a^2*d) - (B*(a - a*sin(c + d*x))^2)/(2*a^4*d)]
    @test_int [(cos(c + d*x)^1*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 3, (B*log(1 + sin(c + d*x)))/(a^2*d) - (A - B)/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^1*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 4, ((A + B)*atanh(sin(c + d*x)))/(4*a^2*d) - (A - B)/(4*d*(a + a*sin(c + d*x))^2) - (A + B)/(4*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^3*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 4, ((2*A + B)*atanh(sin(c + d*x)))/(8*a^2*d) - (a*(A - B))/(12*d*(a + a*sin(c + d*x))^3) - A/(8*d*(a + a*sin(c + d*x))^2) + (A + B)/(16*d*(a^2 - a^2*sin(c + d*x))) - (3*A + B)/(16*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^5*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 4, (5*(3*A + B)*atanh(sin(c + d*x)))/(64*a^2*d) + (A + B)/(64*d*(a - a*sin(c + d*x))^2) - (a^2*(A - B))/(32*d*(a + a*sin(c + d*x))^4) - (a*(3*A - B))/(48*d*(a + a*sin(c + d*x))^3) - (3*A)/(32*d*(a + a*sin(c + d*x))^2) + (5*A + 3*B)/(64*d*(a^2 - a^2*sin(c + d*x))) - (5*A + B)/(32*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^7*(A + B*sin(c + d*x)))/(a + a*sin(c + d*x))^2, x, 4, (7*(4*A + B)*atanh(sin(c + d*x)))/(128*a^2*d) + (a*(A + B))/(192*d*(a - a*sin(c + d*x))^3) + (3*A + 2*B)/(128*d*(a - a*sin(c + d*x))^2) - (a^3*(A - B))/(80*d*(a + a*sin(c + d*x))^5) - (a^2*(2*A - B))/(64*d*(a + a*sin(c + d*x))^4) - (a*(5*A - B))/(96*d*(a + a*sin(c + d*x))^3) - (5*A)/(64*d*(a + a*sin(c + d*x))^2) + (3*(7*A + 3*B))/(256*d*(a^2 - a^2*sin(c + d*x))) - (5*(7*A + B))/(256*d*(a^2 + a^2*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(e*cos(e+f*x))^m*(a+a*sin(e+f*x))^(n/2)*(A+B*sin(e+f*x))=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(e*cos(e+f*x))^(m/2)*(a+a*sin(e+f*x))^n*(A+B*sin(e+f*x))=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(e*cos(e+f*x))^(m/2)*(a+a*sin(e+f*x))^(n/2)*(A+B*sin(e+f*x))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*cos(e+f*x))^m*(a+a*sin(e+f*x))^n*(A+B*sin(e+f*x))*with*n*symbolic=#

    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, -((2^((1/2)*(1 + 2*m + p))*a*(B*m + A*(1 + m + p))*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(1 - 2*m - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^((1/2)*(1 - 2*m - p))*(a + a*sin(e + f*x))^(-1 + m))/(f*g*(1 + p)*(1 + m + p))) - (B*(g*cos(e + f*x))^(1 + p)*(a + a*sin(e + f*x))^m)/(f*g*(1 + m + p))]

    @test_int [cos(e + f*x)^7*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, (8*(A - B)*(a + a*sin(e + f*x))^(4 + m))/(a^4*f*(4 + m)) - (4*(3*A - 5*B)*(a + a*sin(e + f*x))^(5 + m))/(a^5*f*(5 + m)) + (6*(A - 3*B)*(a + a*sin(e + f*x))^(6 + m))/(a^6*f*(6 + m)) - ((A - 7*B)*(a + a*sin(e + f*x))^(7 + m))/(a^7*f*(7 + m)) - (B*(a + a*sin(e + f*x))^(8 + m))/(a^8*f*(8 + m))]
    @test_int [cos(e + f*x)^5*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, (4*(A - B)*(a + a*sin(e + f*x))^(3 + m))/(a^3*f*(3 + m)) - (4*(A - 2*B)*(a + a*sin(e + f*x))^(4 + m))/(a^4*f*(4 + m)) + ((A - 5*B)*(a + a*sin(e + f*x))^(5 + m))/(a^5*f*(5 + m)) + (B*(a + a*sin(e + f*x))^(6 + m))/(a^6*f*(6 + m))]
    @test_int [cos(e + f*x)^3*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, (2*(A - B)*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m)) - ((A - 3*B)*(a + a*sin(e + f*x))^(3 + m))/(a^3*f*(3 + m)) - (B*(a + a*sin(e + f*x))^(4 + m))/(a^4*f*(4 + m))]
    @test_int [cos(e + f*x)^1*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, ((A - B)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(1 + m)) + (B*(a + a*sin(e + f*x))^(2 + m))/(a^2*f*(2 + m))]
    @test_int [sec(e + f*x)^1*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, ((A - B)*(a + a*sin(e + f*x))^m)/(2*f*m) + ((A + B)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^(1 + m))/(4*a*f*(1 + m))]
    @test_int [sec(e + f*x)^3*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, -((a*(A*(2 - m) - B*m)*HypergeometricFunctions._₂F₁(1, -1 + m, m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^(-1 + m))/(4*f*(1 - m))) + (a^2*(A + B)*(a + a*sin(e + f*x))^(-1 + m))/(2*f*(a - a*sin(e + f*x)))]
    @test_int [sec(e + f*x)^5*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 3, -((a^2*(A*(4 - m) - B*m)*HypergeometricFunctions._₂F₁(2, -2 + m, -1 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^(-2 + m))/(16*f*(2 - m))) + (a^4*(A + B)*(a + a*sin(e + f*x))^(-2 + m))/(4*f*(a - a*sin(e + f*x))^2)]

    @test_int [cos(e + f*x)^6*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, -((2^(7/2 + m)*a^3*(B*m + A*(7 + m))*cos(e + f*x)^7*HypergeometricFunctions._₂F₁(7/2, -(5/2) - m, 9/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-3 + m))/(7*f*(7 + m))) - (B*cos(e + f*x)^7*(a + a*sin(e + f*x))^m)/(f*(7 + m))]
    @test_int [cos(e + f*x)^4*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, -((2^(5/2 + m)*a^2*(B*m + A*(5 + m))*cos(e + f*x)^5*HypergeometricFunctions._₂F₁(5/2, -(3/2) - m, 7/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-2 + m))/(5*f*(5 + m))) - (B*cos(e + f*x)^5*(a + a*sin(e + f*x))^m)/(f*(5 + m))]
    @test_int [cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, -((2^(3/2 + m)*a*(B*m + A*(3 + m))*cos(e + f*x)^3*HypergeometricFunctions._₂F₁(3/2, -(1/2) - m, 5/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^(-1 + m))/(3*f*(3 + m))) - (B*cos(e + f*x)^3*(a + a*sin(e + f*x))^m)/(f*(3 + m))]
    @test_int [sec(e + f*x)^2*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, (B*sec(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 - m)) + (2^(-(1/2) + m)*(A*(1 - m) - B*m)*HypergeometricFunctions._₂F₁(-(1/2), 3/2 - m, 1/2, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^m)/(f*(1 - m))]
    @test_int [sec(e + f*x)^4*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, (B*sec(e + f*x)^3*(a + a*sin(e + f*x))^m)/(f*(3 - m)) + (2^(-(3/2) + m)*(A*(3 - m) - B*m)*HypergeometricFunctions._₂F₁(-(3/2), 5/2 - m, -(1/2), (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)^3*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(1 + m))/(3*a*f*(3 - m))]
    @test_int [sec(e + f*x)^6*(a + a*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 4, (B*sec(e + f*x)^5*(a + a*sin(e + f*x))^m)/(f*(5 - m)) + (2^(-(5/2) + m)*(A*(5 - m) - B*m)*HypergeometricFunctions._₂F₁(-(5/2), 7/2 - m, -(3/2), (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)^5*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(2 + m))/(5*a^2*f*(5 - m))]

    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p + 4), x, 4, If(13>=8, ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-3 - p))/(c*f*g*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(c^2*f*g*(3 + p)*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c^3*f*g*(1 + p)*(3 + p)*(5 + p)*(7 + p)), ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-3 - p))/(c*f*g*(35 + 12*p + p^2)) + (2*(3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(c^2*f*g*(5 + p)*(21 + 10*p + p^2)) + (2*(3*A - B*(4 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c^3*f*g*(5 + 6*p + p^2)*(21 + 10*p + p^2)))]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p + 3), x, 3, If(13>=8, ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-3 - p))/(f*g*(5 + p)) + ((2*A - B*(3 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(c*f*g*(3 + p)*(5 + p)) + ((2*A - B*(3 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c^2*f*g*(1 + p)*(3 + p)*(5 + p)), ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-3 - p))/(f*g*(5 + p)) + ((2*A - B*(3 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(c*f*g*(15 + 8*p + p^2)) + ((2*A - B*(3 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c^2*f*g*(3 + p)*(5 + 6*p + p^2)))]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p + 2), x, 2, If(13>=8, ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(f*g*(3 + p)) + ((A - B*(2 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c*f*g*(1 + p)*(3 + p)), ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-2 - p))/(f*g*(3 + p)) + ((A - B*(2 + p))*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(c*f*g*(3 + 4*p + p^2)))]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p + 1), x, 4, ((A + B)*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(-1 - p))/(f*g*(1 + p)) - (2^(1/2 - p/2)*B*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1 + p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((1 + p)/2)*(c - c*sin(e + f*x))^(-1 - p))/(f*g*(1 + p))]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p + 0), x, 4, (2^(1/2 - p/2)*c*(A + B*p)*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1 + p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((1 + p)/2)*(c - c*sin(e + f*x))^(-1 - p))/(f*g*(1 + p)) - (B*(g*cos(e + f*x))^(1 + p))/((c - c*sin(e + f*x))^p*(f*g))]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p - 1), x, 4, (2^(1/2 - p/2)*c^2*(2*A - B*(1 - p))*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-1 + p), (1 + p)/2, (3 + p)/2, (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((1 + p)/2)*(c - c*sin(e + f*x))^(-1 - p))/(f*g*(1 + p)) - (B*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(1 - p))/(2*f*g)]
    @test_int [(g*cos(e + f*x))^p*(A + B*sin(e + f*x))/(c - c*sin(e + f*x))^(p - 2), x, 4, (2^(5/2 - p/2)*c^3*(3*A - B*(2 - p))*(g*cos(e + f*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-3 + p), (1 + p)/2, (3 + p)/2, (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^((1 + p)/2)*(c - c*sin(e + f*x))^(-1 - p))/(3*f*g*(1 + p)) - (B*(g*cos(e + f*x))^(1 + p)*(c - c*sin(e + f*x))^(2 - p))/(3*f*g)]

    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(A*m - A*(m + p + 1)*sin(e + f*x)), x, 1, (A*(g*cos(e + f*x))^(1 + p)*(a + a*sin(e + f*x))^m)/(f*g)]
    @test_int [(g*cos(e + f*x))^p*(a - a*sin(e + f*x))^m*(A*m + A*(m + p + 1)*sin(e + f*x)), x, 1, -((A*(g*cos(e + f*x))^(1 + p)*(a - a*sin(e + f*x))^m)/(f*g))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m, n*and*p*symbolic=#

    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 4, (1/(a*f*(1 + 2*m + p)))*((2^((1 + p)/2)*g*AppellF1((1/2)*(1 + 2*m + p), (1 - p)/2, -n, (1/2)*(3 + 2*m + p), (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))^((1 - p)/2)*(a + a*sin(e + f*x))^(1 + m)*(c + d*sin(e + f*x))^n)/((c + d*sin(e + f*x))/(c - d))^n)]

    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((2^(5/2 + p/2)*a^2*AppellF1((1 + p)/2, (1/2)*(-3 - p), -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^(2 + (1/2)*(-5 - p))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*g*(1 + p)))), -((2^((5 + p)/2)*a^2*g*AppellF1((1 + p)/2, (1/2)*(-3 - p), -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*(1 + p))))]
    @test_int [(g*cos(e + f*x))^p*(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((2^(3/2 + p/2)*a*AppellF1((1 + p)/2, (1/2)*(-1 - p), -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^((1/2)*(-1 - p))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*g*(1 + p)))), -((2^((3 + p)/2)*a*g*AppellF1((1 + p)/2, (1/2)*(-1 - p), -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*(1 + p))))]
    @test_int [(g*cos(e + f*x))^p/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((2^(-(1/2) + p/2)*AppellF1((1 + p)/2, (3 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^(-1 + (1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a*f*g*(1 + p)))), -((2^(-(1/2) + p/2)*g*AppellF1((1 + p)/2, (3 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a*f*(1 + p))))]
    @test_int [(g*cos(e + f*x))^p/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((2^(-(3/2) + p/2)*AppellF1((1 + p)/2, (5 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^(-2 + (3 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^2*f*g*(1 + p)))), -((2^((1/2)*(-3 + p))*g*AppellF1((1 + p)/2, (5 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^2*f*(1 + p))))]
    @test_int [(g*cos(e + f*x))^p/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((2^(-(5/2) + p/2)*AppellF1((1 + p)/2, (7 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^(-3 + (5 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^3*f*g*(1 + p)))), -((2^((1/2)*(-5 + p))*g*AppellF1((1 + p)/2, (7 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^3*f*(1 + p))))]
    @test_int [(g*cos(e + f*x))^p/(a + a*sin(e + f*x))^4*(c + d*sin(e + f*x))^n, x, 3, -((2^(-(7/2) + p/2)*AppellF1((1 + p)/2, (9 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(1 + p)*(1 + sin(e + f*x))^(-4 + (7 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^4*f*g*(1 + p)))), -((2^((1/2)*(-7 + p))*g*AppellF1((1 + p)/2, (9 - p)/2, -n, (3 + p)/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*(g*cos(e + f*x))^(-1 + p)*(1 - sin(e + f*x))*(1 + sin(e + f*x))^((1 - p)/2)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(a^4*f*(1 + p))))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#
    #==#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m, n*and*p*symbolic=#

    @test_int [(g*sec(e + f*x))^p*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 5, (1/(a*f*(1 + 2*m - p)))*((2^(1/2 - p/2)*AppellF1((1/2)*(1 + 2*m - p), (1 + p)/2, -n, (1/2)*(3 + 2*m - p), (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*sec(e + f*x)*(g*sec(e + f*x))^p*(1 - sin(e + f*x))^((1 + p)/2)*(a + a*sin(e + f*x))^(1 + m)*(c + d*sin(e + f*x))^n)/((c + d*sin(e + f*x))/(c - d))^n)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section:: =#
    #=Integrands*of*the*form*cos(e+f*x)^1*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*(a + b*sin(c + d*x)), x, 8, (b*x)/16 - (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)^5)/(5*d) + (b*cos(c + d*x)*sin(c + d*x))/(16*d) - (b*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (b*cos(c + d*x)^3*sin(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x)), x, 7, (a*x)/8 - (b*cos(c + d*x)^3)/(3*d) + (b*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x)), x, 6, (b*x)/8 - (a*cos(c + d*x)^3)/(3*d) + (b*cos(c + d*x)*sin(c + d*x))/(8*d) - (b*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x)), x, 6, (b*x)/2 - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (b*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x)), x, 7, -a*x - (b*atanh(cos(c + d*x)))/d + (b*cos(c + d*x))/d - (a*cot(c + d*x))/d]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x)), x, 5, (-b)*x + (a*atanh(cos(c + d*x)))/(2*d) - (b*cot(c + d*x))/d - (a*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + b*sin(c + d*x)), x, 5, (b*atanh(cos(c + d*x)))/(2*d) - (a*cot(c + d*x)^3)/(3*d) - (b*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + b*sin(c + d*x)), x, 6, (a*atanh(cos(c + d*x)))/(8*d) - (b*cot(c + d*x)^3)/(3*d) + (a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + b*sin(c + d*x)), x, 7, (b*atanh(cos(c + d*x)))/(8*d) - (a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^5)/(5*d) + (b*cot(c + d*x)*csc(c + d*x))/(8*d) - (b*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 10, (a*b*x)/8 - ((7*a^2 + 4*b^2)*cos(c + d*x))/(35*d) + ((7*a^2 + 4*b^2)*cos(c + d*x)^3)/(105*d) - (a*b*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*b*cos(c + d*x)*sin(c + d*x)^3)/(12*d) + ((2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(35*d) + (a*b*cos(c + d*x)*sin(c + d*x)^5)/(21*d) + (cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^2)/(7*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 9, (1/16)*(2*a^2 + b^2)*x - (2*a*b*cos(c + d*x))/(5*d) + (2*a*b*cos(c + d*x)^3)/(15*d) - ((2*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(24*d) + (a*b*cos(c + d*x)*sin(c + d*x)^4)/(15*d) + (cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^2)/(6*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 5, (a*b*x)/4 - ((a^2 + 4*b^2)*cos(c + d*x)^3)/(30*d) + (a*b*cos(c + d*x)*sin(c + d*x))/(4*d) - (a*cos(c + d*x)^3*(a + b*sin(c + d*x)))/(10*d) - (cos(c + d*x)^3*(a + b*sin(c + d*x))^2)/(5*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, a*b*x - (a^2*atanh(cos(c + d*x)))/d + ((2*a^2 - b^2)*cos(c + d*x))/(3*d) + (a*b*cos(c + d*x)*sin(c + d*x))/(3*d) + (cos(c + d*x)*(a + b*sin(c + d*x))^2)/(3*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 9, (-a^2)*x + (b^2*x)/2 - (2*a*b*atanh(cos(c + d*x)))/d + (2*a*b*cos(c + d*x))/d - (a^2*cot(c + d*x))/d + (b^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 6, -2*a*b*x + ((a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*d) + (3*b^2*cos(c + d*x))/(2*d) - (a*b*cot(c + d*x))/d - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^2)/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 6, (-b^2)*x + (a*b*atanh(cos(c + d*x)))/d + ((a^2 - 2*b^2)*cot(c + d*x))/(3*d) - (a*b*cot(c + d*x)*csc(c + d*x))/(3*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^2)/(3*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 8, ((a^2 + 4*b^2)*atanh(cos(c + d*x)))/(8*d) + (2*a*b*cot(c + d*x))/(3*d) + ((a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*b*cot(c + d*x)*csc(c + d*x)^2)/(6*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^2)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 9, (a*b*atanh(cos(c + d*x)))/(4*d) + ((2*a^2 + 5*b^2)*cot(c + d*x))/(15*d) + (a*b*cot(c + d*x)*csc(c + d*x))/(4*d) + ((a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(15*d) - (a*b*cot(c + d*x)*csc(c + d*x)^3)/(10*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^2)/(5*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^7*(a + b*sin(c + d*x))^2, x, 9, ((a^2 + 2*b^2)*atanh(cos(c + d*x)))/(16*d) + (2*a*b*cot(c + d*x))/(5*d) + (2*a*b*cot(c + d*x)^3)/(15*d) + ((a^2 + 2*b^2)*cot(c + d*x)*csc(c + d*x))/(16*d) + ((a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a*b*cot(c + d*x)*csc(c + d*x)^4)/(15*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^2)/(6*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 10, (1/16)*a*(2*a^2 + 3*b^2)*x - (b*(21*a^2 + 4*b^2)*cos(c + d*x))/(35*d) + (b*(21*a^2 + 4*b^2)*cos(c + d*x)^3)/(105*d) - (a*(2*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(2*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x)^3)/(56*d) + (b*(a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(35*d) + (a*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^2)/(14*d) + (cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^3)/(7*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 6, (1/16)*b*(6*a^2 + b^2)*x - (a*(2*a^2 + 33*b^2)*cos(c + d*x)^3)/(120*d) + (b*(6*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) - ((2*a^2 + 5*b^2)*cos(c + d*x)^3*(a + b*sin(c + d*x)))/(40*d) - (a*cos(c + d*x)^3*(a + b*sin(c + d*x))^2)/(10*d) - (cos(c + d*x)^3*(a + b*sin(c + d*x))^3)/(6*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 7, (1/8)*b*(12*a^2 + b^2)*x - (a^3*atanh(cos(c + d*x)))/d + (a*(a^2 - 2*b^2)*cos(c + d*x))/(2*d) + (b*(2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(4*d) + (cos(c + d*x)*(a + b*sin(c + d*x))^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 11, (-a^3)*x + (3/2)*a*b^2*x - (3*a^2*b*atanh(cos(c + d*x)))/d + (3*a^2*b*cos(c + d*x))/d - (b^3*cos(c + d*x)^3)/(3*d) - (a^3*cot(c + d*x))/d + (3*a*b^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 7, (-(1/2))*b*(6*a^2 - b^2)*x + (a*(a^2 - 6*b^2)*atanh(cos(c + d*x)))/(2*d) + (15*a*b^2*cos(c + d*x))/(2*d) + (5*b^3*cos(c + d*x)*sin(c + d*x))/(2*d) - (3*b*cot(c + d*x)*(a + b*sin(c + d*x))^2)/(2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^3)/(2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 7, -3*a*b^2*x + (b*(3*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*d) + (11*b^3*cos(c + d*x))/(6*d) + (a*(a^2 - 3*b^2)*cot(c + d*x))/(3*d) - (b*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^2)/(2*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^3)/(3*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5*(a + b*sin(c + d*x))^3, x, 7, (-b^3)*x + (a*(a^2 + 12*b^2)*atanh(cos(c + d*x)))/(8*d) + (b*(2*a^2 - b^2)*cot(c + d*x))/(2*d) + (a*(a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x))/(8*d) - (b*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^2)/(4*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^3)/(4*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6*(a + b*sin(c + d*x))^3, x, 9, (b*(3*a^2 + 4*b^2)*atanh(cos(c + d*x)))/(8*d) + (a*(2*a^2 + 15*b^2)*cot(c + d*x))/(15*d) + (3*b*(5*a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x))/(40*d) + (a*(2*a^2 - 3*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(30*d) - (3*b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^2)/(20*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^3)/(5*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^7*(a + b*sin(c + d*x))^3, x, 10, (a*(a^2 + 6*b^2)*atanh(cos(c + d*x)))/(16*d) + (b*(6*a^2 + 5*b^2)*cot(c + d*x))/(15*d) + (a*(a^2 + 6*b^2)*cot(c + d*x)*csc(c + d*x))/(16*d) + (b*(3*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^2)/(15*d) + (a*(5*a^2 - 6*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(120*d) - (b*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^2)/(10*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^3)/(6*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 9, (a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) + ((12*a^2 - b^2)*cos(c + d*x))/(3*b^4*d) - (2*a*cos(c + d*x)*sin(c + d*x))/(b^3*d) + (4*cos(c + d*x)*sin(c + d*x)^2)/(3*b^2*d) - (cos(c + d*x)*sin(c + d*x)^3)/(b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 8, -(((6*a^2 - b^2)*x)/(2*b^4)) + (2*a*(3*a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) - (3*a*cos(c + d*x))/(b^3*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) - (cos(c + d*x)*sin(c + d*x)^2)/(b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 5, (2*a*x)/b^3 - (2*(2*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) + (cos(c + d*x)*(2*a + b*sin(c + d*x)))/(b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 8, -((2*b*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d)) - atanh(cos(c + d*x))/(a^2*d) + cos(c + d*x)/(a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 8, -((2*(a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d)) + (2*b*atanh(cos(c + d*x)))/(a^3*d) - (2*cot(c + d*x))/(a^2*d) + cot(c + d*x)/(a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 9, (2*b*(2*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d) + ((a^2 - 6*b^2)*atanh(cos(c + d*x)))/(2*a^4*d) + (3*b*cot(c + d*x))/(a^3*d) - (3*cot(c + d*x)*csc(c + d*x))/(2*a^2*d) + (cot(c + d*x)*csc(c + d*x))/(a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 10, -((2*b^2*(3*a^2 - 4*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*sqrt(a^2 - b^2)*d)) - (b*(a^2 - 4*b^2)*atanh(cos(c + d*x)))/(a^5*d) + ((a^2 - 12*b^2)*cot(c + d*x))/(3*a^4*d) + (2*b*cot(c + d*x)*csc(c + d*x))/(a^3*d) - (4*cot(c + d*x)*csc(c + d*x)^2)/(3*a^2*d) + (cot(c + d*x)*csc(c + d*x)^2)/(a*d*(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 9, -(((12*a^2 - b^2)*x)/(2*b^5)) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(3/2)*d) - (a*(12*a^2 - 11*b^2)*cos(c + d*x))/(2*b^4*(a^2 - b^2)*d) + ((6*a^2 - 5*b^2)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)*d) - (cos(c + d*x)*sin(c + d*x)^3)/(2*b*d*(a + b*sin(c + d*x))^2) - ((4*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(2*b^2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 8, (3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) + (3*cos(c + d*x))/(2*b^3*d) - (cos(c + d*x)*sin(c + d*x)^2)/(2*b*d*(a + b*sin(c + d*x))^2) + (a*(3*a^2 - 2*b^2)*cos(c + d*x))/(2*b^3*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 6, -(x/b^3) + (a*(2*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) - (a*cos(c + d*x)^3)/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (cos(c + d*x)*(2*(a^2 - b^2) + a*b*sin(c + d*x)))/(2*b^2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 8, -((b*(3*a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d)) - atanh(cos(c + d*x))/(a^3*d) + cos(c + d*x)/(2*a*d*(a + b*sin(c + d*x))^2) + ((a^2 - 2*b^2)*cos(c + d*x))/(2*a^2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 9, -(((2*a^4 - 9*a^2*b^2 + 6*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2)*d)) + (3*b*atanh(cos(c + d*x)))/(a^4*d) - ((5*a^2 - 6*b^2)*cot(c + d*x))/(2*a^3*(a^2 - b^2)*d) + cot(c + d*x)/(2*a*d*(a + b*sin(c + d*x))^2) + ((2*a^2 - 3*b^2)*cot(c + d*x))/(2*a^2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 10, (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(3/2)*d) + ((a^2 - 12*b^2)*atanh(cos(c + d*x)))/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*cot(c + d*x))/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*cot(c + d*x)*csc(c + d*x))/(2*a^3*(a^2 - b^2)*d) + (cot(c + d*x)*csc(c + d*x))/(2*a*d*(a + b*sin(c + d*x))^2) + ((3*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(2*a^2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [cos(e + f*x)^2/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^(5/2)), x, 5, (2*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(3*a*d*f*(a + b*sin(e + f*x))^(3/2)) + (4*b*cos(e + f*x))/(3*a*(a^2 - b^2)*f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))) - (4*b*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.E(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(3*a^3*sqrt(a + b)*sqrt(d)*f) - (4*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(3*a^2*sqrt(a + b)*sqrt(d)*f)]

    #= ::Section:: =#
    #=Integrands*of*the*form*cos(e+f*x)^3*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^4*(a + b*sin(c + d*x)), x, 9, (3*a*x)/128 - (b*cos(c + d*x)^5)/(5*d) + (2*b*cos(c + d*x)^7)/(7*d) - (b*cos(c + d*x)^9)/(9*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(128*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(16*d) - (a*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^3*(a + b*sin(c + d*x)), x, 9, (3*b*x)/128 - (a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)^7)/(7*d) + (3*b*cos(c + d*x)*sin(c + d*x))/(128*d) + (b*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (b*cos(c + d*x)^5*sin(c + d*x))/(16*d) - (b*cos(c + d*x)^5*sin(c + d*x)^3)/(8*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + b*sin(c + d*x)), x, 8, (a*x)/16 - (b*cos(c + d*x)^5)/(5*d) + (b*cos(c + d*x)^7)/(7*d) + (a*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + b*sin(c + d*x)), x, 7, (b*x)/16 - (a*cos(c + d*x)^5)/(5*d) + (b*cos(c + d*x)*sin(c + d*x))/(16*d) + (b*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (b*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + b*sin(c + d*x)), x, 8, (3*b*x)/8 - (a*atanh(cos(c + d*x)))/d + (a*cos(c + d*x))/d + (a*cos(c + d*x)^3)/(3*d) + (3*b*cos(c + d*x)*sin(c + d*x))/(8*d) + (b*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + b*sin(c + d*x)), x, 9, -((3*a*x)/2) - (b*atanh(cos(c + d*x)))/d + (b*cos(c + d*x))/d + (b*cos(c + d*x)^3)/(3*d) - (3*a*cot(c + d*x))/(2*d) + (a*cos(c + d*x)^2*cot(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + b*sin(c + d*x)), x, 9, -((3*b*x)/2) + (3*a*atanh(cos(c + d*x)))/(2*d) - (3*a*cos(c + d*x))/(2*d) - (3*b*cot(c + d*x))/(2*d) + (b*cos(c + d*x)^2*cot(c + d*x))/(2*d) - (a*cos(c + d*x)*cot(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + b*sin(c + d*x)), x, 9, a*x + (3*b*atanh(cos(c + d*x)))/(2*d) - (3*b*cos(c + d*x))/(2*d) + (a*cot(c + d*x))/d - (b*cos(c + d*x)*cot(c + d*x)^2)/(2*d) - (a*cot(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + b*sin(c + d*x)), x, 7, b*x - (3*a*atanh(cos(c + d*x)))/(8*d) + (b*cot(c + d*x))/d - (b*cot(c + d*x)^3)/(3*d) + (3*a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + b*sin(c + d*x)), x, 6, -((3*b*atanh(cos(c + d*x)))/(8*d)) - (a*cot(c + d*x)^5)/(5*d) + (3*b*cot(c + d*x)*csc(c + d*x))/(8*d) - (b*cot(c + d*x)^3*csc(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + b*sin(c + d*x)), x, 7, -((a*atanh(cos(c + d*x)))/(16*d)) - (b*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)*csc(c + d*x))/(16*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (a*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + b*sin(c + d*x)), x, 8, -((b*atanh(cos(c + d*x)))/(16*d)) - (a*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)^7)/(7*d) - (b*cot(c + d*x)*csc(c + d*x))/(16*d) + (b*cot(c + d*x)*csc(c + d*x)^3)/(8*d) - (b*cot(c + d*x)^3*csc(c + d*x)^3)/(6*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + b*sin(c + d*x)), x, 9, -((3*a*atanh(cos(c + d*x)))/(128*d)) - (b*cot(c + d*x)^5)/(5*d) - (b*cot(c + d*x)^7)/(7*d) - (3*a*cot(c + d*x)*csc(c + d*x))/(128*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(64*d) + (a*cot(c + d*x)*csc(c + d*x)^5)/(16*d) - (a*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 10, (3*a*b*x)/64 - ((9*a^2 + 4*b^2)*cos(c + d*x))/(105*d) + ((9*a^2 + 4*b^2)*cos(c + d*x)^3)/(315*d) - (3*a*b*cos(c + d*x)*sin(c + d*x))/(64*d) - (a*b*cos(c + d*x)*sin(c + d*x)^3)/(32*d) - ((15*a^4 - 44*a^2*b^2 + 6*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(630*b^2*d) - (a*(10*a^2 - 29*b^2)*cos(c + d*x)*sin(c + d*x)^5)/(504*b*d) - (5*(3*a^2 - 8*b^2)*cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^2)/(252*b^2*d) + (a*cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^3)/(12*b^2*d) - (cos(c + d*x)*sin(c + d*x)^5*(a + b*sin(c + d*x))^3)/(9*b*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 9, (1/128)*(8*a^2 + 3*b^2)*x - (6*a*b*cos(c + d*x))/(35*d) + (2*a*b*cos(c + d*x)^3)/(35*d) - ((8*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(128*d) - ((40*a^4 - 140*a^2*b^2 + 21*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(1344*b^2*d) - (a*(20*a^2 - 69*b^2)*cos(c + d*x)*sin(c + d*x)^4)/(840*b*d) - ((20*a^2 - 63*b^2)*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^2)/(336*b^2*d) + (5*a*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^3)/(56*b^2*d) - (cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^3)/(8*b*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, (a*b*x)/8 - ((a^2 + 6*b^2)*cos(c + d*x)^5)/(105*d) + (a*b*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(12*d) - (a*cos(c + d*x)^5*(a + b*sin(c + d*x)))/(21*d) - (cos(c + d*x)^5*(a + b*sin(c + d*x))^2)/(7*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, (3*a*b*x)/4 - (a^2*atanh(cos(c + d*x)))/d + (a^2*cos(c + d*x))/d + (a^2*cos(c + d*x)^3)/(3*d) - (b^2*cos(c + d*x)^5)/(5*d) + (3*a*b*cos(c + d*x)*sin(c + d*x))/(4*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(2*d), (3*a*b*x)/4 - (a^2*atanh(cos(c + d*x)))/d - ((a^4 - 14*a^2*b^2 + 3*b^4)*cos(c + d*x))/(15*b^2*d) - (a*(2*a^2 - 27*b^2)*cos(c + d*x)*sin(c + d*x))/(60*b*d) - ((a^2 - 12*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(30*b^2*d) + (a*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(10*b^2*d) - (cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^3)/(5*b*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 6, (-(3/8))*(4*a^2 - b^2)*x - (2*a*b*atanh(cos(c + d*x)))/d + (a*(a^2 + 28*b^2)*cos(c + d*x))/(6*b*d) + ((2*a^2 + 39*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((a^2 + 12*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(12*a*b*d) - (cos(c + d*x)*(a + b*sin(c + d*x))^3)/(4*b*d) - (cot(c + d*x)*(a + b*sin(c + d*x))^3)/(a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 6, -3*a*b*x + ((3*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*d) - ((4*a^2 - 23*b^2)*cos(c + d*x))/(6*d) - (b*(a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x))/(3*a*d) - ((2*a^2 - 3*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(6*a^2*d) - (b*cot(c + d*x)*(a + b*sin(c + d*x))^3)/(2*a^2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^3)/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 13, a^2*x - (3*b^2*x)/2 + (3*a*b*atanh(cos(c + d*x)))/d - (3*a*b*cos(c + d*x))/d + (a^2*cot(c + d*x))/d - (3*b^2*cot(c + d*x))/(2*d) + (b^2*cos(c + d*x)^2*cot(c + d*x))/(2*d) - (a*b*cos(c + d*x)*cot(c + d*x)^2)/d - (a^2*cot(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 6, 2*a*b*x - (3*(a^2 - 4*b^2)*atanh(cos(c + d*x)))/(8*d) - (b^2*(39*a^2 + 2*b^2)*cos(c + d*x))/(24*a^2*d) + (17*a*b*cot(c + d*x))/(12*d) + (5*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^2)/(8*d) + (b*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^3)/(12*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^3)/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 6, b^2*x - (3*a*b*atanh(cos(c + d*x)))/(4*d) - ((3*a^4 - 14*a^2*b^2 + b^4)*cot(c + d*x))/(15*a^2*d) + (b*(27*a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x))/(60*a*d) + ((12*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^2)/(30*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^3)/(10*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^3)/(5*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + b*sin(c + d*x))^2, x, 8, -(((a^2 + 6*b^2)*atanh(cos(c + d*x)))/(16*d)) - (2*a*b*cot(c + d*x))/(5*d) - ((15*a^4 - 80*a^2*b^2 + 12*b^4)*cot(c + d*x)*csc(c + d*x))/(240*a^2*d) + (b*(13*a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(60*a*d) + ((35*a^2 - 6*b^2)*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^2)/(120*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^3)/(10*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^3)/(6*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + b*sin(c + d*x))^2, x, 9, -((a*b*atanh(cos(c + d*x)))/(8*d)) - ((2*a^2 + 7*b^2)*cot(c + d*x))/(35*d) - (a*b*cot(c + d*x)*csc(c + d*x))/(8*d) - ((3*a^4 - 18*a^2*b^2 + 4*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(105*a^2*d) + (b*(53*a^2 - 12*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(420*a*d) + (2*(4*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^2)/(35*a^2*d) + (2*b*cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^3)/(21*a^2*d) - (cot(c + d*x)*csc(c + d*x)^6*(a + b*sin(c + d*x))^3)/(7*a*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 10, (1/128)*a*(8*a^2 + 9*b^2)*x - (b*(27*a^2 + 4*b^2)*cos(c + d*x))/(105*d) + (b*(27*a^2 + 4*b^2)*cos(c + d*x)^3)/(315*d) - (a*(8*a^2 + 9*b^2)*cos(c + d*x)*sin(c + d*x))/(128*d) - (a*(40*a^4 - 188*a^2*b^2 + 189*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(4032*b^2*d) - ((20*a^4 - 93*a^2*b^2 + 24*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(2520*b*d) - (a*(20*a^2 - 87*b^2)*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^2)/(1008*b^2*d) - (5*(a^2 - 4*b^2)*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^3)/(126*b^2*d) + (5*a*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^4)/(72*b^2*d) - (cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^4)/(9*b*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 7, (3/128)*b*(8*a^2 + b^2)*x - (a*(2*a^2 + 61*b^2)*cos(c + d*x)^5)/(560*d) + (3*b*(8*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(128*d) + (b*(8*a^2 + b^2)*cos(c + d*x)^3*sin(c + d*x))/(64*d) - ((2*a^2 + 7*b^2)*cos(c + d*x)^5*(a + b*sin(c + d*x)))/(112*d) - (3*a*cos(c + d*x)^5*(a + b*sin(c + d*x))^2)/(56*d) - (cos(c + d*x)^5*(a + b*sin(c + d*x))^3)/(8*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 7, (1/16)*b*(18*a^2 + b^2)*x - (a^3*atanh(cos(c + d*x)))/d - (a*(2*a^4 - 43*a^2*b^2 + 36*b^4)*cos(c + d*x))/(60*b^2*d) - ((4*a^4 - 84*a^2*b^2 + 15*b^4)*cos(c + d*x)*sin(c + d*x))/(240*b*d) - (a*(2*a^2 - 39*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(120*b^2*d) - ((2*a^2 - 35*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(120*b^2*d) + (a*cos(c + d*x)*(a + b*sin(c + d*x))^4)/(15*b^2*d) - (cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^4)/(6*b*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 7, (-(3/8))*a*(4*a^2 - 3*b^2)*x - (3*a^2*b*atanh(cos(c + d*x)))/d + ((a^4 + 56*a^2*b^2 - 2*b^4)*cos(c + d*x))/(10*b*d) + (a*(2*a^2 + 83*b^2)*cos(c + d*x)*sin(c + d*x))/(40*d) + ((a^2 + 28*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(20*b*d) + ((a^2 + 20*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(20*a*b*d) - (cos(c + d*x)*(a + b*sin(c + d*x))^4)/(5*b*d) - (cot(c + d*x)*(a + b*sin(c + d*x))^4)/(a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 7, (-(3/8))*b*(12*a^2 - b^2)*x + (3*a*(a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*d) - (a*(a^2 - 17*b^2)*cos(c + d*x))/(2*d) - (b*(2*a^2 - 21*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) - ((a^2 - 6*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(4*a*d) - ((a^2 - 4*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(4*a^2*d) - (b*cot(c + d*x)*(a + b*sin(c + d*x))^4)/(a^2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^4)/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 17, a^3*x - (9/2)*a*b^2*x + (9*a^2*b*atanh(cos(c + d*x)))/(2*d) - (b^3*atanh(cos(c + d*x)))/d - (9*a^2*b*cos(c + d*x))/(2*d) + (b^3*cos(c + d*x))/d + (b^3*cos(c + d*x)^3)/(3*d) + (a^3*cot(c + d*x))/d - (9*a*b^2*cot(c + d*x))/(2*d) + (3*a*b^2*cos(c + d*x)^2*cot(c + d*x))/(2*d) - (3*a^2*b*cos(c + d*x)*cot(c + d*x)^2)/(2*d) - (a^3*cot(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5*(a + b*sin(c + d*x))^3, x, 7, (3/2)*b*(2*a^2 - b^2)*x - (3*a*(a^2 - 12*b^2)*atanh(cos(c + d*x)))/(8*d) - (b^2*(73*a^2 - 2*b^2)*cos(c + d*x))/(8*a*d) - (13*b^3*cos(c + d*x)*sin(c + d*x))/(4*d) + (17*b*cot(c + d*x)*(a + b*sin(c + d*x))^2)/(8*d) + (5*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^3)/(8*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^4)/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6*(a + b*sin(c + d*x))^3, x, 7, 3*a*b^2*x - (3*b*(3*a^2 - 4*b^2)*atanh(cos(c + d*x)))/(8*d) - (b^3*(83*a^2 + 2*b^2)*cos(c + d*x))/(40*a^2*d) - (a*(4*a^2 - 29*b^2)*cot(c + d*x))/(20*d) + (27*b*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^2)/(40*d) + (2*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^3)/(5*d) + (b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^4)/(20*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^4)/(5*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^7*(a + b*sin(c + d*x))^3, x, 7, b^3*x - (a*(a^2 + 18*b^2)*atanh(cos(c + d*x)))/(16*d) - (b*(36*a^4 - 43*a^2*b^2 + 2*b^4)*cot(c + d*x))/(60*a^2*d) - ((15*a^4 - 84*a^2*b^2 + 4*b^4)*cot(c + d*x)*csc(c + d*x))/(240*a*d) + (b*(39*a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^2)/(120*a^2*d) + ((35*a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^3)/(120*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^4)/(15*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^4)/(6*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^8*(a + b*sin(c + d*x))^3, x, 9, -((3*b*(a^2 + 2*b^2)*atanh(cos(c + d*x)))/(16*d)) - (a*(2*a^2 + 21*b^2)*cot(c + d*x))/(35*d) - (b*(105*a^4 - 116*a^2*b^2 + 12*b^4)*cot(c + d*x)*csc(c + d*x))/(560*a^2*d) - ((4*a^4 - 19*a^2*b^2 + 2*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(140*a*d) + (b*(53*a^2 - 6*b^2)*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^2)/(280*a^2*d) + ((8*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^3)/(35*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^4)/(14*a^2*d) - (cot(c + d*x)*csc(c + d*x)^6*(a + b*sin(c + d*x))^4)/(7*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^9*(a + b*sin(c + d*x))^3, x, 10, -((3*a*(a^2 + 8*b^2)*atanh(cos(c + d*x)))/(128*d)) - (b*(6*a^2 + 7*b^2)*cot(c + d*x))/(35*d) - (3*a*(a^2 + 8*b^2)*cot(c + d*x)*csc(c + d*x))/(128*d) - (b*(24*a^4 - 25*a^2*b^2 + 4*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(280*a^2*d) - ((35*a^4 - 148*a^2*b^2 + 24*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(2240*a*d) + (3*b*(23*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^2)/(560*a^2*d) + ((21*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^3)/(112*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^6*(a + b*sin(c + d*x))^4)/(14*a^2*d) - (cot(c + d*x)*csc(c + d*x)^7*(a + b*sin(c + d*x))^4)/(8*a*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 9, -((3*a*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(4*b^7)) + (6*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*sqrt(a^2 - b^2)*d) - ((30*a^4 - 25*a^2*b^2 + b^4)*cos(c + d*x))/(5*b^6*d) + (3*a*(4*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x))/(4*b^5*d) - ((10*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(5*b^4*d) + ((3*a^2 - 2*b^2)*cos(c + d*x)*sin(c + d*x)^3)/(2*a*b^3*d) - (cos(c + d*x)*sin(c + d*x)^4)/(5*b^2*d) - ((a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(a*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 8, ((40*a^4 - 36*a^2*b^2 + 3*b^4)*x)/(8*b^6) - (2*a*(5*a^4 - 7*a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*sqrt(a^2 - b^2)*d) + (a*(15*a^2 - 11*b^2)*cos(c + d*x))/(3*b^5*d) - ((20*a^2 - 13*b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^4*d) + ((5*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(3*a*b^3*d) - (cos(c + d*x)*sin(c + d*x)^3)/(4*b^2*d) - ((a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(a*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 6, -((a*(4*a^2 - 3*b^2)*x)/b^5) + (2*(4*a^4 - 5*a^2*b^2 + b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) + (cos(c + d*x)^3*(4*a + b*sin(c + d*x)))/(3*b^2*d*(a + b*sin(c + d*x))) - (cos(c + d*x)*(4*a^2 - b^2 - 2*a*b*sin(c + d*x)))/(b^4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 6, -((2*a*x)/b^3) + (2*sqrt(a^2 - b^2)*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^3*d) - atanh(cos(c + d*x))/(a^2*d) - cos(c + d*x)/(b^2*d) - ((a^2 - b^2)*cos(c + d*x))/(a*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 6, x/b^2 - (2*(a^4 + a^2*b^2 - 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^2*sqrt(a^2 - b^2)*d) + (2*b*atanh(cos(c + d*x)))/(a^3*d) + ((a^2 - 2*b^2)*cos(c + d*x))/(a^2*b*d*(a + b*sin(c + d*x))) - cot(c + d*x)/(a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 7, (6*b*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*d) + (3*(a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^4*d) - cos(c + d*x)/(2*a^2*d*(1 - cos(c + d*x)^2)) + (2*b*cot(c + d*x))/(a^3*d) - ((a^2 - b^2)*cos(c + d*x))/(a^3*d*(a + b*sin(c + d*x))), (6*b*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*d) + (3*(a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^4*d) - ((a^2 - 3*b^2)*cot(c + d*x))/(a^3*b*d) + ((2*a^2 - 3*b^2)*cot(c + d*x))/(2*a^2*b*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 8, (2*(a^4 - 5*a^2*b^2 + 4*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*sqrt(a^2 - b^2)*d) - (b*(3*a^2 - 4*b^2)*atanh(cos(c + d*x)))/(a^5*d) + ((7*a^2 - 12*b^2)*cot(c + d*x))/(3*a^4*d) - ((a^2 - 2*b^2)*cot(c + d*x)*csc(c + d*x))/(a^3*b*d) + ((3*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(3*a^2*b*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + b*sin(c + d*x))^2, x, 9, -((2*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*sqrt(a^2 - b^2)*d)) - ((3*a^4 - 36*a^2*b^2 + 40*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) - (b*(11*a^2 - 15*b^2)*cot(c + d*x))/(3*a^5*d) + ((13*a^2 - 20*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^4*d) - ((3*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(3*a^3*b*d) + ((4*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(4*a^2*b*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d*(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 9, (3*(40*a^4 - 24*a^2*b^2 + b^4)*x)/(8*b^7) - (3*a*(10*a^4 - 11*a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*sqrt(a^2 - b^2)*d) + (a*(30*a^2 - 13*b^2)*cos(c + d*x))/(2*b^6*d) - (3*(20*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^5*d) + ((10*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(2*a*b^4*d) - ((15*a^2 - 4*b^2)*cos(c + d*x)*sin(c + d*x)^3)/(4*a^2*b^3*d) - ((a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(2*a*b^2*d*(a + b*sin(c + d*x))^2) + ((7*a^2 - 2*b^2)*cos(c + d*x)*sin(c + d*x)^4)/(2*a^2*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 8, (a*(9 - (20*a^2)/b^2)*x)/(2*b^4) + ((20*a^4 - 19*a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*sqrt(a^2 - b^2)*d) - ((60*a^2 - 17*b^2)*cos(c + d*x))/(6*b^5*d) + ((5*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(a*b^4*d) - ((20*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(6*a^2*b^3*d) - ((a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(2*a*b^2*d*(a + b*sin(c + d*x))^2) + ((6*a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(2*a^2*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 6, (3*(4*a^2 - b^2)*x)/(2*b^5) - (3*a*(4*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) + (cos(c + d*x)^3*(2*a + b*sin(c + d*x)))/(2*b^2*d*(a + b*sin(c + d*x))^2) + (3*cos(c + d*x)*(4*a^2 - b^2 + 2*a*b*sin(c + d*x)))/(2*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 6, x/b^3 - ((2*a^4 - a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^3*sqrt(a^2 - b^2)*d) - atanh(cos(c + d*x))/(a^3*d) - ((a^2 - b^2)*cos(c + d*x))/(2*a*b^2*d*(a + b*sin(c + d*x))^2) + ((3*a^2 + 2*b^2)*cos(c + d*x))/(2*a^2*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 7, -((3*(a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d)) + (3*b*atanh(cos(c + d*x)))/(a^4*d) + ((a^2 - 3*b^2)*cos(c + d*x))/(2*a^2*b*d*(a + b*sin(c + d*x))^2) - cot(c + d*x)/(a*d*(a + b*sin(c + d*x))^2) - ((a^2 + 6*b^2)*cos(c + d*x))/(2*a^3*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 8, (3*b*(3*a^2 - 4*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*sqrt(a^2 - b^2)*d) + (3*(a^2 - 4*b^2)*atanh(cos(c + d*x)))/(2*a^5*d) - ((a^2 - 12*b^2)*cot(c + d*x))/(2*a^4*b*d) + ((a^2 - 2*b^2)*cot(c + d*x))/(2*a^2*b*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*(a + b*sin(c + d*x))^2) - (3*b*cot(c + d*x))/(a^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + b*sin(c + d*x))^3, x, 9, ((2*a^4 - 19*a^2*b^2 + 20*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*sqrt(a^2 - b^2)*d) - (b*(9*a^2 - 20*b^2)*atanh(cos(c + d*x)))/(2*a^6*d) + ((17*a^2 - 60*b^2)*cot(c + d*x))/(6*a^5*d) - ((a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x))/(a^4*b*d) + ((3*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x))/(6*a^2*b*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*(a + b*sin(c + d*x))^2) + ((3*a^2 - 20*b^2)*cot(c + d*x)*csc(c + d*x))/(6*a^3*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + b*sin(c + d*x))^3, x, 10, -((3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^7*sqrt(a^2 - b^2)*d)) - (3*(a^4 - 24*a^2*b^2 + 40*b^4)*atanh(cos(c + d*x)))/(8*a^7*d) - (b*(13*a^2 - 30*b^2)*cot(c + d*x))/(2*a^6*d) + (3*(7*a^2 - 20*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^5*d) - ((3*a^2 - 10*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(2*a^4*b*d) + ((2*a^2 - 3*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(4*a^2*b*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d*(a + b*sin(c + d*x))^2) + ((4*a^2 - 15*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(4*a^3*b*d*(a + b*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)), x, 10, (16*a*(160*a^4 - 279*a^2*b^2 + 27*b^4)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(45045*b^5*d) - (8*(480*a^4 - 937*a^2*b^2 + 231*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(45045*b^5*d) + (8*a*(40*a^2 - 81*b^2)*cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(3003*b^4*d) - (10*(16*a^2 - 33*b^2)*cos(c + d*x)*sin(c + d*x)^2*(a + b*sin(c + d*x))^(3/2))/(1287*b^3*d) + (20*a*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(143*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^(3/2))/(13*b*d) - (8*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*b^6)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(45045*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (16*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(45045*b^6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*sqrt(a + b*sin(c + d*x)), x, 8, (-2*cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(11*d) + (8*a*(32*a^4 - 93*a^2*b^2 + 93*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3465*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3465*b^5*d*sqrt(a + b*sin(c + d*x))) - (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(8*a^2 - 9*b^2 - 7*a*b*sin(c + d*x)))/(693*b^2*d) + (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(32*a^4 - 69*a^2*b^2 + 45*b^4 - 24*a*b*(a^2 - 2*b^2)*sin(c + d*x)))/(3465*b^4*d)]
    @test_int [cos(c + d*x)^3*cot(c + d*x)^1*sqrt(a + b*sin(c + d*x)), x, 10, (-2*(8*a^2 - 45*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(105*b^2*d) + (8*a*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(35*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(7*b*d) + (2*a*(8*a^2 - 51*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(105*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*(8*a^4 - 53*a^2*b^2 - 60*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(105*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*a*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*cot(c + d*x)^2*sqrt(a + b*sin(c + d*x)), x, 10, ((4*a^2 + 15*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15*a*b*d) - (2*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(5*b*d) - (cot(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(a*d) - ((4*a^2 + 57*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (a*(4*a^2 + 11*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*b^2*d*sqrt(a + b*sin(c + d*x))) + (b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1*cot(c + d*x)^3*sqrt(a + b*sin(c + d*x)), x, 10, -((8*a^2 + 3*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(12*a^2*d) + (b*cot(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(2*a*d) + ((8*a^2 - 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(12*a*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((8*a^2 + 31*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(12*b*d*sqrt(a + b*sin(c + d*x))) - ((12*a^2 + b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^0*cot(c + d*x)^4*sqrt(a + b*sin(c + d*x)), x, 10, ((32*a^2 - 3*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(24*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(3/2))/(3*a*d) + ((80*a^2 + 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(24*a^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((32*a^2 + b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(24*a*d*sqrt(a + b*sin(c + d*x))) - (b*(12*a^2 - b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^1*sqrt(a + b*sin(c + d*x)), x, 11, (b*(68*a^2 - 15*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(192*a^3*d) + (5*(4*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(32*a^2*d) + (5*b*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(3/2))/(24*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(4*a*d) + (b*(68*a^2 - 15*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(192*a^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (b*(196*a^2 + 5*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(192*a^2*d*sqrt(a + b*sin(c + d*x))) + ((48*a^4 + 24*a^2*b^2 - 5*b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*a^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^2*sqrt(a + b*sin(c + d*x)), x, 12, -((384*a^4 + 332*a^2*b^2 - 105*b^4)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(1920*a^4*d) + (b*(108*a^2 - 35*b^2)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(960*a^3*d) + ((96*a^2 - 35*b^2)*cot(c + d*x)*csc(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(240*a^2*d) + (7*b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(40*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^(3/2))/(5*a*d) - ((384*a^4 + 332*a^2*b^2 - 105*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(1920*a^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((384*a^4 + 116*a^2*b^2 - 35*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(1920*a^3*d*sqrt(a + b*sin(c + d*x))) + (b*(48*a^4 - 24*a^2*b^2 + 7*b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(128*a^4*d*sqrt(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x, 11, (8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(45045*b^5*d) + (16*a*(32*a^4 - 47*a^2*b^2 - 27*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*b^2 + 117*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*cos(c + d*x)*sin(c + d*x)^2*(a + b*sin(c + d*x))^(5/2))/(2145*b^3*d) + (4*a*cos(c + d*x)*sin(c + d*x)^3*(a + b*sin(c + d*x))^(5/2))/(39*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)^4*(a + b*sin(c + d*x))^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^4 - 471*b^6)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(45045*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(45045*b^6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + b*sin(c + d*x))^(3/2), x, 9, (-6*a*cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(143*d) - (2*cos(c + d*x)^5*(a + b*sin(c + d*x))^(3/2))/(13*d) + (8*(32*a^6 - 137*a^4*b^2 + 258*a^2*b^4 + 231*b^6)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15015*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*a*(32*a^6 - 145*a^4*b^2 + 290*a^2*b^4 - 177*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15015*b^5*d*sqrt(a + b*sin(c + d*x))) - (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(4*a*(2*a^2 - 5*b^2) - 7*b*(a^2 + 11*b^2)*sin(c + d*x)))/(3003*b^2*d) + (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*(32*a^4 - 113*a^2*b^2 + 177*b^4) - 3*b*(8*a^4 - 27*a^2*b^2 - 77*b^4)*sin(c + d*x)))/(15015*b^4*d)]
    @test_int [cos(c + d*x)^3*cot(c + d*x)^1*(a + b*sin(c + d*x))^(3/2), x, 11, (-2*a*(8*a^2 - 87*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b^2*d) - (2*(8*a^2 - 77*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(315*b^2*d) + (8*a*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(63*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(9*b*d) + (2*(8*a^4 - 93*a^2*b^2 + 84*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*a*(8*a^4 - 95*a^2*b^2 - 228*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*a^2*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*cot(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x, 11, ((4*a^2 + 65*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(35*b*d) + ((4*a^2 + 35*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(35*a*b*d) - (2*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(7*b*d) - (cot(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(a*d) - (a*(4*a^2 + 167*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(35*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^4 + 61*a^2*b^2 + 40*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(35*b^2*d*sqrt(a + b*sin(c + d*x))) + (3*a*b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1*cot(c + d*x)^3*(a + b*sin(c + d*x))^(3/2), x, 11, -((8*a^2 - 15*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(20*a*d) - ((8*a^2 - 5*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(20*a^2*d) - (b*cot(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(2*a*d) + ((8*a^2 - 81*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(20*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (a*(8*a^2 + 37*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(20*b*d*sqrt(a + b*sin(c + d*x))) - (3*(4*a^2 - b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^0*cot(c + d*x)^4*(a + b*sin(c + d*x))^(3/2), x, 11, -(b*(16*a^2 + b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(8*a^2*d) + ((32*a^2 + b^2)*cot(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(24*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(12*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(5/2))/(3*a*d) + ((32*a^2 - b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(8*a*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((16*a^2 + 21*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*d*sqrt(a + b*sin(c + d*x))) - (b*(36*a^2 + b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^1*(a + b*sin(c + d*x))^(3/2), x, 11, (b*(68*a^2 - 3*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(64*a^2*d) + ((20*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(32*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(5/2))/(8*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(5/2))/(4*a*d) + (b*(236*a^2 + 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(64*a^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (b*(20*a^2 + b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*a*d*sqrt(a + b*sin(c + d*x))) + (3*(16*a^4 - 24*a^2*b^2 + b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x, 12, -((128*a^4 - 116*a^2*b^2 + 15*b^4)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(640*a^3*d) + (3*b*(36*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(320*a^2*d) + ((32*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(3/2))/(80*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(5/2))/(8*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^(5/2))/(5*a*d) - ((128*a^4 - 116*a^2*b^2 + 15*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(640*a^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((128*a^4 + 692*a^2*b^2 + 5*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(640*a^2*d*sqrt(a + b*sin(c + d*x))) + (3*b*(48*a^4 + 8*a^2*b^2 - b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(128*a^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^3*(a + b*sin(c + d*x))^(3/2), x, 13, -(b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(7680*a^4*d) - ((240*a^4 - 168*a^2*b^2 + 35*b^4)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3840*a^3*d) + (b*(156*a^2 - 35*b^2)*cot(c + d*x)*csc(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(960*a^2*d) + (7*(4*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(96*a^2*d) + (7*b*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^(5/2))/(60*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^(5/2))/(6*a*d) - (b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(7680*a^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (b*(2544*a^4 + 176*a^2*b^2 - 35*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(7680*a^3*d*sqrt(a + b*sin(c + d*x))) + ((64*a^6 + 144*a^4*b^2 - 36*a^2*b^4 + 7*b^6)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(512*a^4*d*sqrt(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^1*(a + b*sin(c + d*x))^(5/2), x, 10, (-2*(3*a^2 + 13*b^2)*cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(429*d) - (2*a*cos(c + d*x)^5*(a + b*sin(c + d*x))^(3/2))/(39*d) - (2*cos(c + d*x)^5*(a + b*sin(c + d*x))^(5/2))/(15*d) + (8*a*(32*a^6 - 189*a^4*b^2 + 570*a^2*b^4 + 1635*b^6)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(45045*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(45045*b^5*d*sqrt(a + b*sin(c + d*x))) - (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(8*a^4 - 33*a^2*b^2 - 39*b^4 - 7*a*b*(a^2 + 63*b^2)*sin(c + d*x)))/(9009*b^2*d) + (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(32*a^6 - 165*a^4*b^2 + 450*a^2*b^4 + 195*b^6 - 24*a*b*(a^4 - 5*a^2*b^2 - 60*b^4)*sin(c + d*x)))/(45045*b^4*d)]
    @test_int [cos(c + d*x)^3*cot(c + d*x)^1*(a + b*sin(c + d*x))^(5/2), x, 12, (-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(693*b^2*d) - (2*a*(8*a^2 - 131*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(693*b^2*d) - (2*(8*a^2 - 117*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(693*b^2*d) + (8*a*cos(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(99*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(11*b*d) + (2*a*(8*a^4 - 147*a^2*b^2 + 444*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(693*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*(8*a^6 - 149*a^4*b^2 - 516*a^2*b^4 - 36*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(693*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*a^3*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2*cot(c + d*x)^2*(a + b*sin(c + d*x))^(5/2), x, 12, (a*(20*a^2 + 759*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b*d) + ((20*a^2 + 469*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(315*b*d) + ((4*a^2 + 63*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(63*a*b*d) - (2*cos(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(9*b*d) - (cot(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(a*d) - ((20*a^4 + 1689*a^2*b^2 - 168*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (a*(20*a^4 + 739*a^2*b^2 + 816*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^2*d*sqrt(a + b*sin(c + d*x))) + (5*a^2*b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1*cot(c + d*x)^3*(a + b*sin(c + d*x))^(5/2), x, 12, -((8*a^2 - 73*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(28*d) - ((8*a^2 - 35*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(28*a*d) - ((8*a^2 - 21*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(28*a^2*d) - (3*b*cot(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(2*a*d) + (a*(8*a^2 - 247*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(28*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((8*a^4 + 3*a^2*b^2 - 32*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(28*b*d*sqrt(a + b*sin(c + d*x))) - (3*a*(4*a^2 - 5*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^0*cot(c + d*x)^4*(a + b*sin(c + d*x))^(5/2), x, 12, -(b*(96*a^2 - 25*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(40*a*d) - (b*(208*a^2 - 25*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(120*a^2*d) + ((32*a^2 - 3*b^2)*cot(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(24*a^2*d) - (b*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(7/2))/(12*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(7/2))/(3*a*d) + ((176*a^2 - 167*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(40*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (a*(96*a^2 + 179*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(40*d*sqrt(a + b*sin(c + d*x))) - (5*b*(12*a^2 - b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^1*(a + b*sin(c + d*x))^(5/2), x, 12, -(b^2*(196*a^2 + 5*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(64*a^2*d) + (5*b*(68*a^2 + b^2)*cot(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(5/2))/(96*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(7/2))/(24*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(64*a*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (b*(148*a^2 + 169*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*d*sqrt(a + b*sin(c + d*x))) + ((48*a^4 - 360*a^2*b^2 - 5*b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*a*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^2*(a + b*sin(c + d*x))^(5/2), x, 12, -((128*a^4 - 580*a^2*b^2 + 15*b^4)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(640*a^2*d) + (b*(36*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(64*a^2*d) + ((32*a^2 - b^2)*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(5/2))/(80*a^2*d) + (3*b*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(7/2))/(40*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^(7/2))/(5*a*d) - ((128*a^4 - 2476*a^2*b^2 - 15*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(640*a^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((128*a^4 + 492*a^2*b^2 - 5*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(640*a*d*sqrt(a + b*sin(c + d*x))) + (3*b*(80*a^4 - 40*a^2*b^2 + b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(128*a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cot(c + d*x)^4*csc(c + d*x)^3*(a + b*sin(c + d*x))^(5/2), x, 13, -(b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(1536*a^3*d) - ((16*a^4 - 56*a^2*b^2 + 5*b^4)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(256*a^2*d) + (b*(52*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2*(a + b*sin(c + d*x))^(3/2))/(192*a^2*d) + ((28*a^2 - 3*b^2)*cot(c + d*x)*csc(c + d*x)^3*(a + b*sin(c + d*x))^(5/2))/(96*a^2*d) + (b*cot(c + d*x)*csc(c + d*x)^4*(a + b*sin(c + d*x))^(7/2))/(12*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5*(a + b*sin(c + d*x))^(7/2))/(6*a*d) - (b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(1536*a^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (b*(816*a^4 + 1696*a^2*b^2 + 5*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(1536*a^2*d*sqrt(a + b*sin(c + d*x))) + ((64*a^6 + 720*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(512*a^3*d*sqrt(a + b*sin(c + d*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(cos(c + d*x)^4*sin(c + d*x)^3)/sqrt(a + b*sin(c + d*x)), x, 10, (64*a*(80*a^4 - 118*a^2*b^2 + 17*b^4)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15015*b^6*d) - (8*(480*a^4 - 683*a^2*b^2 + 77*b^4)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15015*b^5*d) + (4*a*(160*a^2 - 223*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(3003*b^4*d) - (10*(8*a^2 - 11*b^2)*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(429*b^3*d) + (24*a*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + b*sin(c + d*x)))/(143*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(13*b*d) + (8*(1280*a^6 - 2048*a^4*b^2 + 453*a^2*b^4 + 231*b^6)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15015*b^7*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*a*(1280*a^6 - 2368*a^4*b^2 + 875*a^2*b^4 + 213*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15015*b^7*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^2)/sqrt(a + b*sin(c + d*x)), x, 9, (-8*(160*a^4 - 247*a^2*b^2 + 45*b^4)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3465*b^5*d) + (8*a*(120*a^2 - 179*b^2)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3465*b^4*d) - (2*(80*a^2 - 117*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(693*b^3*d) + (20*a*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(99*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + b*sin(c + d*x)))/(11*b*d) - (16*a*(160*a^4 - 267*a^2*b^2 + 69*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3465*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3465*b^6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^1)/sqrt(a + b*sin(c + d*x)), x, 7, (-2*cos(c + d*x)^3*(8*a - 7*b*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(63*b^2*d) + (8*(32*a^4 - 57*a^2*b^2 + 21*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^5*d*sqrt(a + b*sin(c + d*x))) + (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*sin(c + d*x)))/(315*b^4*d)]
    @test_int [(cos(c + d*x)^3*cot(c + d*x)^1)/sqrt(a + b*sin(c + d*x)), x, 9, (8*a*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15*b^2*d) - (2*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(5*b*d) + (2*(8*a^2 - 21*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*a*(8*a^2 - 23*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^2*cot(c + d*x)^2)/sqrt(a + b*sin(c + d*x)), x, 9, (-2*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3*b*d) - (cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(a*d) - ((4*a^2 + 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*a*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^2 - 7*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*b^2*d*sqrt(a + b*sin(c + d*x))) - (b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^1*cot(c + d*x)^3)/sqrt(a + b*sin(c + d*x)), x, 9, (3*b*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(2*a*d) + ((8*a^2 + 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(4*a^2*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((8*a^2 + b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a*b*d*sqrt(a + b*sin(c + d*x))) - (3*(4*a^2 - b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^0*cot(c + d*x)^4)/sqrt(a + b*sin(c + d*x)), x, 10, ((32*a^2 - 15*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(24*a^3*d) + (5*b*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(12*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(3*a*d) + ((32*a^2 - 15*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(24*a^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((16*a^2 + 5*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(24*a^2*d*sqrt(a + b*sin(c + d*x))) + (b*(12*a^2 - 5*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cot(c + d*x)^4*csc(c + d*x)^1)/sqrt(a + b*sin(c + d*x)), x, 11, -(b*(188*a^2 - 105*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(192*a^4*d) + (5*(12*a^2 - 7*b^2)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(96*a^3*d) + (7*b*cot(c + d*x)*csc(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(24*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(4*a*d) - (b*(188*a^2 - 105*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(192*a^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (b*(68*a^2 - 35*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(192*a^3*d*sqrt(a + b*sin(c + d*x))) + ((48*a^4 - 72*a^2*b^2 + 35*b^4)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(64*a^4*d*sqrt(a + b*sin(c + d*x)))]

    @test_int [(cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(3/2), x, 10, (-2*(a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(a*b^2*d*sqrt(a + b*sin(c + d*x))) - (8*(640*a^4 - 592*a^2*b^2 + 15*b^4)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(1155*b^6*d) + (8*a*(480*a^2 - 419*b^2)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(1155*b^5*d) - (20*(32*a^2 - 27*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(231*b^4*d) + (2*(40*a^2 - 33*b^2)*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(33*a*b^3*d) - (2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + b*sin(c + d*x)))/(11*b^2*d) - (8*a*(1280*a^4 - 1344*a^2*b^2 + 123*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(1155*b^7*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(1155*b^7*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(3/2), x, 9, (-2*(a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(a*b^2*d*sqrt(a + b*sin(c + d*x))) + (8*a*(160*a^2 - 139*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b^5*d) - (16*(60*a^2 - 49*b^2)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b^4*d) + (2*(80*a^2 - 63*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(63*a*b^3*d) - (2*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(9*b^2*d) + (8*(320*a^4 - 318*a^2*b^2 + 21*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (16*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^1)/(a + b*sin(c + d*x))^(3/2), x, 7, (-8*a*(32*a^2 - 29*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(35*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(32*a^4 - 37*a^2*b^2 + 5*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(35*b^5*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)^3*(8*a + b*sin(c + d*x)))/(7*b^2*d*sqrt(a + b*sin(c + d*x))) - (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(32*a^2 - 5*b^2 - 24*a*b*sin(c + d*x)))/(35*b^4*d)]
    @test_int [(cos(c + d*x)^3*cot(c + d*x)^1)/(a + b*sin(c + d*x))^(3/2), x, 9, (-2*(a^2 - b^2)*cos(c + d*x))/(a*b^2*d*sqrt(a + b*sin(c + d*x))) - (2*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3*b^2*d) - (2*(8*a^2 - 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*a*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (2*(8*a^2 - 5*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x))^(3/2), x, 9, ((2*a^2 - 3*b^2)*cos(c + d*x))/(a^2*b*d*sqrt(a + b*sin(c + d*x))) - cot(c + d*x)/(a*d*sqrt(a + b*sin(c + d*x))) + ((4*a^2 - 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(a^2*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((4*a^2 - b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a*b^2*d*sqrt(a + b*sin(c + d*x))) - (3*b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^1*cot(c + d*x)^3)/(a + b*sin(c + d*x))^(3/2), x, 10, ((4*a^2 - 5*b^2)*cot(c + d*x))/(2*a^2*b*d*sqrt(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*sqrt(a + b*sin(c + d*x))) - ((8*a^2 - 15*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(4*a^3*b*d) - ((8*a^2 - 15*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(4*a^3*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((8*a^2 - 5*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a^2*b*d*sqrt(a + b*sin(c + d*x))) - (3*(4*a^2 - 5*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^0*cot(c + d*x)^4)/(a + b*sin(c + d*x))^(3/2), x, 11, ((6*a^2 - 7*b^2)*cot(c + d*x)*csc(c + d*x))/(3*a^2*b*d*sqrt(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*sqrt(a + b*sin(c + d*x))) + (5*(16*a^2 - 21*b^2)*cot(c + d*x)*sqrt(a + b*sin(c + d*x)))/(24*a^4*d) - ((24*a^2 - 35*b^2)*cot(c + d*x)*csc(c + d*x)*sqrt(a + b*sin(c + d*x)))/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(24*a^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((32*a^2 - 35*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(24*a^3*d*sqrt(a + b*sin(c + d*x))) + (b*(36*a^2 - 35*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a^4*d*sqrt(a + b*sin(c + d*x)))]

    @test_int [(cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2), x, 10, (-2*(a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^4)/(3*a*b^2*d*(a + b*sin(c + d*x))^(3/2)) + (2*(13*a^2 - 5*b^2)*cos(c + d*x)*sin(c + d*x)^4)/(3*a^2*b^2*d*sqrt(a + b*sin(c + d*x))) + (128*a*(40*a^2 - 19*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b^6*d) - (8*(480*a^2 - 203*b^2)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(315*b^5*d) + (4*(160*a^2 - 63*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(63*a*b^4*d) - (10*(8*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(9*a^2*b^3*d) + (8*(1280*a^4 - 768*a^2*b^2 + 21*b^4)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^7*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*a*(1280*a^4 - 1088*a^2*b^2 + 123*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^7*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(5/2), x, 9, (-2*(a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^3)/(3*a*b^2*d*(a + b*sin(c + d*x))^(3/2)) + (2*(11*a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x)^3)/(3*a^2*b^2*d*sqrt(a + b*sin(c + d*x))) - (8*(32*a^2 - 11*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(21*b^5*d) + (8*(24*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x)*sqrt(a + b*sin(c + d*x)))/(21*a*b^4*d) - (2*(80*a^2 - 21*b^2)*cos(c + d*x)*sin(c + d*x)^2*sqrt(a + b*sin(c + d*x)))/(21*a^2*b^3*d) - (16*a*(32*a^2 - 15*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(21*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(64*a^4 - 46*a^2*b^2 + 3*b^4)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(21*b^6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^4*sin(c + d*x)^1)/(a + b*sin(c + d*x))^(5/2), x, 7, (8*(32*a^2 - 9*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*b^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (8*a*(32*a^2 - 17*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*b^5*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)^3*(8*a + 3*b*sin(c + d*x)))/(15*b^2*d*(a + b*sin(c + d*x))^(3/2)) + (4*cos(c + d*x)*(32*a^2 - 9*b^2 + 8*a*b*sin(c + d*x)))/(15*b^4*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^3*cot(c + d*x)^1)/(a + b*sin(c + d*x))^(5/2), x, 9, (-2*(a^2 - b^2)*cos(c + d*x))/(3*a*b^2*d*(a + b*sin(c + d*x))^(3/2)) + (2*(5*a^2 + 3*b^2)*cos(c + d*x))/(3*a^2*b^2*d*sqrt(a + b*sin(c + d*x))) + (2*(8*a^2 + 3*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*a^2*b^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*(8*a^2 + b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*a*b^3*d*sqrt(a + b*sin(c + d*x))) + (2*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x))^(5/2), x, 10, ((2*a^2 - 5*b^2)*cos(c + d*x))/(3*a^2*b*d*(a + b*sin(c + d*x))^(3/2)) - cot(c + d*x)/(a*d*(a + b*sin(c + d*x))^(3/2)) - ((4*a^2 + 15*b^2)*cos(c + d*x))/(3*a^3*b*d*sqrt(a + b*sin(c + d*x))) - ((4*a^2 + 15*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*a^3*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^2 + 5*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*a^2*b^2*d*sqrt(a + b*sin(c + d*x))) - (5*b*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(a^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^1*cot(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2), x, 11, ((4*a^2 - 7*b^2)*cot(c + d*x))/(6*a^2*b*d*(a + b*sin(c + d*x))^(3/2)) - (cot(c + d*x)*csc(c + d*x))/(2*a*d*(a + b*sin(c + d*x))^(3/2)) - ((8*a^2 - 105*b^2)*cos(c + d*x))/(12*a^4*d*sqrt(a + b*sin(c + d*x))) - ((8*a^2 - 35*b^2)*cot(c + d*x))/(12*a^3*b*d*sqrt(a + b*sin(c + d*x))) - ((8*a^2 - 105*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(12*a^4*b*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((8*a^2 - 35*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(12*a^3*b*d*sqrt(a + b*sin(c + d*x))) - ((12*a^2 - 35*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(4*a^4*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^0*cot(c + d*x)^4)/(a + b*sin(c + d*x))^(5/2), x, 12, ((2*a^2 - 3*b^2)*cot(c + d*x)*csc(c + d*x))/(3*a^2*b*d*(a + b*sin(c + d*x))^(3/2)) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*(a + b*sin(c + d*x))^(3/2)) + (b*(32*a^2 - 105*b^2)*cos(c + d*x))/(8*a^5*d*sqrt(a + b*sin(c + d*x))) + ((16*a^2 - 35*b^2)*cot(c + d*x))/(8*a^4*d*sqrt(a + b*sin(c + d*x))) - ((8*a^2 - 21*b^2)*cot(c + d*x)*csc(c + d*x))/(12*a^3*b*d*sqrt(a + b*sin(c + d*x))) + ((32*a^2 - 105*b^2)*Elliptic.E((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(8*a^5*d*sqrt((a + b*sin(c + d*x))/(a + b))) - ((16*a^2 - 35*b^2)*Elliptic.F((c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a^4*d*sqrt(a + b*sin(c + d*x))) + (15*b*(4*a^2 - 7*b^2)*Elliptic.Pi(2, (c - pi/2 + d*x)/2, (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(8*a^5*d*sqrt(a + b*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [cos(e + f*x)^4/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^(9/2)), x, 8, (2*cos(e + f*x)^3*sqrt(d*sin(e + f*x)))/(7*a*d*f*(a + b*sin(e + f*x))^(7/2)) + (12*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(35*a^2*d*f*(a + b*sin(e + f*x))^(5/2)) + (8*(a^2 - 2*b^2)*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(35*a^3*(a^2 - b^2)*d*f*(a + b*sin(e + f*x))^(3/2)) + (32*b*(2*a^2 - b^2)*cos(e + f*x))/(35*a^3*(a^2 - b^2)^2*f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))) - (32*b*(2*a^2 - b^2)*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.E(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(35*a^5*(a - b)*(a + b)^(3/2)*sqrt(d)*f) - (8*(5*a^2 - 3*a*b - 4*b^2)*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(35*a^4*(a - b)*(a + b)^(3/2)*sqrt(d)*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*(d*sin(e+f*x))^(n/3)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [(cos(c + d*x)^4*sin(c + d*x)^(1/3))/sqrt(a + b*sin(c + d*x)), x, 0, Unintegrable((cos(c + d*x)^4*sin(c + d*x)^(1/3))/sqrt(a + b*sin(c + d*x)), x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^4*sin(e+f*x)^n*(a+b*sin(e+f*x))^m*with*n*and/or*p*symbolic=#

    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^p, x, 0, Unintegrable(cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^p, x)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^(-3 - p)*(a + b*sin(c + d*x))^p, x, 0, Unintegrable(cos(c + d*x)^4*sin(c + d*x)^(-3 - p)*(a + b*sin(c + d*x))^p, x)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^(-4 - p)*(a + b*sin(c + d*x))^p, x, 0, Unintegrable(cos(c + d*x)^4*sin(c + d*x)^(-4 - p)*(a + b*sin(c + d*x))^p, x)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^3, x, 8, If(13>=8, -((3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n))) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*sqrt(cos(c + d*x)^2)) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*cos(c + d*x)*sin(c + d*x)^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2*(7 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*(7 + n)*sqrt(cos(c + d*x)^2)) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^3)/(b^2*d*(5 + n)*(6 + n)*(7 + n)) + (a*(3 + n)*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^4)/(b^2*d*(6 + n)*(7 + n)) - (cos(c + d*x)*sin(c + d*x)^(2 + n)*(a + b*sin(c + d*x))^4)/(b*d*(7 + n)), -((3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n))/(b^2*d*(5 + n)*(6 + n)*(7 + n)*(8 + 6*n + n^2))) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(6 + n)*(8 + 14*n + 7*n^2 + n^3)*sqrt(cos(c + d*x)^2)) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*cos(c + d*x)*sin(c + d*x)^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2*(7 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(5 + n)*(7 + n)*(6 + 5*n + n^2)*sqrt(cos(c + d*x)^2)) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^3)/(b^2*d*(5 + n)*(6 + n)*(7 + n)) + (a*(3 + n)*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^4)/(b^2*d*(6 + n)*(7 + n)) - (cos(c + d*x)*sin(c + d*x)^(2 + n)*(a + b*sin(c + d*x))^4)/(b*d*(7 + n)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^2, x, 7, If(13>=8, -(((3*b^4*(5 + n) + 2*a^4*(6 + 5*n + n^2) - 2*a^2*b^2*(40 + 13*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n))) + (3*(b^2*(1 + n) + a^2*(6 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*sqrt(cos(c + d*x)^2)) - (2*a*(a^2*(6 + 5*n + n^2) - b^2*(39 + 13*n + n^2))*cos(c + d*x)*sin(c + d*x)^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)) + (6*a*b*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*sqrt(cos(c + d*x)^2)) - ((a^2*(2 + n)*(3 + n) - b^2*(5 + n)*(7 + n))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)) + (a*(3 + n)*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^3)/(b^2*d*(5 + n)*(6 + n)) - (cos(c + d*x)*sin(c + d*x)^(2 + n)*(a + b*sin(c + d*x))^3)/(b*d*(6 + n)), -(((3*b^4*(5 + n) + 2*a^4*(6 + 5*n + n^2) - 2*a^2*b^2*(40 + 13*n + n^2))*cos(c + d*x)*sin(c + d*x)^(1 + n))/(b^2*d*(5 + n)*(6 + n)*(8 + 6*n + n^2))) + (3*(b^2*(1 + n) + a^2*(6 + n))*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(4 + n)*(6 + n)*(2 + 3*n + n^2)*sqrt(cos(c + d*x)^2)) - (2*a*(a^2*(6 + 5*n + n^2) - b^2*(39 + 13*n + n^2))*cos(c + d*x)*sin(c + d*x)^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)) + (6*a*b*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(5 + n)*(6 + 5*n + n^2)*sqrt(cos(c + d*x)^2)) - ((a^2*(2 + n)*(3 + n) - b^2*(5 + n)*(7 + n))*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)) + (a*(3 + n)*cos(c + d*x)*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^3)/(b^2*d*(5 + n)*(6 + n)) - (cos(c + d*x)*sin(c + d*x)^(2 + n)*(a + b*sin(c + d*x))^3)/(b*d*(6 + n)))]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^1, x, 3, (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)*sqrt(cos(c + d*x)^2)) + (b*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n)*sqrt(cos(c + d*x)^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^5*(a + b*sin(c + d*x)), x, 4, (a*sin(c + d*x)^6)/(6*d) + (b*sin(c + d*x)^7)/(7*d) - (a*sin(c + d*x)^8)/(4*d) - (2*b*sin(c + d*x)^9)/(9*d) + (a*sin(c + d*x)^10)/(10*d) + (b*sin(c + d*x)^11)/(11*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^4*(a + b*sin(c + d*x)), x, 4, (a*sin(c + d*x)^5)/(5*d) + (b*sin(c + d*x)^6)/(6*d) - (2*a*sin(c + d*x)^7)/(7*d) - (b*sin(c + d*x)^8)/(4*d) + (a*sin(c + d*x)^9)/(9*d) + (b*sin(c + d*x)^10)/(10*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x)), x, 7, -((a*cos(c + d*x)^6)/(6*d)) + (a*cos(c + d*x)^8)/(8*d) + (b*sin(c + d*x)^5)/(5*d) - (2*b*sin(c + d*x)^7)/(7*d) + (b*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2*(a + b*sin(c + d*x)), x, 7, -((b*cos(c + d*x)^6)/(6*d)) + (b*cos(c + d*x)^8)/(8*d) + (a*sin(c + d*x)^3)/(3*d) - (2*a*sin(c + d*x)^5)/(5*d) + (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1*(a + b*sin(c + d*x)), x, 6, -((a*cos(c + d*x)^6)/(6*d)) + (b*sin(c + d*x)^3)/(3*d) - (2*b*sin(c + d*x)^5)/(5*d) + (b*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1*(a + b*sin(c + d*x)), x, 4, (a*log(sin(c + d*x)))/d + (b*sin(c + d*x))/d - (a*sin(c + d*x)^2)/d - (2*b*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^4)/(4*d) + (b*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2*(a + b*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (b*log(sin(c + d*x)))/d - (2*a*sin(c + d*x))/d - (b*sin(c + d*x)^2)/d + (a*sin(c + d*x)^3)/(3*d) + (b*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3*(a + b*sin(c + d*x)), x, 4, -((b*csc(c + d*x))/d) - (a*csc(c + d*x)^2)/(2*d) - (2*a*log(sin(c + d*x)))/d - (2*b*sin(c + d*x))/d + (a*sin(c + d*x)^2)/(2*d) + (b*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + b*sin(c + d*x)), x, 4, (2*a*csc(c + d*x))/d - (b*csc(c + d*x)^2)/(2*d) - (a*csc(c + d*x)^3)/(3*d) - (2*b*log(sin(c + d*x)))/d + (a*sin(c + d*x))/d + (b*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + b*sin(c + d*x)), x, 3, (2*b*csc(c + d*x))/d + (a*csc(c + d*x)^2)/d - (b*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^4)/(4*d) + (a*log(sin(c + d*x)))/d + (b*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + b*sin(c + d*x)), x, 4, -((a*csc(c + d*x))/d) + (b*csc(c + d*x)^2)/d + (2*a*csc(c + d*x)^3)/(3*d) - (b*csc(c + d*x)^4)/(4*d) - (a*csc(c + d*x)^5)/(5*d) + (b*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7*(a + b*sin(c + d*x)), x, 6, -((a*cot(c + d*x)^6)/(6*d)) - (b*csc(c + d*x))/d + (2*b*csc(c + d*x)^3)/(3*d) - (b*csc(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^8*(a + b*sin(c + d*x)), x, 6, -((b*cot(c + d*x)^6)/(6*d)) - (a*csc(c + d*x)^3)/(3*d) + (2*a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^9*(a + b*sin(c + d*x)), x, 7, -((a*cot(c + d*x)^6)/(6*d)) - (a*cot(c + d*x)^8)/(8*d) - (b*csc(c + d*x)^3)/(3*d) + (2*b*csc(c + d*x)^5)/(5*d) - (b*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^10*(a + b*sin(c + d*x)), x, 7, -((b*cot(c + d*x)^6)/(6*d)) - (b*cot(c + d*x)^8)/(8*d) - (a*csc(c + d*x)^5)/(5*d) + (2*a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^11*(a + b*sin(c + d*x)), x, 4, -((b*csc(c + d*x)^5)/(5*d)) - (a*csc(c + d*x)^6)/(6*d) + (2*b*csc(c + d*x)^7)/(7*d) + (a*csc(c + d*x)^8)/(4*d) - (b*csc(c + d*x)^9)/(9*d) - (a*csc(c + d*x)^10)/(10*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^12*(a + b*sin(c + d*x)), x, 4, -((b*csc(c + d*x)^6)/(6*d)) - (a*csc(c + d*x)^7)/(7*d) + (b*csc(c + d*x)^8)/(4*d) + (2*a*csc(c + d*x)^9)/(9*d) - (b*csc(c + d*x)^10)/(10*d) - (a*csc(c + d*x)^11)/(11*d)]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 4, (a^2*sin(c + d*x)^3)/(3*d) + (a*b*sin(c + d*x)^4)/(2*d) - ((2*a^2 - b^2)*sin(c + d*x)^5)/(5*d) - (2*a*b*sin(c + d*x)^6)/(3*d) + ((a^2 - 2*b^2)*sin(c + d*x)^7)/(7*d) + (a*b*sin(c + d*x)^8)/(4*d) + (b^2*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 4, (a^2*sin(c + d*x)^2)/(2*d) + (2*a*b*sin(c + d*x)^3)/(3*d) - ((2*a^2 - b^2)*sin(c + d*x)^4)/(4*d) - (4*a*b*sin(c + d*x)^5)/(5*d) + ((a^2 - 2*b^2)*sin(c + d*x)^6)/(6*d) + (2*a*b*sin(c + d*x)^7)/(7*d) + (b^2*sin(c + d*x)^8)/(8*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 4, (a^2*log(sin(c + d*x)))/d + (2*a*b*sin(c + d*x))/d - ((2*a^2 - b^2)*sin(c + d*x)^2)/(2*d) - (4*a*b*sin(c + d*x)^3)/(3*d) + ((a^2 - 2*b^2)*sin(c + d*x)^4)/(4*d) + (2*a*b*sin(c + d*x)^5)/(5*d) + (b^2*sin(c + d*x)^6)/(6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 4, -((a^2*csc(c + d*x))/d) + (2*a*b*log(sin(c + d*x)))/d - ((2*a^2 - b^2)*sin(c + d*x))/d - (2*a*b*sin(c + d*x)^2)/d + ((a^2 - 2*b^2)*sin(c + d*x)^3)/(3*d) + (a*b*sin(c + d*x)^4)/(2*d) + (b^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 4, -((2*a*b*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/(2*d) - ((2*a^2 - b^2)*log(sin(c + d*x)))/d - (4*a*b*sin(c + d*x))/d + ((a^2 - 2*b^2)*sin(c + d*x)^2)/(2*d) + (2*a*b*sin(c + d*x)^3)/(3*d) + (b^2*sin(c + d*x)^4)/(4*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 4, ((2*a^2 - b^2)*csc(c + d*x))/d - (a*b*csc(c + d*x)^2)/d - (a^2*csc(c + d*x)^3)/(3*d) - (4*a*b*log(sin(c + d*x)))/d + ((a^2 - 2*b^2)*sin(c + d*x))/d + (a*b*sin(c + d*x)^2)/d + (b^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 3, (4*a*b*csc(c + d*x))/d + ((2*a^2 - b^2)*csc(c + d*x)^2)/(2*d) - (2*a*b*csc(c + d*x)^3)/(3*d) - (a^2*csc(c + d*x)^4)/(4*d) + ((a^2 - 2*b^2)*log(sin(c + d*x)))/d + (2*a*b*sin(c + d*x))/d + (b^2*sin(c + d*x)^2)/(2*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 4, -(((a^2 - 2*b^2)*csc(c + d*x))/d) + (2*a*b*csc(c + d*x)^2)/d + ((2*a^2 - b^2)*csc(c + d*x)^3)/(3*d) - (a*b*csc(c + d*x)^4)/(2*d) - (a^2*csc(c + d*x)^5)/(5*d) + (2*a*b*log(sin(c + d*x)))/d + (b^2*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7*(a + b*sin(c + d*x))^2, x, 4, -((2*a*b*csc(c + d*x))/d) - ((a^2 - 2*b^2)*csc(c + d*x)^2)/(2*d) + (4*a*b*csc(c + d*x)^3)/(3*d) + ((2*a^2 - b^2)*csc(c + d*x)^4)/(4*d) - (2*a*b*csc(c + d*x)^5)/(5*d) - (a^2*csc(c + d*x)^6)/(6*d) + (b^2*log(sin(c + d*x)))/d]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^8*(a + b*sin(c + d*x))^2, x, 4, -((b^2*csc(c + d*x))/d) - (a*b*csc(c + d*x)^2)/d - ((a^2 - 2*b^2)*csc(c + d*x)^3)/(3*d) + (a*b*csc(c + d*x)^4)/d + ((2*a^2 - b^2)*csc(c + d*x)^5)/(5*d) - (a*b*csc(c + d*x)^6)/(3*d) - (a^2*csc(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^9*(a + b*sin(c + d*x))^2, x, 4, -((b^2*csc(c + d*x)^2)/(2*d)) - (2*a*b*csc(c + d*x)^3)/(3*d) - ((a^2 - 2*b^2)*csc(c + d*x)^4)/(4*d) + (4*a*b*csc(c + d*x)^5)/(5*d) + ((2*a^2 - b^2)*csc(c + d*x)^6)/(6*d) - (2*a*b*csc(c + d*x)^7)/(7*d) - (a^2*csc(c + d*x)^8)/(8*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 4, (a^2*(7*a^4 - 10*a^2*b^2 + 3*b^4)*log(a + b*sin(c + d*x)))/(b^8*d) - (2*a*(3*a^4 - 4*a^2*b^2 + b^4)*sin(c + d*x))/(b^7*d) + ((5*a^4 - 6*a^2*b^2 + b^4)*sin(c + d*x)^2)/(2*b^6*d) - (4*a*(a^2 - b^2)*sin(c + d*x)^3)/(3*b^5*d) + ((3*a^2 - 2*b^2)*sin(c + d*x)^4)/(4*b^4*d) - (2*a*sin(c + d*x)^5)/(5*b^3*d) + sin(c + d*x)^6/(6*b^2*d) + (a^3*(a^2 - b^2)^2)/(b^8*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 4, -((2*a*(3*a^4 - 4*a^2*b^2 + b^4)*log(a + b*sin(c + d*x)))/(b^7*d)) + ((5*a^4 - 6*a^2*b^2 + b^4)*sin(c + d*x))/(b^6*d) - (2*a*(a^2 - b^2)*sin(c + d*x)^2)/(b^5*d) - ((2 - (3*a^2)/b^2)*sin(c + d*x)^3)/(3*b^2*d) - (a*sin(c + d*x)^4)/(2*b^3*d) + sin(c + d*x)^5/(5*b^2*d) - (a^2*(a^2 - b^2)^2)/(b^7*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 4, ((5*a^4 - 6*a^2*b^2 + b^4)*log(a + b*sin(c + d*x)))/(b^6*d) - (4*a*(a^2 - b^2)*sin(c + d*x))/(b^5*d) + ((3*a^2 - 2*b^2)*sin(c + d*x)^2)/(2*b^4*d) - (2*a*sin(c + d*x)^3)/(3*b^3*d) + sin(c + d*x)^4/(4*b^2*d) + (a*(a^2 - b^2)^2)/(b^6*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 4, log(sin(c + d*x))/(a^2*d) + ((a^2 - b^2)*(3*a^2 + b^2)*log(a + b*sin(c + d*x)))/(a^2*b^4*d) - (2*a*sin(c + d*x))/(b^3*d) + sin(c + d*x)^2/(2*b^2*d) + (a^2 - b^2)^2/(a*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 4, -(csc(c + d*x)/(a^2*d)) - (2*b*log(sin(c + d*x)))/(a^3*d) - (2*(a^4 - b^4)*log(a + b*sin(c + d*x)))/(a^3*b^3*d) + sin(c + d*x)/(b^2*d) - (a^2 - b^2)^2/(a^2*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 4, (2*b*csc(c + d*x))/(a^3*d) - csc(c + d*x)^2/(2*a^2*d) - ((2*a^2 - 3*b^2)*log(sin(c + d*x)))/(a^4*d) + ((a^4 + 2*a^2*b^2 - 3*b^4)*log(a + b*sin(c + d*x)))/(a^4*b^2*d) + (a^2 - b^2)^2/(a^3*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 4, ((2*a^2 - 3*b^2)*csc(c + d*x))/(a^4*d) + (b*csc(c + d*x)^2)/(a^3*d) - csc(c + d*x)^3/(3*a^2*d) + (4*b*(a^2 - b^2)*log(sin(c + d*x)))/(a^5*d) - (4*b*(a^2 - b^2)*log(a + b*sin(c + d*x)))/(a^5*d) - (a^2 - b^2)^2/(a^4*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + b*sin(c + d*x))^2, x, 3, (-4*b*(a^2 - b^2)*csc(c + d*x))/(a^5*d) + ((2*a^2 - 3*b^2)*csc(c + d*x)^2)/(2*a^4*d) + (2*b*csc(c + d*x)^3)/(3*a^3*d) - csc(c + d*x)^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*log(sin(c + d*x)))/(a^6*d) - ((a^4 - 6*a^2*b^2 + 5*b^4)*log(a + b*sin(c + d*x)))/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + b*sin(c + d*x))^2, x, 4, -(((a^4 - 6*a^2*b^2 + 5*b^4)*csc(c + d*x))/(a^6*d)) - (2*b*(a^2 - b^2)*csc(c + d*x)^2)/(a^5*d) + ((2*a^2 - 3*b^2)*csc(c + d*x)^3)/(3*a^4*d) + (b*csc(c + d*x)^4)/(2*a^3*d) - csc(c + d*x)^5/(5*a^2*d) - (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*log(sin(c + d*x)))/(a^7*d) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*log(a + b*sin(c + d*x)))/(a^7*d) - (b*(a^2 - b^2)^2)/(a^6*d*(a + b*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [cos(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^2, x, 3, (a^2*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (2*a*b*sin(c + d*x)^(2 + n))/(d*(2 + n)) - ((2*a^2 - b^2)*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (4*a*b*sin(c + d*x)^(4 + n))/(d*(4 + n)) + ((a^2 - 2*b^2)*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (2*a*b*sin(c + d*x)^(6 + n))/(d*(6 + n)) + (b^2*sin(c + d*x)^(7 + n))/(d*(7 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^1, x, 3, (a*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (b*sin(c + d*x)^(2 + n))/(d*(2 + n)) - (2*a*sin(c + d*x)^(3 + n))/(d*(3 + n)) - (2*b*sin(c + d*x)^(4 + n))/(d*(4 + n)) + (a*sin(c + d*x)^(5 + n))/(d*(5 + n)) + (b*sin(c + d*x)^(6 + n))/(d*(6 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + b*sin(c + d*x))^1, x, 5, -((a*(a^2 - 2*b^2)*sin(c + d*x)^(1 + n))/(b^4*d*(1 + n))) + ((a^2 - b^2)^2*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -((b*sin(c + d*x))/a))*sin(c + d*x)^(1 + n))/(a*b^4*d*(1 + n)) + ((a^2 - 2*b^2)*sin(c + d*x)^(2 + n))/(b^3*d*(2 + n)) - (a*sin(c + d*x)^(3 + n))/(b^2*d*(3 + n)) + sin(c + d*x)^(4 + n)/(b*d*(4 + n))]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^n/(a + b*sin(c + d*x))^2, x, 5, ((3*a^2 - 2*b^2)*sin(c + d*x)^(1 + n))/(b^4*d*(1 + n)) + ((a^2 - b^2)*(b^2*n - a^2*(4 + n))*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -((b*sin(c + d*x))/a))*sin(c + d*x)^(1 + n))/(a^2*b^4*d*(1 + n)) - (2*a*sin(c + d*x)^(2 + n))/(b^3*d*(2 + n)) + sin(c + d*x)^(3 + n)/(b^2*d*(3 + n)) + ((a^2 - b^2)^2*sin(c + d*x)^(1 + n))/(a*b^4*d*(a + b*sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^6*sin(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 12, (5*a*b*x)/512 - ((a^2 + b^2)*cos(c + d*x)^7)/(7*d) + ((2*a^2 + 3*b^2)*cos(c + d*x)^9)/(9*d) - ((a^2 + 3*b^2)*cos(c + d*x)^11)/(11*d) + (b^2*cos(c + d*x)^13)/(13*d) + (5*a*b*cos(c + d*x)*sin(c + d*x))/(512*d) + (5*a*b*cos(c + d*x)^3*sin(c + d*x))/(768*d) + (a*b*cos(c + d*x)^5*sin(c + d*x))/(192*d) - (a*b*cos(c + d*x)^7*sin(c + d*x))/(32*d) - (a*b*cos(c + d*x)^7*sin(c + d*x)^3)/(12*d) - (a*b*cos(c + d*x)^7*sin(c + d*x)^5)/(6*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 12, ((12*a^2 + 5*b^2)*x)/1024 - (2*a*b*cos(c + d*x)^7)/(7*d) + (4*a*b*cos(c + d*x)^9)/(9*d) - (2*a*b*cos(c + d*x)^11)/(11*d) + ((12*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(1024*d) + ((12*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(1536*d) + ((12*a^2 + 5*b^2)*cos(c + d*x)^5*sin(c + d*x))/(1920*d) - ((44*a^2 + 45*b^2)*cos(c + d*x)^7*sin(c + d*x))/(320*d) + ((12*a^2 + 25*b^2)*cos(c + d*x)^9*sin(c + d*x))/(120*d) - (b^2*cos(c + d*x)^11*sin(c + d*x))/(12*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 11, (3*a*b*x)/128 - ((a^2 + b^2)*cos(c + d*x)^7)/(7*d) + ((a^2 + 2*b^2)*cos(c + d*x)^9)/(9*d) - (b^2*cos(c + d*x)^11)/(11*d) + (3*a*b*cos(c + d*x)*sin(c + d*x))/(128*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(64*d) + (a*b*cos(c + d*x)^5*sin(c + d*x))/(80*d) - (3*a*b*cos(c + d*x)^7*sin(c + d*x))/(40*d) - (a*b*cos(c + d*x)^7*sin(c + d*x)^3)/(5*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 11, (1/256)*(10*a^2 + 3*b^2)*x - (2*a*b*cos(c + d*x)^7)/(7*d) + (2*a*b*cos(c + d*x)^9)/(9*d) + ((10*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(256*d) + ((10*a^2 + 3*b^2)*cos(c + d*x)^3*sin(c + d*x))/(384*d) + ((10*a^2 + 3*b^2)*cos(c + d*x)^5*sin(c + d*x))/(480*d) - ((10*a^2 + 11*b^2)*cos(c + d*x)^7*sin(c + d*x))/(80*d) + (b^2*cos(c + d*x)^9*sin(c + d*x))/(10*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 7, (5*a*b*x)/64 - ((a^2 + 8*b^2)*cos(c + d*x)^7)/(252*d) + (5*a*b*cos(c + d*x)*sin(c + d*x))/(64*d) + (5*a*b*cos(c + d*x)^3*sin(c + d*x))/(96*d) + (a*b*cos(c + d*x)^5*sin(c + d*x))/(24*d) - (a*cos(c + d*x)^7*(a + b*sin(c + d*x)))/(36*d) - (cos(c + d*x)^7*(a + b*sin(c + d*x))^2)/(9*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, If(13<11, 10, 9), (5*a*b*x)/8 - (a^2*atanh(cos(c + d*x)))/d + (a^2*cos(c + d*x))/d + (a^2*cos(c + d*x)^3)/(3*d) + (a^2*cos(c + d*x)^5)/(5*d) - (b^2*cos(c + d*x)^7)/(7*d) + (5*a*b*cos(c + d*x)*sin(c + d*x))/(8*d) + (5*a*b*cos(c + d*x)^3*sin(c + d*x))/(12*d) + (a*b*cos(c + d*x)^5*sin(c + d*x))/(3*d)]

    @test_int [cos(c + d*x)^6*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, If(13<11, 11, 12), (1/16)*-5*(6*a^2 - b^2)*x - (2*a*b*atanh(cos(c + d*x)))/d + (2*a*b*cos(c + d*x))/d + (2*a*b*cos(c + d*x)^3)/(3*d) + (2*a*b*cos(c + d*x)^5)/(5*d) - (a^2*cot(c + d*x))/d - ((14*a^2 - 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) - ((6*a^2 - 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (b^2*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 11, (1/4)*-15*a*b*x + ((5*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*d) - ((2*a^2 - b^2)*cos(c + d*x))/d - ((a^2 - b^2)*cos(c + d*x)^3)/(3*d) + (b^2*cos(c + d*x)^5)/(5*d) - (15*a*b*cot(c + d*x))/(4*d) + (5*a*b*cos(c + d*x)^2*cot(c + d*x))/(4*d) + (a*b*cos(c + d*x)^4*cot(c + d*x))/(2*d) - (a^2*cot(c + d*x)*csc(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 12, (5/8)*(4*a^2 - 3*b^2)*x + (5*a*b*atanh(cos(c + d*x)))/d - (5*a*b*cos(c + d*x))/d - (5*a*b*cos(c + d*x)^3)/(3*d) + ((2*a^2 - b^2)*cot(c + d*x))/d - (a*b*cos(c + d*x)^3*cot(c + d*x)^2)/d - (a^2*cot(c + d*x)^3)/(3*d) + ((4*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) - (b^2*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 12, 5*a*b*x - (5*(3*a^2 - 4*b^2)*atanh(cos(c + d*x)))/(8*d) + ((a^2 - 2*b^2)*cos(c + d*x))/d - (b^2*cos(c + d*x)^3)/(3*d) + (5*a*b*cot(c + d*x))/d - (5*a*b*cot(c + d*x)^3)/(3*d) + (a*b*cos(c + d*x)^2*cot(c + d*x)^3)/d + ((9*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*d) - (a^2*cot(c + d*x)*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 16, -(a^2*x) + (5*b^2*x)/2 - (15*a*b*atanh(cos(c + d*x)))/(4*d) + (15*a*b*cos(c + d*x))/(4*d) - (a^2*cot(c + d*x))/d + (5*b^2*cot(c + d*x))/(2*d) + (5*a*b*cos(c + d*x)*cot(c + d*x)^2)/(4*d) + (a^2*cot(c + d*x)^3)/(3*d) - (5*b^2*cot(c + d*x)^3)/(6*d) + (b^2*cos(c + d*x)^2*cot(c + d*x)^3)/(2*d) - (a*b*cos(c + d*x)*cot(c + d*x)^4)/(2*d) - (a^2*cot(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7*(a + b*sin(c + d*x))^2, x, 11, -2*a*b*x + (5*(a^2 - 6*b^2)*atanh(cos(c + d*x)))/(16*d) + (b^2*cos(c + d*x))/d - (2*a*b*cot(c + d*x))/d + (2*a*b*cot(c + d*x)^3)/(3*d) - (2*a*b*cot(c + d*x)^5)/(5*d) - ((11*a^2 - 18*b^2)*cot(c + d*x)*csc(c + d*x))/(16*d) + ((13*a^2 - 6*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a^2*cot(c + d*x)*csc(c + d*x)^5)/(6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8*(a + b*sin(c + d*x))^2, x, 9, (-b^2)*x + (5*a*b*atanh(cos(c + d*x)))/(8*d) - (b^2*cot(c + d*x))/d + (b^2*cot(c + d*x)^3)/(3*d) - (b^2*cot(c + d*x)^5)/(5*d) - (a^2*cot(c + d*x)^7)/(7*d) - (5*a*b*cot(c + d*x)*csc(c + d*x))/(8*d) + (5*a*b*cot(c + d*x)^3*csc(c + d*x))/(12*d) - (a*b*cot(c + d*x)^5*csc(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^9*(a + b*sin(c + d*x))^2, x, 9, (5*(a^2 + 8*b^2)*atanh(cos(c + d*x)))/(128*d) - (2*a*b*cot(c + d*x)^7)/(7*d) + ((5*a^2 - 88*b^2)*cot(c + d*x)*csc(c + d*x))/(128*d) - ((59*a^2 - 104*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(192*d) + ((17*a^2 - 8*b^2)*cot(c + d*x)*csc(c + d*x)^5)/(48*d) - (a^2*cot(c + d*x)*csc(c + d*x)^7)/(8*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^10*(a + b*sin(c + d*x))^2, x, 9, (5*a*b*atanh(cos(c + d*x)))/(64*d) - ((a^2 + b^2)*cot(c + d*x)^7)/(7*d) - (a^2*cot(c + d*x)^9)/(9*d) + (5*a*b*cot(c + d*x)*csc(c + d*x))/(64*d) - (5*a*b*cot(c + d*x)*csc(c + d*x)^3)/(32*d) + (5*a*b*cot(c + d*x)^3*csc(c + d*x)^3)/(24*d) - (a*b*cot(c + d*x)^5*csc(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^11*(a + b*sin(c + d*x))^2, x, 11, ((3*a^2 + 10*b^2)*atanh(cos(c + d*x)))/(256*d) - (2*a*b*cot(c + d*x)^7)/(7*d) - (2*a*b*cot(c + d*x)^9)/(9*d) + ((3*a^2 + 10*b^2)*cot(c + d*x)*csc(c + d*x))/(256*d) + ((3*a^2 - 118*b^2)*cot(c + d*x)*csc(c + d*x)^3)/(384*d) - ((93*a^2 - 170*b^2)*cot(c + d*x)*csc(c + d*x)^5)/(480*d) + ((21*a^2 - 10*b^2)*cot(c + d*x)*csc(c + d*x)^7)/(80*d) - (a^2*cot(c + d*x)*csc(c + d*x)^9)/(10*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^12*(a + b*sin(c + d*x))^2, x, 10, (3*a*b*atanh(cos(c + d*x)))/(128*d) - ((a^2 + b^2)*cot(c + d*x)^7)/(7*d) - ((2*a^2 + b^2)*cot(c + d*x)^9)/(9*d) - (a^2*cot(c + d*x)^11)/(11*d) + (3*a*b*cot(c + d*x)*csc(c + d*x))/(128*d) + (a*b*cot(c + d*x)*csc(c + d*x)^3)/(64*d) - (a*b*cot(c + d*x)*csc(c + d*x)^5)/(16*d) + (a*b*cot(c + d*x)^3*csc(c + d*x)^5)/(8*d) - (a*b*cot(c + d*x)^5*csc(c + d*x)^5)/(5*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 11, (a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*cos(c + d*x))/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*cos(c + d*x)*sin(c + d*x))/(8*b^7*d) + ((280*a^4 - 441*a^2*b^2 + 150*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(140*a^2*b^4*d) + (cos(c + d*x)*sin(c + d*x)^4)/(4*a*d*(a + b*sin(c + d*x))) - (3*b*cos(c + d*x)*sin(c + d*x)^5)/(20*a^2*d*(a + b*sin(c + d*x))) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*cos(c + d*x)*sin(c + d*x)^5)/(15*a^2*b^3*d*(a + b*sin(c + d*x))) - (4*a*cos(c + d*x)*sin(c + d*x)^6)/(21*b^2*d*(a + b*sin(c + d*x))) + (cos(c + d*x)*sin(c + d*x)^7)/(7*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 10, -(((112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*x)/(16*b^8)) + (2*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^8*d) - (a*(105*a^4 - 170*a^2*b^2 + 61*b^4)*cos(c + d*x))/(15*b^7*d) + ((56*a^4 - 86*a^2*b^2 + 27*b^4)*cos(c + d*x)*sin(c + d*x))/(16*b^6*d) - ((35*a^4 - 52*a^2*b^2 + 15*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(15*a*b^5*d) + ((42*a^4 - 61*a^2*b^2 + 16*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(24*a^2*b^4*d) + (cos(c + d*x)*sin(c + d*x)^3)/(3*a*d*(a + b*sin(c + d*x))) - (b*cos(c + d*x)*sin(c + d*x)^4)/(6*a^2*d*(a + b*sin(c + d*x))) - ((14*a^4 - 20*a^2*b^2 + 5*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(10*a^2*b^3*d*(a + b*sin(c + d*x))) - (7*a*cos(c + d*x)*sin(c + d*x)^5)/(30*b^2*d*(a + b*sin(c + d*x))) + (cos(c + d*x)*sin(c + d*x)^6)/(6*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 7, (a*(24*a^4 - 40*a^2*b^2 + 15*b^4)*x)/(4*b^7) - (2*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*d) + (cos(c + d*x)^5*(6*a + b*sin(c + d*x)))/(5*b^2*d*(a + b*sin(c + d*x))) - (cos(c + d*x)^3*(2*(6*a^2 - b^2) - 9*a*b*sin(c + d*x)))/(6*b^4*d) + (cos(c + d*x)*(4*(6*a^4 - 7*a^2*b^2 + b^4) - a*b*(12*a^2 - 11*b^2)*sin(c + d*x)))/(4*b^6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 16, (a*x)/b^3 + (2*a*(2*a^2 - 3*b^2)*x)/b^5 + (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*d) - (2*(a^2 - b^2)^(3/2)*(5*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^5*d) - atanh(cos(c + d*x))/(a^2*d) + cos(c + d*x)/(b^2*d) + (3*(a^2 - b^2)*cos(c + d*x))/(b^4*d) - cos(c + d*x)^3/(3*b^2*d) - (a*cos(c + d*x)*sin(c + d*x))/(b^3*d) + ((a^2 - b^2)^2*cos(c + d*x))/(a*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 16, -(x/(2*b^2)) - (3*(a^2 - b^2)*x)/b^4 - (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b^4*d) + (4*(2*a^6 - 3*a^4*b^2 + b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^4*sqrt(a^2 - b^2)*d) + (2*b*atanh(cos(c + d*x)))/(a^3*d) - (2*a*cos(c + d*x))/(b^3*d) - cot(c + d*x)/(a^2*d) + (cos(c + d*x)*sin(c + d*x))/(2*b^2*d) - ((a^2 - b^2)^2*cos(c + d*x))/(a^2*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 16, (2*a*x)/b^3 + (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^3*d) - (6*(a^2 - b^2)^(3/2)*(a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*b^3*d) - atanh(cos(c + d*x))/(2*a^2*d) + (3*(a^2 - b^2)*atanh(cos(c + d*x)))/(a^4*d) + cos(c + d*x)/(b^2*d) + (2*b*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d) + ((a^2 - b^2)^2*cos(c + d*x))/(a^3*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 17, -(x/b^2) - (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^2*d) + (4*(a^6 - 3*a^2*b^4 + 2*b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*b^2*sqrt(a^2 - b^2)*d) + (b*atanh(cos(c + d*x)))/(a^3*d) - (2*b*(3*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(a^5*d) - cot(c + d*x)/(a^2*d) + (3*(a^2 - b^2)*cot(c + d*x))/(a^4*d) - cot(c + d*x)^3/(3*a^2*d) + (b*cot(c + d*x)*csc(c + d*x))/(a^3*d) - ((a^2 - b^2)^2*cos(c + d*x))/(a^4*b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + b*sin(c + d*x))^2, x, 9, -((10*b*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d)) - (5*(3*a^4 - 12*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) + ((3*a^4 - 20*a^2*b^2 + 15*b^4)*cot(c + d*x))/(3*a^5*b*d) + (5*(5*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^4*d) - cot(c + d*x)/(b*d*(a + b*sin(c + d*x))) - ((6*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x))/(3*a^3*d*(a + b*sin(c + d*x))) + (5*b*cot(c + d*x)*csc(c + d*x)^2)/(12*a^2*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + b*sin(c + d*x))^2, x, 10, -((2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^7*d)) + (b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*atanh(cos(c + d*x)))/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*cot(c + d*x))/(15*a^6*d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*cot(c + d*x)*csc(c + d*x))/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(30*a^4*b^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*b*d*(a + b*sin(c + d*x))) + (a*cot(c + d*x)*csc(c + d*x)^2)/(6*b^2*d*(a + b*sin(c + d*x))) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(6*a^3*b^2*d*(a + b*sin(c + d*x))) + (3*b*cot(c + d*x)*csc(c + d*x)^3)/(10*a^2*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7/(a + b*sin(c + d*x))^2, x, 11, (2*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^8*d) + ((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*atanh(cos(c + d*x)))/(16*a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*cot(c + d*x))/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*cot(c + d*x)*csc(c + d*x))/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(24*a^4*b^2*d) - (cot(c + d*x)*csc(c + d*x)^2)/(3*b*d*(a + b*sin(c + d*x))) + (a*cot(c + d*x)*csc(c + d*x)^3)/(6*b^2*d*(a + b*sin(c + d*x))) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(10*a^3*b^2*d*(a + b*sin(c + d*x))) + (7*b*cot(c + d*x)*csc(c + d*x)^4)/(30*a^2*d*(a + b*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a*d*(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 11, -(((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/(16*b^9)) + (a*sqrt(a^2 - b^2)*(56*a^4 - 47*a^2*b^2 + 6*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^9*d) - (a*(840*a^4 - 985*a^2*b^2 + 213*b^4)*cos(c + d*x))/(30*b^8*d) + ((224*a^4 - 244*a^2*b^2 + 43*b^4)*cos(c + d*x)*sin(c + d*x))/(16*b^7*d) - ((280*a^4 - 291*a^2*b^2 + 45*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(30*a*b^6*d) + ((168*a^4 - 169*a^2*b^2 + 24*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(24*a^2*b^5*d) + (cos(c + d*x)*sin(c + d*x)^4)/(4*a*d*(a + b*sin(c + d*x))^2) - (b*cos(c + d*x)*sin(c + d*x)^5)/(10*a^2*d*(a + b*sin(c + d*x))^2) - ((56*a^4 - 60*a^2*b^2 + 9*b^4)*cos(c + d*x)*sin(c + d*x)^5)/(60*a^2*b^3*d*(a + b*sin(c + d*x))^2) - (4*a*cos(c + d*x)*sin(c + d*x)^6)/(15*b^2*d*(a + b*sin(c + d*x))^2) + (cos(c + d*x)*sin(c + d*x)^7)/(6*b*d*(a + b*sin(c + d*x))^2) - ((112*a^4 - 110*a^2*b^2 + 15*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(20*a^2*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 10, (a*(168*a^4 - 200*a^2*b^2 + 45*b^4)*x)/(8*b^8) - (sqrt(a^2 - b^2)*(42*a^4 - 29*a^2*b^2 + 2*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^8*d) + ((630*a^4 - 645*a^2*b^2 + 91*b^4)*cos(c + d*x))/(30*b^7*d) - ((84*a^4 - 79*a^2*b^2 + 8*b^4)*cos(c + d*x)*sin(c + d*x))/(8*a*b^6*d) + ((210*a^4 - 187*a^2*b^2 + 15*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(30*a^2*b^5*d) + (cos(c + d*x)*sin(c + d*x)^3)/(3*a*d*(a + b*sin(c + d*x))^2) - (b*cos(c + d*x)*sin(c + d*x)^4)/(12*a^2*d*(a + b*sin(c + d*x))^2) - ((63*a^4 - 60*a^2*b^2 + 5*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(60*a^2*b^3*d*(a + b*sin(c + d*x))^2) - (7*a*cos(c + d*x)*sin(c + d*x)^5)/(20*b^2*d*(a + b*sin(c + d*x))^2) + (cos(c + d*x)*sin(c + d*x)^6)/(5*b*d*(a + b*sin(c + d*x))^2) - ((63*a^4 - 54*a^2*b^2 + 4*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(12*a^2*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 7, -((15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7)) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*sqrt(a^2 - b^2)*d) + (cos(c + d*x)^5*(3*a + b*sin(c + d*x)))/(4*b^2*d*(a + b*sin(c + d*x))^2) + (5*cos(c + d*x)^3*(4*a^2 - b^2 + a*b*sin(c + d*x)))/(4*b^4*d*(a + b*sin(c + d*x))) - (15*cos(c + d*x)*(4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*sin(c + d*x)))/(8*b^6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 20, -(x/(2*b^3)) - (3*(2*a^2 - b^2)*x)/b^5 + (sqrt(a^2 - b^2)*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b^5*d) - (2*sqrt(a^2 - b^2)*(5*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b^5*d) + (2*(10*a^6 - 9*a^4*b^2 - b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^5*sqrt(a^2 - b^2)*d) - atanh(cos(c + d*x))/(a^3*d) - (3*a*cos(c + d*x))/(b^4*d) + (cos(c + d*x)*sin(c + d*x))/(2*b^3*d) + ((a^2 - b^2)^2*cos(c + d*x))/(2*a*b^4*d*(a + b*sin(c + d*x))^2) + (3*(a^2 - b^2)*cos(c + d*x))/(2*b^4*d*(a + b*sin(c + d*x))) - ((a^2 - b^2)*(5*a^2 + b^2)*cos(c + d*x))/(a^2*b^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 20, (3*a*x)/b^4 + (3*sqrt(a^2 - b^2)*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^4*d) - (6*(2*a^6 - a^4*b^2 - b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*b^4*sqrt(a^2 - b^2)*d) + (3*b*atanh(cos(c + d*x)))/(a^4*d) + cos(c + d*x)/(b^3*d) - cot(c + d*x)/(a^3*d) - ((a^2 - b^2)^2*cos(c + d*x))/(2*a^2*b^3*d*(a + b*sin(c + d*x))^2) - (3*(a^2 - b^2)*cos(c + d*x))/(2*a*b^3*d*(a + b*sin(c + d*x))) + (2*(a^2 - b^2)*(2*a^2 + b^2)*cos(c + d*x))/(a^3*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 21, -(x/b^3) - (6*sqrt(a^2 - b^2)*(a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^3*d) + (sqrt(a^2 - b^2)*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^3*d) + (6*(a^6 + a^2*b^4 - 2*b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*b^3*sqrt(a^2 - b^2)*d) - atanh(cos(c + d*x))/(2*a^3*d) + (3*(a^2 - 2*b^2)*atanh(cos(c + d*x)))/(a^5*d) + (3*b*cot(c + d*x))/(a^4*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d) + ((a^2 - b^2)^2*cos(c + d*x))/(2*a^3*b^2*d*(a + b*sin(c + d*x))^2) + (3*(a^2 - b^2)*cos(c + d*x))/(2*a^2*b^2*d*(a + b*sin(c + d*x))) - (3*(a^4 - b^4)*cos(c + d*x))/(a^4*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + b*sin(c + d*x))^3, x, 9, (5*(a^2 - 4*b^2)*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d) - (5*b*(3*a^2 - 4*b^2)*atanh(cos(c + d*x)))/(2*a^6*d) + ((3*a^4 + 35*a^2*b^2 - 60*b^4)*cot(c + d*x))/(6*a^5*b^2*d) - cos(c + d*x)/(b*d*(a + b*sin(c + d*x))^2) - (a*cot(c + d*x))/(2*b^2*d*(a + b*sin(c + d*x))^2) - ((3*a^2 - 5*b^2)*cot(c + d*x))/(3*a^3*d*(a + b*sin(c + d*x))^2) + (5*b*cot(c + d*x)*csc(c + d*x))/(6*a^2*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d*(a + b*sin(c + d*x))^2) - (5*(a^2 - 2*b^2)*cot(c + d*x))/(2*a^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + b*sin(c + d*x))^3, x, 10, -((15*b*(a^2 - 2*b^2)*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^7*d)) - (15*(a^4 - 8*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^7*d) + ((a^4 - 25*a^2*b^2 + 30*b^4)*cot(c + d*x))/(2*a^6*b*d) + (15*(3*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^5*d) - cot(c + d*x)/(2*b*d*(a + b*sin(c + d*x))^2) - ((4*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x))/(4*a^3*d*(a + b*sin(c + d*x))^2) + (b*cot(c + d*x)*csc(c + d*x)^2)/(2*a^2*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d*(a + b*sin(c + d*x))^2) - ((7*a^2 - 10*b^2)*cot(c + d*x)*csc(c + d*x))/(2*a^4*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + b*sin(c + d*x))^3, x, 11, -((sqrt(a^2 - b^2)*(2*a^4 - 29*a^2*b^2 + 42*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^8*d)) + (b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*atanh(cos(c + d*x)))/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*cot(c + d*x))/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*cot(c + d*x)*csc(c + d*x))/(8*a^6*b*d) - ((15*a^4 - 187*a^2*b^2 + 210*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(30*a^5*b^2*d) - (cot(c + d*x)*csc(c + d*x))/(3*b*d*(a + b*sin(c + d*x))^2) + (a*cot(c + d*x)*csc(c + d*x)^2)/(12*b^2*d*(a + b*sin(c + d*x))^2) + ((5*a^4 - 60*a^2*b^2 + 63*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(60*a^3*b^2*d*(a + b*sin(c + d*x))^2) + (7*b*cot(c + d*x)*csc(c + d*x)^3)/(20*a^2*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d*(a + b*sin(c + d*x))^2) + ((4*a^4 - 54*a^2*b^2 + 63*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(12*a^4*b^2*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8/(a + b*sin(c + d*x))^3, x, 13, -((3*b^2*sqrt(a^2 - b^2)*(4*a^4 - 23*a^2*b^2 + 24*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^10*d)) - (3*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*atanh(cos(c + d*x)))/(16*a^10*d) + ((10*a^6 - 889*a^4*b^2 + 3255*a^2*b^4 - 2520*b^6)*cot(c + d*x))/(70*a^9*d) + (3*b*(27*a^4 - 116*a^2*b^2 + 96*b^4)*cot(c + d*x)*csc(c + d*x))/(16*a^8*d) - ((205*a^4 - 973*a^2*b^2 + 840*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(70*a^7*d) + ((16*a^4 - 81*a^2*b^2 + 72*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(8*a^6*b*d) - (3*(35*a^4 - 185*a^2*b^2 + 168*b^4)*cot(c + d*x)*csc(c + d*x)^4)/(70*a^5*b^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(5*b*d*(a + b*sin(c + d*x))^2) + (a*cot(c + d*x)*csc(c + d*x)^4)/(10*b^2*d*(a + b*sin(c + d*x))^2) + ((7*a^4 - 35*a^2*b^2 + 30*b^4)*cot(c + d*x)*csc(c + d*x)^4)/(35*a^3*b^2*d*(a + b*sin(c + d*x))^2) + (3*b*cot(c + d*x)*csc(c + d*x)^5)/(14*a^2*d*(a + b*sin(c + d*x))^2) - (cot(c + d*x)*csc(c + d*x)^6)/(7*a*d*(a + b*sin(c + d*x))^2) + ((12*a^4 - 65*a^2*b^2 + 60*b^4)*cot(c + d*x)*csc(c + d*x)^4)/(10*a^4*b^2*d*(a + b*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^6*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [cos(e + f*x)^6/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^(13/2)), x, 8, (2*cos(e + f*x)^5*sqrt(d*sin(e + f*x)))/(11*a*d*f*(a + b*sin(e + f*x))^(11/2)) - (20*(a^2 - b^2)*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(99*a^2*b^2*d*f*(a + b*sin(e + f*x))^(9/2)) + (80*(3*a^2 + 2*b^2)*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(693*a^3*b^2*d*f*(a + b*sin(e + f*x))^(7/2)) - (4*(5*a^4 - 17*a^2*b^2 + 16*b^4)*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(231*a^4*b^2*(a^2 - b^2)*d*f*(a + b*sin(e + f*x))^(5/2)) - (8*(5*a^6 - 22*a^4*b^2 + 65*a^2*b^4 - 32*b^6)*cos(e + f*x)*sqrt(d*sin(e + f*x)))/(693*a^5*b^2*(a^2 - b^2)^2*d*f*(a + b*sin(e + f*x))^(3/2)) + (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*cos(e + f*x))/(693*a^5*(a^2 - b^2)^3*f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))) - (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.E(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(693*a^7*(a - b)^2*(a + b)^(5/2)*sqrt(d)*f) - (16*(45*a^4 - 48*a^3*b - 69*a^2*b^2 + 24*a*b^3 + 32*b^4)*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(693*a^6*(a - b)^2*(a + b)^(5/2)*sqrt(d)*f)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^2=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(5/2)*sqrt(d*sin(e + f*x))), x, 9, (2*(a^2 + b^2)*sqrt(d*sin(e + f*x)))/(3*d*f*g*(g*cos(e + f*x))^(3/2)) + (4*a*b*(d*sin(e + f*x))^(3/2))/(3*d^2*f*g*(g*cos(e + f*x))^(3/2)) + ((2*a^2 - b^2)*Elliptic.F((1/4)*(4*e - pi) + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(3*f*g^2*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))), (2*(a^2 + b^2)*sqrt(d*sin(e + f*x)))/(3*d*f*g*(g*cos(e + f*x))^(3/2)) + (4*a*b*(d*sin(e + f*x))^(3/2))/(3*d^2*f*g*(g*cos(e + f*x))^(3/2)) - (2*(2*a^2 - b^2)*(1 - csc(e + f*x)^2)^(3/4)*Elliptic.F((1/2)*acsc(sin(e + f*x)), 2)*(d*sin(e + f*x))^(3/2))/(3*d^2*f*g*(g*cos(e + f*x))^(3/2))]

    @test_int [(a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(7/2)*sqrt(d*sin(e + f*x))), x, 6, (8*a^2*sqrt(d*sin(e + f*x)))/(5*d*f*g^3*sqrt(g*cos(e + f*x))) + (8*a*b*(d*sin(e + f*x))^(3/2))/(5*d^2*f*g^3*sqrt(g*cos(e + f*x))) + (2*sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^2)/(5*d*f*g*(g*cos(e + f*x))^(5/2)) - (8*a*b*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*d*f*g^4*sqrt(sin(2*e + 2*f*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(d*sin(e+f*x))^n / (a+b*sin(e+f*x))^1=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*sin(e+f*x)^n / (a+b*sin(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [cos(c + d*x)*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, -((a^3*log(a + b*sin(c + d*x)))/(b^4*d)) + (a^2*sin(c + d*x))/(b^3*d) - (a*sin(c + d*x)^2)/(2*b^2*d) + sin(c + d*x)^3/(3*b*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, (a^2*log(a + b*sin(c + d*x)))/(b^3*d) - (a*sin(c + d*x))/(b^2*d) + sin(c + d*x)^2/(2*b*d)]
    @test_int [cos(c + d*x)*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, -((a*log(a + b*sin(c + d*x)))/(b^2*d)) + sin(c + d*x)/(b*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - log(a + b*sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (b*log(sin(c + d*x)))/(a^2*d) + (b*log(a + b*sin(c + d*x)))/(a^2*d)]
    @test_int [cos(c + d*x)*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) + (b^2*log(sin(c + d*x)))/(a^3*d) - (b^2*log(a + b*sin(c + d*x)))/(a^3*d)]

    @test_int [cos(c + d*x)^2*sin(c + d*x)^4/(a + b*sin(c + d*x)), x, 10, (a*(8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^6) - (2*a^4*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*d) + ((15*a^4 - 5*a^2*b^2 - 2*b^4)*cos(c + d*x))/(15*b^5*d) - (a*(4*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^4*d) + ((5*a^2 - b^2)*cos(c + d*x)*sin(c + d*x)^2)/(15*b^3*d) - (a*cos(c + d*x)*sin(c + d*x)^3)/(4*b^2*d) + (cos(c + d*x)*sin(c + d*x)^4)/(5*b*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 9, -(((8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^5)) + (2*a^3*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*d) - (a*(3*a^2 - b^2)*cos(c + d*x))/(3*b^4*d) + ((4*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^3*d) - (a*cos(c + d*x)*sin(c + d*x)^2)/(3*b^2*d) + (cos(c + d*x)*sin(c + d*x)^3)/(4*b*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 8, (a*(2*a^2 - b^2)*x)/(2*b^4) - (2*a^2*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*d) + ((3*a^2 - b^2)*cos(c + d*x))/(3*b^3*d) - (a*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (cos(c + d*x)*sin(c + d*x)^2)/(3*b*d)]
    @test_int [cos(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 5, -(((2*a^2 - b^2)*x)/(2*b^3)) + (2*a*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*d) - (cos(c + d*x)*(2*a - b*sin(c + d*x)))/(2*b^2*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, -(x/b) + (2*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b*d) - atanh(cos(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 7, -((2*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*d)) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 8, (2*b*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*d) + ((a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^3*d) + (b*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^4/(a + b*sin(c + d*x)), x, 9, -((2*b^2*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*d)) - (b*(a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^4*d) + ((a^2 - 3*b^2)*cot(c + d*x))/(3*a^3*d) + (b*cot(c + d*x)*csc(c + d*x))/(2*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^5/(a + b*sin(c + d*x)), x, 10, (2*b^3*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*d) + ((a^4 + 4*a^2*b^2 - 8*b^4)*atanh(cos(c + d*x)))/(8*a^5*d) - (b*(a^2 - 3*b^2)*cot(c + d*x))/(3*a^4*d) + ((a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) + (b*cot(c + d*x)*csc(c + d*x)^2)/(3*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^2*csc(c + d*x)^6/(a + b*sin(c + d*x)), x, 11, -((2*b^4*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d)) - (b*(a^4 + 4*a^2*b^2 - 8*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) + ((2*a^4 + 5*a^2*b^2 - 15*b^4)*cot(c + d*x))/(15*a^5*d) - (b*(a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^4*d) + ((a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(15*a^3*d) + (b*cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d)]

    @test_int [cos(c + d*x)^3*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (a^3*(a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^6*d) - (a^2*(a^2 - b^2)*sin(c + d*x))/(b^5*d) + (a*(a^2 - b^2)*sin(c + d*x)^2)/(2*b^4*d) - ((a^2 - b^2)*sin(c + d*x)^3)/(3*b^3*d) + (a*sin(c + d*x)^4)/(4*b^2*d) - sin(c + d*x)^5/(5*b*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -((a^2*(a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^5*d)) + (a*(a^2 - b^2)*sin(c + d*x))/(b^4*d) - ((a^2 - b^2)*sin(c + d*x)^2)/(2*b^3*d) + (a*sin(c + d*x)^3)/(3*b^2*d) - sin(c + d*x)^4/(4*b*d)]
    @test_int [cos(c + d*x)^3*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, (a*(a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^4*d) - ((a^2 - b^2)*sin(c + d*x))/(b^3*d) + (a*sin(c + d*x)^2)/(2*b^2*d) - sin(c + d*x)^3/(3*b*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) + ((a^2 - b^2)*log(a + b*sin(c + d*x)))/(a*b^2*d) - sin(c + d*x)/(b*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (b*log(sin(c + d*x)))/(a^2*d) - ((1 - b^2/a^2)*log(a + b*sin(c + d*x)))/(b*d)]
    @test_int [cos(c + d*x)^3*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 3, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) - ((a^2 - b^2)*log(sin(c + d*x)))/(a^3*d) + ((a^2 - b^2)*log(a + b*sin(c + d*x)))/(a^3*d)]

    @test_int [cos(c + d*x)^4*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 9, ((16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*x)/(16*b^7) - (2*a^3*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*d) + (a*(15*a^4 - 20*a^2*b^2 + 3*b^4)*cos(c + d*x))/(15*b^6*d) - ((8*a^4 - 10*a^2*b^2 + b^4)*cos(c + d*x)*sin(c + d*x))/(16*b^5*d) + (a*(5*a^2 - 6*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(15*b^4*d) - ((6*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x)^3)/(24*b^3*d) + (a*cos(c + d*x)*sin(c + d*x)^4)/(5*b^2*d) - (cos(c + d*x)*sin(c + d*x)^5)/(6*b*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 8, -((a*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6)) + (2*a^2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*d) - ((15*a^4 - 20*a^2*b^2 + 3*b^4)*cos(c + d*x))/(15*b^5*d) + (a*(4*a^2 - 5*b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^4*d) - ((5*a^2 - 6*b^2)*cos(c + d*x)*sin(c + d*x)^2)/(15*b^3*d) + (a*cos(c + d*x)*sin(c + d*x)^3)/(4*b^2*d) - (cos(c + d*x)*sin(c + d*x)^4)/(5*b*d)]
    @test_int [cos(c + d*x)^4*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, ((8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*d) - (cos(c + d*x)^3*(4*a - 3*b*sin(c + d*x)))/(12*b^2*d) + (cos(c + d*x)*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*sin(c + d*x)))/(8*b^4*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, ((2*a^2 - 3*b^2)*x)/(2*b^3) - (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b^3*d) - atanh(cos(c + d*x))/(a*d) + (a*cos(c + d*x))/(b^2*d) - (cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 6, -((a*x)/b^2) + (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^2*d) + (b*atanh(cos(c + d*x)))/(a^2*d) - cos(c + d*x)/(b*d) - cot(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 6, x/b - (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b*d) + ((3*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^3*d) + (b*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^4/(a + b*sin(c + d*x)), x, 7, (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*d) - (b*(3*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^4*d) + ((4*a^2 - 3*b^2)*cot(c + d*x))/(3*a^3*d) + (b*cot(c + d*x)*csc(c + d*x))/(2*a^2*d) - (cot(c + d*x)*csc(c + d*x)^2)/(3*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^5/(a + b*sin(c + d*x)), x, 8, -((2*b*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*d)) - ((3*a^4 - 12*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^5*d) - (b*(4*a^2 - 3*b^2)*cot(c + d*x))/(3*a^4*d) + ((5*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) + (b*cot(c + d*x)*csc(c + d*x)^2)/(3*a^2*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^4*csc(c + d*x)^6/(a + b*sin(c + d*x)), x, 9, (2*b^2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d) + (b*(3*a^4 - 12*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) - ((3*a^4 - 20*a^2*b^2 + 15*b^4)*cot(c + d*x))/(15*a^5*d) - (b*(5*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^4*d) + ((6*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(15*a^3*d) + (b*cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d)]

    @test_int [cos(c + d*x)^5*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, -((a^3*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(b^8*d)) + (a^2*(a^2 - b^2)^2*sin(c + d*x))/(b^7*d) - (a*(a^2 - b^2)^2*sin(c + d*x)^2)/(2*b^6*d) + ((a^2 - b^2)^2*sin(c + d*x)^3)/(3*b^5*d) - (a*(a^2 - 2*b^2)*sin(c + d*x)^4)/(4*b^4*d) + ((a^2 - 2*b^2)*sin(c + d*x)^5)/(5*b^3*d) - (a*sin(c + d*x)^6)/(6*b^2*d) + sin(c + d*x)^7/(7*b*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, (a^2*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(b^7*d) - (a*(a^2 - b^2)^2*sin(c + d*x))/(b^6*d) + ((a^2 - b^2)^2*sin(c + d*x)^2)/(2*b^5*d) - (a*(a^2 - 2*b^2)*sin(c + d*x)^3)/(3*b^4*d) + ((a^2 - 2*b^2)*sin(c + d*x)^4)/(4*b^3*d) - (a*sin(c + d*x)^5)/(5*b^2*d) + sin(c + d*x)^6/(6*b*d)]
    @test_int [cos(c + d*x)^5*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, -((a*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(b^6*d)) + ((a^2 - b^2)^2*sin(c + d*x))/(b^5*d) - (a*(a^2 - 2*b^2)*sin(c + d*x)^2)/(2*b^4*d) + ((a^2 - 2*b^2)*sin(c + d*x)^3)/(3*b^3*d) - (a*sin(c + d*x)^4)/(4*b^2*d) + sin(c + d*x)^5/(5*b*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) - ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a*b^4*d) + ((a^2 - 2*b^2)*sin(c + d*x))/(b^3*d) - (a*sin(c + d*x)^2)/(2*b^2*d) + sin(c + d*x)^3/(3*b*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (b*log(sin(c + d*x)))/(a^2*d) + ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^2*b^3*d) - (a*sin(c + d*x))/(b^2*d) + sin(c + d*x)^2/(2*b*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) - ((2*a^2 - b^2)*log(sin(c + d*x)))/(a^3*d) - ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^3*b^2*d) + sin(c + d*x)/(b*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^4/(a + b*sin(c + d*x)), x, 4, ((2*a^2 - b^2)*csc(c + d*x))/(a^3*d) + (b*csc(c + d*x)^2)/(2*a^2*d) - csc(c + d*x)^3/(3*a*d) + (b*(2*a^2 - b^2)*log(sin(c + d*x)))/(a^4*d) + ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^4*b*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^5/(a + b*sin(c + d*x)), x, 3, -((b*(2*a^2 - b^2)*csc(c + d*x))/(a^4*d)) + ((2*a^2 - b^2)*csc(c + d*x)^2)/(2*a^3*d) + (b*csc(c + d*x)^3)/(3*a^2*d) - csc(c + d*x)^4/(4*a*d) + ((a^2 - b^2)^2*log(sin(c + d*x)))/(a^5*d) - ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^5*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^6/(a + b*sin(c + d*x)), x, 4, -(((a^2 - b^2)^2*csc(c + d*x))/(a^5*d)) - (b*(2*a^2 - b^2)*csc(c + d*x)^2)/(2*a^4*d) + ((2*a^2 - b^2)*csc(c + d*x)^3)/(3*a^3*d) + (b*csc(c + d*x)^4)/(4*a^2*d) - csc(c + d*x)^5/(5*a*d) - (b*(a^2 - b^2)^2*log(sin(c + d*x)))/(a^6*d) + (b*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^6*d)]
    @test_int [cos(c + d*x)^5*csc(c + d*x)^7/(a + b*sin(c + d*x)), x, 4, (b*(a^2 - b^2)^2*csc(c + d*x))/(a^6*d) - ((a^2 - b^2)^2*csc(c + d*x)^2)/(2*a^5*d) - (b*(2*a^2 - b^2)*csc(c + d*x)^3)/(3*a^4*d) + ((2*a^2 - b^2)*csc(c + d*x)^4)/(4*a^3*d) + (b*csc(c + d*x)^5)/(5*a^2*d) - csc(c + d*x)^6/(6*a*d) + (b^2*(a^2 - b^2)^2*log(sin(c + d*x)))/(a^7*d) - (b^2*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^7*d)]

    @test_int [cos(c + d*x)^6*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 11, -(((128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*x)/(128*b^9)) + (2*a^3*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^9*d) - (a*(105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*cos(c + d*x))/(105*b^8*d) + ((64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*cos(c + d*x)*sin(c + d*x))/(128*b^7*d) - (a*(35*a^4 - 77*a^2*b^2 + 45*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(105*b^6*d) + ((48*a^4 - 104*a^2*b^2 + 59*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(192*b^5*d) + (cos(c + d*x)*sin(c + d*x)^4)/(4*a*d) - ((28*a^4 - 60*a^2*b^2 + 35*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(140*a*b^4*d) - (b*cos(c + d*x)*sin(c + d*x)^5)/(5*a^2*d) + ((40*a^4 - 85*a^2*b^2 + 48*b^4)*cos(c + d*x)*sin(c + d*x)^5)/(240*a^2*b^3*d) - (a*cos(c + d*x)*sin(c + d*x)^6)/(7*b^2*d) + (cos(c + d*x)*sin(c + d*x)^7)/(8*b*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 10, (a*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^8) - (2*a^2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^8*d) + ((105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*cos(c + d*x))/(105*b^7*d) - (a*(8*a^4 - 18*a^2*b^2 + 11*b^4)*cos(c + d*x)*sin(c + d*x))/(16*b^6*d) + ((35*a^4 - 77*a^2*b^2 + 45*b^4)*cos(c + d*x)*sin(c + d*x)^2)/(105*b^5*d) + (cos(c + d*x)*sin(c + d*x)^3)/(3*a*d) - ((6*a^4 - 13*a^2*b^2 + 8*b^4)*cos(c + d*x)*sin(c + d*x)^3)/(24*a*b^4*d) - (b*cos(c + d*x)*sin(c + d*x)^4)/(4*a^2*d) + ((28*a^4 - 60*a^2*b^2 + 35*b^4)*cos(c + d*x)*sin(c + d*x)^4)/(140*a^2*b^3*d) - (a*cos(c + d*x)*sin(c + d*x)^5)/(6*b^2*d) + (cos(c + d*x)*sin(c + d*x)^6)/(7*b*d)]
    @test_int [cos(c + d*x)^6*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 7, -(((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^7)) + (2*a*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^7*d) - (cos(c + d*x)^5*(6*a - 5*b*sin(c + d*x)))/(30*b^2*d) + (cos(c + d*x)^3*(8*a*(a^2 - b^2) - b*(6*a^2 - 5*b^2)*sin(c + d*x)))/(24*b^4*d) - (cos(c + d*x)*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^2*b^2 + 5*b^4)*sin(c + d*x)))/(16*b^6*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 14, -((3*x)/(8*b)) - ((a^2 - 3*b^2)*x)/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*x)/b^5 + (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b^5*d) - atanh(cos(c + d*x))/(a*d) - (a*cos(c + d*x))/(b^2*d) - (a*(a^2 - 3*b^2)*cos(c + d*x))/(b^4*d) + (a*cos(c + d*x)^3)/(3*b^2*d) + (3*cos(c + d*x)*sin(c + d*x))/(8*b*d) + ((a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x))/(2*b^3*d) + (cos(c + d*x)*sin(c + d*x)^3)/(4*b*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 13, (a*x)/(2*b^2) + (a*(a^2 - 3*b^2)*x)/b^4 - (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*b^4*d) + (b*atanh(cos(c + d*x)))/(a^2*d) + cos(c + d*x)/(b*d) + ((a^2 - 3*b^2)*cos(c + d*x))/(b^3*d) - cos(c + d*x)^3/(3*b*d) - cot(c + d*x)/(a*d) - (a*cos(c + d*x)*sin(c + d*x))/(2*b^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 6, -(((2*a^2 - 5*b^2)*x)/(2*b^3)) + (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*b^3*d) + ((5*a^2 - 2*b^2)*atanh(cos(c + d*x)))/(2*a^3*d) - (a*cos(c + d*x))/(b^2*d) + (b*cot(c + d*x))/(a^2*d) - (cot(c + d*x)*csc(c + d*x))/(2*a*d) + (cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^4/(a + b*sin(c + d*x)), x, 13, (a*x)/b^2 - (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*b^2*d) + (b*atanh(cos(c + d*x)))/(2*a^2*d) - (b*(3*a^2 - b^2)*atanh(cos(c + d*x)))/(a^4*d) + cos(c + d*x)/(b*d) - cot(c + d*x)/(a*d) + ((3*a^2 - b^2)*cot(c + d*x))/(a^3*d) - cot(c + d*x)^3/(3*a*d) + (b*cot(c + d*x)*csc(c + d*x))/(2*a^2*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^5/(a + b*sin(c + d*x)), x, 15, -(x/b) + (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*b*d) - ((15*a^4 - 20*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^5*d) + (b*(-2*a^2 + b^2)*cot(c + d*x))/(a^4*d) + (b*cot(c + d*x)^3)/(3*a^2*d) + ((7*a^2 - 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^3*d) - (cot(c + d*x)^3*csc(c + d*x))/(4*a*d), -(x/b) + (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*b*d) - (3*atanh(cos(c + d*x)))/(8*a*d) + ((3*a^2 - b^2)*atanh(cos(c + d*x)))/(2*a^3*d) - ((3*a^4 - 3*a^2*b^2 + b^4)*atanh(cos(c + d*x)))/(a^5*d) + (b*cot(c + d*x))/(a^2*d) - (b*(3*a^2 - b^2)*cot(c + d*x))/(a^4*d) + (b*cot(c + d*x)^3)/(3*a^2*d) - (3*cot(c + d*x)*csc(c + d*x))/(8*a*d) + ((3*a^2 - b^2)*cot(c + d*x)*csc(c + d*x))/(2*a^3*d) - (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^6/(a + b*sin(c + d*x)), x, 9, -((2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d)) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*cot(c + d*x))/(15*a^5*d) + (b*(-9*a^2 + 4*b^2)*cot(c + d*x)*csc(c + d*x))/(8*a^4*d) + ((11*a^2 - 5*b^2)*cot(c + d*x)*csc(c + d*x)^2)/(15*a^3*d) + (b*cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d), -((2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*d)) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*atanh(cos(c + d*x)))/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*cot(c + d*x))/(15*a^5*d) - (cot(c + d*x)*csc(c + d*x))/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*cot(c + d*x)*csc(c + d*x))/(8*a^4*b*d) + (a*cot(c + d*x)*csc(c + d*x)^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(30*a^3*b^2*d) + (b*cot(c + d*x)*csc(c + d*x)^3)/(4*a^2*d) - (cot(c + d*x)*csc(c + d*x)^4)/(5*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^7/(a + b*sin(c + d*x)), x, 10, (2*b*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^7*d) + ((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*atanh(cos(c + d*x)))/(16*a^7*d) + (b*(23*a^4 - 35*a^2*b^2 + 15*b^4)*cot(c + d*x))/(15*a^6*d) - ((11*a^4 - 18*a^2*b^2 + 8*b^4)*cot(c + d*x)*csc(c + d*x))/(16*a^5*d) - (cot(c + d*x)*csc(c + d*x)^2)/(2*b*d) + ((15*a^4 - 22*a^2*b^2 + 10*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(30*a^4*b*d) + (a*cot(c + d*x)*csc(c + d*x)^3)/(3*b^2*d) - ((8*a^4 - 13*a^2*b^2 + 6*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(24*a^3*b^2*d) + (b*cot(c + d*x)*csc(c + d*x)^4)/(5*a^2*d) - (cot(c + d*x)*csc(c + d*x)^5)/(6*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^8/(a + b*sin(c + d*x)), x, 11, -((2*b^2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^8*d)) - (b*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*atanh(cos(c + d*x)))/(16*a^8*d) + ((15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*cot(c + d*x))/(105*a^7*d) + (b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*cot(c + d*x)*csc(c + d*x))/(16*a^6*d) - ((45*a^4 - 77*a^2*b^2 + 35*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(105*a^5*d) - (cot(c + d*x)*csc(c + d*x)^3)/(3*b*d) + ((8*a^4 - 13*a^2*b^2 + 6*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(24*a^4*b*d) + (a*cot(c + d*x)*csc(c + d*x)^4)/(4*b^2*d) - ((35*a^4 - 60*a^2*b^2 + 28*b^4)*cot(c + d*x)*csc(c + d*x)^4)/(140*a^3*b^2*d) + (b*cot(c + d*x)*csc(c + d*x)^5)/(6*a^2*d) - (cot(c + d*x)*csc(c + d*x)^6)/(7*a*d)]
    @test_int [cos(c + d*x)^6*csc(c + d*x)^9/(a + b*sin(c + d*x)), x, 12, (2*b^3*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^9*d) + ((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*atanh(cos(c + d*x)))/(128*a^9*d) - (b*(15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*cot(c + d*x))/(105*a^8*d) + ((5*a^6 - 88*a^4*b^2 + 144*a^2*b^4 - 64*b^6)*cot(c + d*x)*csc(c + d*x))/(128*a^7*d) + (b*(45*a^4 - 77*a^2*b^2 + 35*b^4)*cot(c + d*x)*csc(c + d*x)^2)/(105*a^6*d) - ((59*a^4 - 104*a^2*b^2 + 48*b^4)*cot(c + d*x)*csc(c + d*x)^3)/(192*a^5*d) - (cot(c + d*x)*csc(c + d*x)^4)/(4*b*d) + ((35*a^4 - 60*a^2*b^2 + 28*b^4)*cot(c + d*x)*csc(c + d*x)^4)/(140*a^4*b*d) + (a*cot(c + d*x)*csc(c + d*x)^5)/(5*b^2*d) - ((48*a^4 - 85*a^2*b^2 + 40*b^4)*cot(c + d*x)*csc(c + d*x)^5)/(240*a^3*b^2*d) + (b*cot(c + d*x)*csc(c + d*x)^6)/(7*a^2*d) - (cot(c + d*x)*csc(c + d*x)^7)/(8*a*d)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [sec(c + d*x)*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, -log(1 - sin(c + d*x))/(2*(a + b)*d) - log(1 + sin(c + d*x))/(2*(a - b)*d) + (a^3*log(a + b*sin(c + d*x)))/(b^2*(a^2 - b^2)*d) - sin(c + d*x)/(b*d)]
    @test_int [sec(c + d*x)*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -log(1 - sin(c + d*x))/(2*(a + b)*d) + log(1 + sin(c + d*x))/(2*(a - b)*d) - (a^2*log(a + b*sin(c + d*x)))/(b*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 3, -log(1 - sin(c + d*x))/(2*(a + b)*d) - log(1 + sin(c + d*x))/(2*(a - b)*d) + (a*log(a + b*sin(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, -log(1 - sin(c + d*x))/(2*(a + b)*d) + log(sin(c + d*x))/(a*d) - log(1 + sin(c + d*x))/(2*(a - b)*d) + (b^2*log(a + b*sin(c + d*x)))/(a*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - log(1 - sin(c + d*x))/(2*(a + b)*d) - (b*log(sin(c + d*x)))/(a^2*d) + log(1 + sin(c + d*x))/(2*(a - b)*d) - (b^3*log(a + b*sin(c + d*x)))/(a^2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) - log(1 - sin(c + d*x))/(2*(a + b)*d) + ((a^2 + b^2)*log(sin(c + d*x)))/(a^3*d) - log(1 + sin(c + d*x))/(2*(a - b)*d) + (b^4*log(a + b*sin(c + d*x)))/(a^3*(a^2 - b^2)*d)]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^5/(a + b*sin(c + d*x)), x, 14, (3*b*x)/(2*(a^2 - b^2)) - (a^2*(2*a^2 + b^2)*x)/(2*b^3*(a^2 - b^2)) + (2*a^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) + (a*cos(c + d*x))/((a^2 - b^2)*d) - (a^3*cos(c + d*x))/(b^2*(a^2 - b^2)*d) + (a*sec(c + d*x))/((a^2 - b^2)*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d) - (3*b*tan(c + d*x))/(2*(a^2 - b^2)*d) + (b*sin(c + d*x)^2*tan(c + d*x))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + b*sin(c + d*x)), x, 12, -((a*x)/(a^2 - b^2)) + (a^3*x)/(b^2*(a^2 - b^2)) - (2*a^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) + (a^2*cos(c + d*x))/(b*(a^2 - b^2)*d) - (b*cos(c + d*x))/((a^2 - b^2)*d) - (b*sec(c + d*x))/((a^2 - b^2)*d) + (a*tan(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 9, -((a^2*x)/(b*(a^2 - b^2))) + (b*x)/(a^2 - b^2) + (2*a^3*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(3/2)*d) + (a*sec(c + d*x))/((a^2 - b^2)*d) - (b*tan(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 8, -((2*a^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d)) - (b*sec(c + d*x))/((a^2 - b^2)*d) + (a*tan(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 5, (2*a*b*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) + (sec(c + d*x)*(a - b*sin(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 10, (2*b^3*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*(a^2 - b^2)^(3/2)*d) - atanh(cos(c + d*x))/(a*d) + sec(c + d*x)/(a*d) + (b*sec(c + d*x)*(b - a*sin(c + d*x)))/(a*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 13, -((2*b^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d) - (b*sec(c + d*x))/((a^2 - b^2)*d) + (a*tan(c + d*x))/((a^2 - b^2)*d), -((2*b^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d) - (b*sec(c + d*x))/(a^2*d) - (b^2*sec(c + d*x)*(b - a*sin(c + d*x)))/(a^2*(a^2 - b^2)*d) + tan(c + d*x)/(a*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 17, (2*b^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - ((3*a^2 + 2*b^2)*atanh(cos(c + d*x)))/(2*a^3*d) + (b*cot(c + d*x))/(a^2*d) + ((3*a^2 - b^2)*sec(c + d*x))/(2*a*(a^2 - b^2)*d) - (csc(c + d*x)^2*sec(c + d*x))/(2*a*d) - (b*tan(c + d*x))/((a^2 - b^2)*d), (2*b^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - (3*atanh(cos(c + d*x)))/(2*a*d) - (b^2*atanh(cos(c + d*x)))/(a^3*d) + (b*cot(c + d*x))/(a^2*d) + (3*sec(c + d*x))/(2*a*d) + (b^2*sec(c + d*x))/(a^3*d) - (csc(c + d*x)^2*sec(c + d*x))/(2*a*d) + (b^3*sec(c + d*x)*(b - a*sin(c + d*x)))/(a^3*(a^2 - b^2)*d) - (b*tan(c + d*x))/(a^2*d)]

    @test_int [sec(c + d*x)^3*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, ((2*a + b)*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) + ((2*a - b)*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) - (a^3*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) + (sec(c + d*x)^2*(a - b*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^3*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 5, (a*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) - (a*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) + (a^2*b*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^3*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 5, -((b*log(1 - sin(c + d*x)))/(4*(a + b)^2*d)) + (b*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) - (a*b^2*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) + (sec(c + d*x)^2*(a - b*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^3*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, -((2*a + 3*b)*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) + log(sin(c + d*x))/(a*d) - ((2*a - 3*b)*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) - (b^4*log(a + b*sin(c + d*x)))/(a*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - sin(c + d*x))) + 1/(4*(a - b)*d*(1 + sin(c + d*x)))]
    @test_int [sec(c + d*x)^3*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - ((3*a + 4*b)*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) - (b*log(sin(c + d*x)))/(a^2*d) + ((3*a - 4*b)*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) + (b^5*log(a + b*sin(c + d*x)))/(a^2*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - sin(c + d*x))) - 1/(4*(a - b)*d*(1 + sin(c + d*x)))]
    @test_int [sec(c + d*x)^3*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) - ((4*a + 5*b)*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) + ((2*a^2 + b^2)*log(sin(c + d*x)))/(a^3*d) - ((4*a - 5*b)*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) - (b^6*log(a + b*sin(c + d*x)))/(a^3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - sin(c + d*x))) + 1/(4*(a - b)*d*(1 + sin(c + d*x)))]

    @test_int [sec(c + d*x)^4*sin(c + d*x)^4/(a + b*sin(c + d*x)), x, 13, (2*a^4*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) + (a^2*b*sec(c + d*x))/((a^2 - b^2)^2*d) + (b*sec(c + d*x))/((a^2 - b^2)*d) - (b*sec(c + d*x)^3)/(3*(a^2 - b^2)*d) - (a^3*tan(c + d*x))/((a^2 - b^2)^2*d) + (a*tan(c + d*x)^3)/(3*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 10, -((2*a^3*b*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (a*sec(c + d*x)^3)/(3*(a^2 - b^2)*d) - (a^2*sec(c + d*x)*(a - b*sin(c + d*x)))/((a^2 - b^2)^2*d) - (b*tan(c + d*x)^3)/(3*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 10, (2*a^2*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (b*sec(c + d*x)^3)/(3*(a^2 - b^2)*d) + (a^2*sec(c + d*x)*(b - a*sin(c + d*x)))/((a^2 - b^2)^2*d) + (a*tan(c + d*x))/((a^2 - b^2)*d) + (a*tan(c + d*x)^3)/(3*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, -((2*a*b^3*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (sec(c + d*x)^3*(a - b*sin(c + d*x)))/(3*(a^2 - b^2)*d) - (sec(c + d*x)*(3*a*b^2 - b*(a^2 + 2*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 12, -((2*b^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*(a^2 - b^2)^(5/2)*d)) - atanh(cos(c + d*x))/(a*d) + sec(c + d*x)/(a*d) + sec(c + d*x)^3/(3*a*d) + (b*sec(c + d*x)^3*(b - a*sin(c + d*x)))/(3*a*(a^2 - b^2)*d) - (b*sec(c + d*x)*(3*b^3 + a*(2*a^2 - 5*b^2)*sin(c + d*x)))/(3*a*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 15, (2*b^6*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(5/2)*d) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d) + (b*(-a^2 + 2*b^2)*sec(c + d*x))/((a^2 - b^2)^2*d) + (b*sec(c + d*x)^3*(-a + b*sin(c + d*x)))/(3*a*(a^2 - b^2)*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*tan(c + d*x))/(3*a*(a^2 - b^2)^2*d) + tan(c + d*x)^3/(3*a*d), (2*b^6*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(5/2)*d) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d) - (b*sec(c + d*x))/(a^2*d) - (b*sec(c + d*x)^3)/(3*a^2*d) - (b^2*sec(c + d*x)^3*(b - a*sin(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)*(3*b^3 + a*(2*a^2 - 5*b^2)*sin(c + d*x)))/(3*a^2*(a^2 - b^2)^2*d) + (2*tan(c + d*x))/(a*d) + tan(c + d*x)^3/(3*a*d)]
    @test_int [sec(c + d*x)^4*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 20, -((2*b^7*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) - (5*atanh(cos(c + d*x)))/(2*a*d) - (b^2*atanh(cos(c + d*x)))/(a^3*d) + (b*cot(c + d*x))/(a^2*d) + (5*sec(c + d*x))/(2*a*d) + (b^2*sec(c + d*x))/(a^3*d) + (5*sec(c + d*x)^3)/(6*a*d) + (b^2*sec(c + d*x)^3)/(3*a^3*d) - (csc(c + d*x)^2*sec(c + d*x)^3)/(2*a*d) + (b^3*sec(c + d*x)^3*(b - a*sin(c + d*x)))/(3*a^3*(a^2 - b^2)*d) - (b^3*sec(c + d*x)*(3*b^3 + a*(2*a^2 - 5*b^2)*sin(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d) - (2*b*tan(c + d*x))/(a^2*d) - (b*tan(c + d*x)^3)/(3*a^2*d)]

    @test_int [sec(c + d*x)^5*sin(c + d*x)^8/(a + b*sin(c + d*x)), x, 6, -(((35*a^2 + 57*a*b + 24*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) + ((35*a^2 - 57*a*b + 24*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (a^8*log(a + b*sin(c + d*x)))/(b^3*(a^2 - b^2)^3*d) + (a*sin(c + d*x))/(b^2*d) - sin(c + d*x)^2/(2*b*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(4*b*(4*a^2 - 3*b^2) - a*(13*a^2 - 9*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^7/(a + b*sin(c + d*x)), x, 6, -(((24*a^2 + 37*a*b + 15*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) - ((24*a^2 - 37*a*b + 15*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (a^7*log(a + b*sin(c + d*x)))/(b^2*(a^2 - b^2)^3*d) - sin(c + d*x)/(b*d) + (sec(c + d*x)^4*(a - b*sin(c + d*x)))/(4*(a^2 - b^2)*d) - (sec(c + d*x)^2*(4*a*(3*a^2 - 2*b^2) - b*(13*a^2 - 9*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^6/(a + b*sin(c + d*x)), x, 6, -(((15*a^2 + 21*a*b + 8*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) + ((15*a^2 - 21*a*b + 8*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (a^6*log(a + b*sin(c + d*x)))/(b*(a^2 - b^2)^3*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(4*b*(3*a^2 - 2*b^2) - a*(9*a^2 - 5*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^5/(a + b*sin(c + d*x)), x, 5, -(((8*a^2 + 9*a*b + 3*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) - ((8*a^2 - 9*a*b + 3*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (a^5*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) + (sec(c + d*x)^4*(a - b*sin(c + d*x)))/(4*(a^2 - b^2)*d) - (sec(c + d*x)^2*(4*a*(2*a^2 - b^2) - b*(9*a^2 - 5*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4/(a + b*sin(c + d*x)), x, 6, -((a*(3*a + b)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) + (a*(3*a - b)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (a^4*b*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(4*b*(2*a^2 - b^2) - a*(5*a^2 - b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 6, (b*(3*a + b)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) - ((3*a - b)*b*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (a^3*b^2*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) + (sec(c + d*x)^4*(a - b*sin(c + d*x)))/(4*(a^2 - b^2)*d) - (sec(c + d*x)^2*(4*a^3 - b*(5*a^2 - b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 6, (a*(a + 3*b)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) - (a*(a - 3*b)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (a^2*b^3*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (a*sec(c + d*x)^2*(4*a*b - (a^2 + 3*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, -((b*(a + 3*b)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d)) + ((a - 3*b)*b*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (a*b^4*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) + (sec(c + d*x)^4*(a - b*sin(c + d*x)))/(4*(a^2 - b^2)*d) - (sec(c + d*x)^2*(4*a*b^2 - b*(a^2 + 3*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1/(a + b*sin(c + d*x)), x, 4, -((8*a^2 + 21*a*b + 15*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) + log(sin(c + d*x))/(a*d) - ((8*a^2 - 21*a*b + 15*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (b^6*log(a + b*sin(c + d*x)))/(a*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - sin(c + d*x))^2) + (5*a + 7*b)/(16*(a + b)^2*d*(1 - sin(c + d*x))) + 1/(16*(a - b)*d*(1 + sin(c + d*x))^2) + (5*a - 7*b)/(16*(a - b)^2*d*(1 + sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - ((15*a^2 + 37*a*b + 24*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) - (b*log(sin(c + d*x)))/(a^2*d) + ((15*a^2 - 37*a*b + 24*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (b^7*log(a + b*sin(c + d*x)))/(a^2*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - sin(c + d*x))^2) + (7*a + 9*b)/(16*(a + b)^2*d*(1 - sin(c + d*x))) - 1/(16*(a - b)*d*(1 + sin(c + d*x))^2) - (7*a - 9*b)/(16*(a - b)^2*d*(1 + sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, (b*csc(c + d*x))/(a^2*d) - csc(c + d*x)^2/(2*a*d) - ((24*a^2 + 57*a*b + 35*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) + ((3*a^2 + b^2)*log(sin(c + d*x)))/(a^3*d) - ((24*a^2 - 57*a*b + 35*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) + (b^8*log(a + b*sin(c + d*x)))/(a^3*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - sin(c + d*x))^2) + (9*a + 11*b)/(16*(a + b)^2*d*(1 - sin(c + d*x))) + 1/(16*(a - b)*d*(1 + sin(c + d*x))^2) + (9*a - 11*b)/(16*(a - b)^2*d*(1 + sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*(d*sin(e+f*x))^(n/2) / (a+b*sin(e+f*x))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(p/2)*sin(e+f*x)^n / (a+b*sin(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(sqrt(g*cos(e + f*x))*sin(e + f*x)^4)/(a + b*sin(e + f*x)), x, 21, (a^4*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (a^4*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (2*a^2*(g*cos(e + f*x))^(3/2))/(3*b^3*f*g) - (2*(g*cos(e + f*x))^(3/2))/(3*b*f*g) + (2*(g*cos(e + f*x))^(7/2))/(7*b*f*g^3) - (2*a^3*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(b^4*f*sqrt(cos(e + f*x))) - (4*a*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*b^2*f*sqrt(cos(e + f*x))) + (a^5*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (a^5*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (2*a*(g*cos(e + f*x))^(3/2)*sin(e + f*x))/(5*b^2*f*g)]
    @test_int [(sqrt(g*cos(e + f*x))*sin(e + f*x)^3)/(a + b*sin(e + f*x)), x, 18, -((a^3*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*(-a^2 + b^2)^(1/4)*f)) + (a^3*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*(-a^2 + b^2)^(1/4)*f) + (2*a*(g*cos(e + f*x))^(3/2))/(3*b^2*f*g) + (2*a^2*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(b^3*f*sqrt(cos(e + f*x))) + (4*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*b*f*sqrt(cos(e + f*x))) - (a^4*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^4*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (a^4*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^4*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (2*(g*cos(e + f*x))^(3/2)*sin(e + f*x))/(5*b*f*g)]
    @test_int [(sqrt(g*cos(e + f*x))*sin(e + f*x)^2)/(a + b*sin(e + f*x)), x, 15, (a^2*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (a^2*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (2*(g*cos(e + f*x))^(3/2))/(3*b*f*g) - (2*a*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(b^2*f*sqrt(cos(e + f*x))) + (a^3*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (a^3*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [(sqrt(g*cos(e + f*x))*sin(e + f*x)^1)/(a + b*sin(e + f*x)), x, 12, -((a*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(1/4)*f)) + (a*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(1/4)*f) + (2*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(b*f*sqrt(cos(e + f*x))) - (a^2*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (a^2*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [(sqrt(g*cos(e + f*x))*csc(e + f*x)^1)/(a + b*sin(e + f*x)), x, 16, (sqrt(g)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f) - (sqrt(b)*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(1/4)*f) - (sqrt(g)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f) + (sqrt(b)*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(1/4)*f) - (g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [(sqrt(g*cos(e + f*x))*csc(e + f*x)^2)/(a + b*sin(e + f*x)), x, 19, -((b*sqrt(g)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f)) + (b^(3/2)*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(1/4)*f) + (b*sqrt(g)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f) - (b^(3/2)*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(1/4)*f) - ((g*cos(e + f*x))^(3/2)*csc(e + f*x))/(a*f*g) - (sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(a*f*sqrt(cos(e + f*x))) + (b*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (b*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [(sqrt(g*cos(e + f*x))*csc(e + f*x)^3)/(a + b*sin(e + f*x)), x, 25, (sqrt(g)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f) + (b^2*sqrt(g)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) - (b^(5/2)*sqrt(g)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*(-a^2 + b^2)^(1/4)*f) - (sqrt(g)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f) - (b^2*sqrt(g)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) + (b^(5/2)*sqrt(g)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*(-a^2 + b^2)^(1/4)*f) + (b*(g*cos(e + f*x))^(3/2)*csc(e + f*x))/(a^2*f*g) - ((g*cos(e + f*x))^(3/2)*csc(e + f*x)^2)/(2*a*f*g) + (b*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(a^2*f*sqrt(cos(e + f*x))) - (b^2*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a^2*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (b^2*g*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a^2*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]

    @test_int [((g*cos(e + f*x))^(3/2)*sin(e + f*x)^3)/(a + b*sin(e + f*x)), x, 24, (a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*f) + (a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*f) - (2*a^3*g*sqrt(g*cos(e + f*x)))/(b^4*f) + (2*a*(g*cos(e + f*x))^(5/2))/(5*b^2*f*g) - (2*a^4*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(b^5*f*sqrt(g*cos(e + f*x))) + (2*a^2*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*b^3*f*sqrt(g*cos(e + f*x))) + (4*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(21*b*f*sqrt(g*cos(e + f*x))) + (a^4*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (a^4*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (2*a^2*g*sqrt(g*cos(e + f*x))*sin(e + f*x))/(3*b^3*f) + (4*g*sqrt(g*cos(e + f*x))*sin(e + f*x))/(21*b*f) - (2*(g*cos(e + f*x))^(5/2)*sin(e + f*x))/(7*b*f*g)]
    @test_int [((g*cos(e + f*x))^(3/2)*sin(e + f*x)^2)/(a + b*sin(e + f*x)), x, 20, -((a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*f)) - (a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*f) + (2*a^2*g*sqrt(g*cos(e + f*x)))/(b^3*f) - (2*(g*cos(e + f*x))^(5/2))/(5*b*f*g) + (2*a^3*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(b^4*f*sqrt(g*cos(e + f*x))) - (2*a*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*b^2*f*sqrt(g*cos(e + f*x))) - (a^3*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^4*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (a^3*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^4*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (2*a*g*sqrt(g*cos(e + f*x))*sin(e + f*x))/(3*b^2*f)]
    @test_int [((g*cos(e + f*x))^(3/2)*sin(e + f*x)^1)/(a + b*sin(e + f*x)), x, 13, (a*(-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*f) + (a*(-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*f) - (2*(3*a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(3*b^3*f*sqrt(g*cos(e + f*x))) + (a^2*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (a^2*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (2*g*sqrt(g*cos(e + f*x))*(3*a - b*sin(e + f*x)))/(3*b^2*f)]
    @test_int [((g*cos(e + f*x))^(3/2)*csc(e + f*x)^1)/(a + b*sin(e + f*x)), x, 21, -((g^(3/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f)) + ((-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*sqrt(b)*f) - (g^(3/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f) + ((-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*sqrt(b)*f) - (2*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(b*f*sqrt(g*cos(e + f*x))) + ((a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + ((a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*csc(e + f*x)^2)/(a + b*sin(e + f*x)), x, 24, (b*g^(3/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f) - (sqrt(b)*(-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*f) + (b*g^(3/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f) - (sqrt(b)*(-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*f) - (g*sqrt(g*cos(e + f*x))*csc(e + f*x))/(a*f) + (g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(a*f*sqrt(g*cos(e + f*x))) - ((a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - ((a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(3/2)*csc(e + f*x)^3)/(a + b*sin(e + f*x)), x, 30, (g^(3/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f) - (b^2*g^(3/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*f) + (g^(3/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f) - (b^2*g^(3/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*f) + (b*g*sqrt(g*cos(e + f*x))*csc(e + f*x))/(a^2*f) - (g*sqrt(g*cos(e + f*x))*csc(e + f*x)^2)/(2*a*f) - (b*g^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(a^2*f*sqrt(g*cos(e + f*x))) + (b*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (b*(a^2 - b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]

    @test_int [((g*cos(e + f*x))^(5/2)*sin(e + f*x)^3)/(a + b*sin(e + f*x)), x, 24, -((a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(11/2)*f)) + (a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(11/2)*f) - (2*a^3*g*(g*cos(e + f*x))^(3/2))/(3*b^4*f) + (2*a*(g*cos(e + f*x))^(7/2))/(7*b^2*f*g) - (2*a^4*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(b^5*f*sqrt(cos(e + f*x))) + (6*a^2*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*b^3*f*sqrt(cos(e + f*x))) + (4*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(15*b*f*sqrt(cos(e + f*x))) + (a^4*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^6*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (a^4*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^6*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (2*a^2*g*(g*cos(e + f*x))^(3/2)*sin(e + f*x))/(5*b^3*f) + (4*g*(g*cos(e + f*x))^(3/2)*sin(e + f*x))/(45*b*f) - (2*(g*cos(e + f*x))^(7/2)*sin(e + f*x))/(9*b*f*g)]
    @test_int [((g*cos(e + f*x))^(5/2)*sin(e + f*x)^2)/(a + b*sin(e + f*x)), x, 20, (a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*f) - (a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(9/2)*f) + (2*a^2*g*(g*cos(e + f*x))^(3/2))/(3*b^3*f) - (2*(g*cos(e + f*x))^(7/2))/(7*b*f*g) + (2*a^3*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(b^4*f*sqrt(cos(e + f*x))) - (6*a*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(5*b^2*f*sqrt(cos(e + f*x))) - (a^3*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (a^3*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^5*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (2*a*g*(g*cos(e + f*x))^(3/2)*sin(e + f*x))/(5*b^2*f)]
    @test_int [((g*cos(e + f*x))^(5/2)*sin(e + f*x)^1)/(a + b*sin(e + f*x)), x, 13, -((a*(-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*f)) + (a*(-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*f) - (2*(5*a^2 - 3*b^2)*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(5*b^3*f*sqrt(cos(e + f*x))) + (a^2*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^4*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (a^2*(a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^4*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (2*g*(g*cos(e + f*x))^(3/2)*(5*a - 3*b*sin(e + f*x)))/(15*b^2*f)]
    @test_int [((g*cos(e + f*x))^(5/2)*csc(e + f*x)^1)/(a + b*sin(e + f*x)), x, 21, (g^(5/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*b^(3/2)*f) - (g^(5/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a*f) + ((-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*b^(3/2)*f) - (2*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(b*f*sqrt(cos(e + f*x))) + ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^2*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^2*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(5/2)*csc(e + f*x)^2)/(a + b*sin(e + f*x)), x, 24, -((b*g^(5/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f)) + ((-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*sqrt(b)*f) + (b*g^(5/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*sqrt(b)*f) - (g*(g*cos(e + f*x))^(3/2)*csc(e + f*x))/(a*f) - (g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*f*sqrt(cos(e + f*x))) - ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*b*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*b*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]
    @test_int [((g*cos(e + f*x))^(5/2)*csc(e + f*x)^3)/(a + b*sin(e + f*x)), x, 30, -((3*g^(5/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f)) + (b^2*g^(5/2)*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) - (sqrt(b)*(-a^2 + b^2)^(3/4)*g^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*f) + (3*g^(5/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f) - (b^2*g^(5/2)*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f) + (sqrt(b)*(-a^2 + b^2)^(3/4)*g^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*f) + (b*g*(g*cos(e + f*x))^(3/2)*csc(e + f*x))/(a^2*f) - (g*(g*cos(e + f*x))^(3/2)*csc(e + f*x)^2)/(2*a*f) + (b*g^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a^2*f*sqrt(cos(e + f*x))) + ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a^2*(b - sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + ((a^2 - b^2)*g^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a^2*(b + sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [sin(e + f*x)^4/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 23, -((a^4*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g))) - (a^4*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - (2*a^2*sqrt(g*cos(e + f*x)))/(b^3*f*g) - (2*sqrt(g*cos(e + f*x)))/(b*f*g) + (2*(g*cos(e + f*x))^(5/2))/(5*b*f*g^3) - (2*a^3*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(b^4*f*sqrt(g*cos(e + f*x))) - (4*a*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*b^2*f*sqrt(g*cos(e + f*x))) + (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^4*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^4*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (2*a*sqrt(g*cos(e + f*x))*sin(e + f*x))/(3*b^2*f*g)]
    @test_int [sin(e + f*x)^3/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 19, (a^3*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (a^3*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (2*a*sqrt(g*cos(e + f*x)))/(b^2*f*g) + (2*a^2*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(b^3*f*sqrt(g*cos(e + f*x))) + (4*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(3*b*f*sqrt(g*cos(e + f*x))) - (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^3*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (2*sqrt(g*cos(e + f*x))*sin(e + f*x))/(3*b*f*g)]
    @test_int [sin(e + f*x)^2/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 15, -((a^2*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g))) - (a^2*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - (2*sqrt(g*cos(e + f*x)))/(b*f*g) - (2*a*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(b^2*f*sqrt(g*cos(e + f*x))) + (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) + (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [sin(e + f*x)^1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 12, (a*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (a*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (2*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(b*f*sqrt(g*cos(e + f*x))) - (a^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b^2 + b*sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) - (a^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [csc(e + f*x)^1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 16, -(atan(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*sqrt(g))) + (b^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - atanh(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*sqrt(g)) + (b^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - (b*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (b*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [csc(e + f*x)^2/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 19, (b*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*sqrt(g)) - (b^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (b*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*sqrt(g)) - (b^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - (sqrt(g*cos(e + f*x))*csc(e + f*x))/(a*f*g) + (sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(a*f*sqrt(g*cos(e + f*x))) + (b^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a*(a^2 - b^2 + b*sqrt(-a^2 + b^2))*f*sqrt(g*cos(e + f*x))) + (b^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]
    @test_int [csc(e + f*x)^3/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x))), x, 25, (-3*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f*sqrt(g)) - (b^2*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f*sqrt(g)) + (b^(7/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*(-a^2 + b^2)^(3/4)*f*sqrt(g)) - (3*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(4*a*f*sqrt(g)) - (b^2*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^3*f*sqrt(g)) + (b^(7/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^3*(-a^2 + b^2)^(3/4)*f*sqrt(g)) + (b*sqrt(g*cos(e + f*x))*csc(e + f*x))/(a^2*f*g) - (sqrt(g*cos(e + f*x))*csc(e + f*x)^2)/(2*a*f*g) - (b*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(a^2*f*sqrt(g*cos(e + f*x))) - (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x))) - (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(a^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*sqrt(g*cos(e + f*x)))]

    @test_int [sin(e + f*x)^4/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 22, (a^4*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^4*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) + (2*a^2*(g*cos(e + f*x))^(3/2))/(3*b*(a^2 - b^2)*f*g^3) - (2*b*(g*cos(e + f*x))^(3/2))/(3*(a^2 - b^2)*f*g^3) - (4*a*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/((a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) + (2*a^3*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(b^2*(a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) - (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^3*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) - (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^3*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (2*a*sin(e + f*x))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]
    @test_int [sin(e + f*x)^3/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 18, -((a^3*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a^3*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*a)/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) - (2*a^2*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/(b*(a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) + (4*b*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/((a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) + (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b^2*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) - (2*b*sin(e + f*x))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]
    @test_int [sin(e + f*x)^2/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 15, (a^2*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^2*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) - (2*a*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/((a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) - (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) - (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (2*a*sin(e + f*x))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]
    @test_int [sin(e + f*x)^1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 13, -((a*sqrt(b)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a*sqrt(b)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*b*sqrt(g*cos(e + f*x))*Elliptic.E((e + f*x)/2, 2))/((a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) + (a^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (a^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (2*(a - b*sin(e + f*x)))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]
    @test_int [csc(e + f*x)^1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 21, atan(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*g^(3/2)) - (b^(5/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - atanh(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*g^(3/2)) + (b^(5/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + 2/(a*f*g*sqrt(g*cos(e + f*x))) + (2*b*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/((a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) + (b^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/((a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (b^2*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/((a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (2*b*(b - a*sin(e + f*x)))/(a*(a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]
    @test_int [csc(e + f*x)^2/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x, 25, -((b*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*g^(3/2))) + (b^(7/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (b*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*g^(3/2)) - (b^(7/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/(a^2*f*g*sqrt(g*cos(e + f*x))) - csc(e + f*x)/(a*f*g*sqrt(g*cos(e + f*x))) - (3*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*f*g^2*sqrt(cos(e + f*x))) - (2*b^2*sqrt(g*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), 2))/(a*(a^2 - b^2)*f*g^2*sqrt(cos(e + f*x))) - (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) - (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*f*g*sqrt(g*cos(e + f*x))) + (3*sin(e + f*x))/(a*f*g*sqrt(g*cos(e + f*x))) - (2*b^2*(b - a*sin(e + f*x)))/(a^2*(a^2 - b^2)*f*g*sqrt(g*cos(e + f*x)))]

    @test_int [sin(e + f*x)^4/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 22, -((a^4*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^4*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2)) + (2*a^2*sqrt(g*cos(e + f*x)))/(b*(a^2 - b^2)*f*g^3) - (2*b*sqrt(g*cos(e + f*x)))/((a^2 - b^2)*f*g^3) - (4*a*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) + (2*a^3*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(b^2*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) - (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^2*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) - (a^5*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(b^2*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (2*a*sin(e + f*x))/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]
    @test_int [sin(e + f*x)^3/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 18, (a^3*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a^3*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(sqrt(b)*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (2*a)/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2)) - (2*a^2*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(b*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) + (4*b*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(3*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) + (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (a^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/(b*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) - (2*b*sin(e + f*x))/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]
    @test_int [sin(e + f*x)^2/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 15, -((a^2*sqrt(b)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^2*sqrt(b)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2)) + (2*a*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(3*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) - (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) - (a^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (2*a*sin(e + f*x))/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]
    @test_int [sin(e + f*x)^1/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 13, (a*b^(3/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a*b^(3/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b*sqrt(cos(e + f*x))*Elliptic.F((e + f*x)/2, 2))/(3*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) + (a^2*b*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (a^2*b*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (e + f*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (2*(a - b*sin(e + f*x)))/(3*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]
    @test_int [csc(e + f*x)^1/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 21, -(atan(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*g^(5/2))) + (b^(7/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - atanh(sqrt(g*cos(e + f*x))/sqrt(g))/(a*f*g^(5/2)) + (b^(7/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + 2/(3*a*f*g*(g*cos(e + f*x))^(3/2)) - (2*b*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) + (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/((a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (b^3*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/((a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (2*b*(b - a*sin(e + f*x)))/(3*a*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]
    @test_int [csc(e + f*x)^2/((g*cos(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x, 25, (b*atan(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*g^(5/2)) - (b^(9/2)*atan((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (b*atanh(sqrt(g*cos(e + f*x))/sqrt(g)))/(a^2*f*g^(5/2)) - (b^(9/2)*atanh((sqrt(b)*sqrt(g*cos(e + f*x)))/((-a^2 + b^2)^(1/4)*sqrt(g))))/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*a^2*f*g*(g*cos(e + f*x))^(3/2)) - csc(e + f*x)/(a*f*g*(g*cos(e + f*x))^(3/2)) + (5*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*a*f*g^2*sqrt(g*cos(e + f*x))) + (2*b^2*sqrt(cos(e + f*x))*Elliptic.F((1/2)*(e + f*x), 2))/(3*a*(a^2 - b^2)*f*g^2*sqrt(g*cos(e + f*x))) - (b^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) - (b^4*sqrt(cos(e + f*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(e + f*x), 2))/(a*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*f*g^2*sqrt(g*cos(e + f*x))) + (5*sin(e + f*x))/(3*a*f*g*(g*cos(e + f*x))^(3/2)) - (2*b^2*(b - a*sin(e + f*x)))/(3*a^2*(a^2 - b^2)*f*g*(g*cos(e + f*x))^(3/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(p/2)*(d*sin(e+f*x))^(n/2) / (a+b*sin(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(5/2), x, 31, (a^2*d^(5/2)*sqrt(g)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^3*f) + (d^(5/2)*sqrt(g)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(4*sqrt(2)*b*f) - (a^2*d^(5/2)*sqrt(g)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^3*f) - (d^(5/2)*sqrt(g)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(4*sqrt(2)*b*f) - (a^2*d^(5/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^3*f) - (d^(5/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(8*sqrt(2)*b*f) + (a^2*d^(5/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^3*f) + (d^(5/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(8*sqrt(2)*b*f) - (2*sqrt(2)*a^3*d^3*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^3*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*a^3*d^3*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^3*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) - (d^2*(g*cos(e + f*x))^(3/2)*sqrt(d*sin(e + f*x)))/(2*b*f*g) - (a*d^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(b^2*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(3/2), x, 19, -((a*d^(3/2)*sqrt(g)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^2*f)) + (a*d^(3/2)*sqrt(g)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^2*f) + (a*d^(3/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^2*f) - (a*d^(3/2)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^2*f) + (2*sqrt(2)*a^2*d^2*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^2*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*a^2*d^2*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^2*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) + (d*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(b*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(1/2), x, 16, (sqrt(d)*sqrt(g)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*f) - (sqrt(d)*sqrt(g)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*f) - (sqrt(d)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*f) + (sqrt(d)*sqrt(g)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*f) - (2*sqrt(2)*a*d*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*a*d*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b*sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(1/2), x, 5, (2*sqrt(2)*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(sqrt(-a + b)*sqrt(a + b)*f*sqrt(d*sin(e + f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(3/2), x, 9, -((2*(g*cos(e + f*x))^(3/2))/(a*d*f*g*sqrt(d*sin(e + f*x)))) - (2*sqrt(2)*b*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*sqrt(-a + b)*sqrt(a + b)*d*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*sqrt(-a + b)*sqrt(a + b)*d*f*sqrt(d*sin(e + f*x))) - (2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a*d^2*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(5/2), x, 11, -((2*(g*cos(e + f*x))^(3/2))/(3*a*d*f*g*(d*sin(e + f*x))^(3/2))) + (2*b*(g*cos(e + f*x))^(3/2))/(a^2*d^2*f*g*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^2*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*sqrt(-a + b)*sqrt(a + b)*d^2*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^2*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*sqrt(-a + b)*sqrt(a + b)*d^2*f*sqrt(d*sin(e + f*x))) + (2*b*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^2*d^3*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(7/2), x, 16, -((2*(g*cos(e + f*x))^(3/2))/(5*a*d*f*g*(d*sin(e + f*x))^(5/2))) + (2*b*(g*cos(e + f*x))^(3/2))/(3*a^2*d^2*f*g*(d*sin(e + f*x))^(3/2)) - (4*(g*cos(e + f*x))^(3/2))/(5*a*d^3*f*g*sqrt(d*sin(e + f*x))) - (2*b^2*(g*cos(e + f*x))^(3/2))/(a^3*d^3*f*g*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^3*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^3*sqrt(-a + b)*sqrt(a + b)*d^3*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^3*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^3*sqrt(-a + b)*sqrt(a + b)*d^3*f*sqrt(d*sin(e + f*x))) - (4*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*a*d^4*f*sqrt(sin(2*e + 2*f*x))) - (2*b^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^3*d^4*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(9/2), x, 19, -((2*(g*cos(e + f*x))^(3/2))/(7*a*d*f*g*(d*sin(e + f*x))^(7/2))) + (2*b*(g*cos(e + f*x))^(3/2))/(5*a^2*d^2*f*g*(d*sin(e + f*x))^(5/2)) - (8*(g*cos(e + f*x))^(3/2))/(21*a*d^3*f*g*(d*sin(e + f*x))^(3/2)) - (2*b^2*(g*cos(e + f*x))^(3/2))/(3*a^3*d^3*f*g*(d*sin(e + f*x))^(3/2)) + (4*b*(g*cos(e + f*x))^(3/2))/(5*a^2*d^4*f*g*sqrt(d*sin(e + f*x))) + (2*b^3*(g*cos(e + f*x))^(3/2))/(a^4*d^4*f*g*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^4*sqrt(g)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^4*sqrt(-a + b)*sqrt(a + b)*d^4*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^4*sqrt(g)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^4*sqrt(-a + b)*sqrt(a + b)*d^4*f*sqrt(d*sin(e + f*x))) + (4*b*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*a^2*d^5*f*sqrt(sin(2*e + 2*f*x))) + (2*b^3*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^4*d^5*f*sqrt(sin(2*e + 2*f*x)))]

    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(3/2), x, 31, (3*d^(3/2)*g^(3/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(4*sqrt(2)*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^3*f) - (3*d^(3/2)*g^(3/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(4*sqrt(2)*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^3*f) + (2*sqrt(2)*a*sqrt(-a^2 + b^2)*d^(3/2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^3*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*a*sqrt(-a^2 + b^2)*d^(3/2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^3*f*sqrt(g*cos(e + f*x))) - (3*d^(3/2)*g^(3/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(8*sqrt(2)*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^3*f) + (3*d^(3/2)*g^(3/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(8*sqrt(2)*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^3*f) - (a*d*g*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))/(b^2*f) + (g*sqrt(g*cos(e + f*x))*(d*sin(e + f*x))^(3/2))/(2*b*f) + (a*d^2*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(2*b^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(1/2), x, 19, -((a*sqrt(d)*g^(3/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^2*f)) + (a*sqrt(d)*g^(3/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^2*f) - (2*sqrt(2)*sqrt(-a^2 + b^2)*sqrt(d)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^2*f*sqrt(g*cos(e + f*x))) + (2*sqrt(2)*sqrt(-a^2 + b^2)*sqrt(d)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^2*f*sqrt(g*cos(e + f*x))) + (a*sqrt(d)*g^(3/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^2*f) - (a*sqrt(d)*g^(3/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^2*f) + (g*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))/(b*f) - (d*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(2*b*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(1/2), x, 18, (g^(3/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b*sqrt(d)*f) - (g^(3/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b*sqrt(d)*f) + (2*sqrt(2)*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a*b*sqrt(d)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a*b*sqrt(d)*f*sqrt(g*cos(e + f*x))) - (g^(3/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b*sqrt(d)*f) + (g^(3/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b*sqrt(d)*f) + (g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(3/2), x, 8, -((2*sqrt(2)*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*d^(3/2)*f*sqrt(g*cos(e + f*x)))) + (2*sqrt(2)*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*d^(3/2)*f*sqrt(g*cos(e + f*x))) - (2*g*sqrt(g*cos(e + f*x)))/(a*d*f*sqrt(d*sin(e + f*x))) - (b*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^2*d*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(5/2), x, 12, (2*sqrt(2)*b*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^3*d^(5/2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*b*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^3*d^(5/2)*f*sqrt(g*cos(e + f*x))) - (2*g*sqrt(g*cos(e + f*x)))/(3*a*d*f*(d*sin(e + f*x))^(3/2)) + (2*b*g*sqrt(g*cos(e + f*x)))/(a^2*d^2*f*sqrt(d*sin(e + f*x))) + (2*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(3*a*d^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) - ((a^2 - b^2)*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^3*d^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(7/2), x, 15, -((2*sqrt(2)*b^2*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^4*d^(7/2)*f*sqrt(g*cos(e + f*x)))) + (2*sqrt(2)*b^2*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^4*d^(7/2)*f*sqrt(g*cos(e + f*x))) - (2*g*sqrt(g*cos(e + f*x)))/(5*a*d*f*(d*sin(e + f*x))^(5/2)) + (2*b*g*sqrt(g*cos(e + f*x)))/(3*a^2*d^2*f*(d*sin(e + f*x))^(3/2)) - (8*g*sqrt(g*cos(e + f*x)))/(5*a*d^3*f*sqrt(d*sin(e + f*x))) + (2*(a^2 - b^2)*g*sqrt(g*cos(e + f*x)))/(a^3*d^3*f*sqrt(d*sin(e + f*x))) - (2*b*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(3*a^2*d^3*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) + (b*(a^2 - b^2)*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^4*d^3*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [(g*cos(e + f*x))^(3/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(9/2), x, 20, (2*sqrt(2)*b^3*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^5*d^(9/2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*b^3*sqrt(-a^2 + b^2)*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^5*d^(9/2)*f*sqrt(g*cos(e + f*x))) - (2*g*sqrt(g*cos(e + f*x)))/(7*a*d*f*(d*sin(e + f*x))^(7/2)) + (2*b*g*sqrt(g*cos(e + f*x)))/(5*a^2*d^2*f*(d*sin(e + f*x))^(5/2)) - (4*g*sqrt(g*cos(e + f*x)))/(7*a*d^3*f*(d*sin(e + f*x))^(3/2)) + (2*(a^2 - b^2)*g*sqrt(g*cos(e + f*x)))/(3*a^3*d^3*f*(d*sin(e + f*x))^(3/2)) + (8*b*g*sqrt(g*cos(e + f*x)))/(5*a^2*d^4*f*sqrt(d*sin(e + f*x))) - (2*b*(a^2 - b^2)*g*sqrt(g*cos(e + f*x)))/(a^4*d^4*f*sqrt(d*sin(e + f*x))) + (4*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(7*a*d^4*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) - (2*(a^2 - b^2)*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(3*a^3*d^4*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) - (b^2*(a^2 - b^2)*g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^5*d^4*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]

    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))*(d*sin(e + f*x))^(1/2), x, 31, -((sqrt(d)*g^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(4*sqrt(2)*b*f)) - ((a^2 - b^2)*sqrt(d)*g^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^3*f) + (sqrt(d)*g^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(4*sqrt(2)*b*f) + ((a^2 - b^2)*sqrt(d)*g^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^3*f) + (sqrt(d)*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(8*sqrt(2)*b*f) + ((a^2 - b^2)*sqrt(d)*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^3*f) - (sqrt(d)*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(8*sqrt(2)*b*f) - ((a^2 - b^2)*sqrt(d)*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^3*f) - (2*sqrt(2)*a*sqrt(-a + b)*sqrt(a + b)*d*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^3*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*a*sqrt(-a + b)*sqrt(a + b)*d*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^3*f*sqrt(d*sin(e + f*x))) + (g*(g*cos(e + f*x))^(3/2)*sqrt(d*sin(e + f*x)))/(2*b*f) + (a*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(b^2*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(1/2), x, 19, (a*g^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^2*sqrt(d)*f) - (a*g^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b^2*sqrt(d)*f) - (a*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^2*sqrt(d)*f) + (a*g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b^2*sqrt(d)*f) + (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^2*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b^2*f*sqrt(d*sin(e + f*x))) - (g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(b*d*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(3/2), x, 20, -((g^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*d^(3/2)*f)) + (g^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*d^(3/2)*f) + (g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*d^(3/2)*f) - (g^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*d^(3/2)*f) - (2*g*(g*cos(e + f*x))^(3/2))/(a*d*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*b*d*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*b*d*f*sqrt(d*sin(e + f*x))) - (2*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a*d^2*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(5/2), x, 10, -((2*g*(g*cos(e + f*x))^(3/2))/(3*a*d*f*(d*sin(e + f*x))^(3/2))) + (2*b*g*(g*cos(e + f*x))^(3/2))/(a^2*d^2*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*d^2*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*d^2*f*sqrt(d*sin(e + f*x))) + (2*b*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^2*d^3*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(7/2), x, 15, -((2*g*(g*cos(e + f*x))^(3/2))/(5*a*d*f*(d*sin(e + f*x))^(5/2))) + (2*b*g*(g*cos(e + f*x))^(3/2))/(3*a^2*d^2*f*(d*sin(e + f*x))^(3/2)) - (4*g*(g*cos(e + f*x))^(3/2))/(5*a*d^3*f*sqrt(d*sin(e + f*x))) + (2*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(a^3*d^3*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^3*d^3*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^3*d^3*f*sqrt(d*sin(e + f*x))) - (4*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*a*d^4*f*sqrt(sin(2*e + 2*f*x))) + (2*(a^2 - b^2)*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^3*d^4*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(9/2), x, 18, -((2*g*(g*cos(e + f*x))^(3/2))/(7*a*d*f*(d*sin(e + f*x))^(7/2))) + (2*b*g*(g*cos(e + f*x))^(3/2))/(5*a^2*d^2*f*(d*sin(e + f*x))^(5/2)) - (8*g*(g*cos(e + f*x))^(3/2))/(21*a*d^3*f*(d*sin(e + f*x))^(3/2)) + (2*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(3*a^3*d^3*f*(d*sin(e + f*x))^(3/2)) + (4*b*g*(g*cos(e + f*x))^(3/2))/(5*a^2*d^4*f*sqrt(d*sin(e + f*x))) - (2*b*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(a^4*d^4*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^2*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^4*d^4*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^2*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^4*d^4*f*sqrt(d*sin(e + f*x))) + (4*b*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*a^2*d^5*f*sqrt(sin(2*e + 2*f*x))) - (2*b*(a^2 - b^2)*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^4*d^5*f*sqrt(sin(2*e + 2*f*x)))]
    @test_int [(g*cos(e + f*x))^(5/2)/(a + b*sin(e + f*x))/(d*sin(e + f*x))^(11/2), x, 24, -((2*g*(g*cos(e + f*x))^(3/2))/(9*a*d*f*(d*sin(e + f*x))^(9/2))) + (2*b*g*(g*cos(e + f*x))^(3/2))/(7*a^2*d^2*f*(d*sin(e + f*x))^(7/2)) - (4*g*(g*cos(e + f*x))^(3/2))/(15*a*d^3*f*(d*sin(e + f*x))^(5/2)) + (2*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(5*a^3*d^3*f*(d*sin(e + f*x))^(5/2)) + (8*b*g*(g*cos(e + f*x))^(3/2))/(21*a^2*d^4*f*(d*sin(e + f*x))^(3/2)) - (2*b*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(3*a^4*d^4*f*(d*sin(e + f*x))^(3/2)) - (8*g*(g*cos(e + f*x))^(3/2))/(15*a*d^5*f*sqrt(d*sin(e + f*x))) + (4*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(5*a^3*d^5*f*sqrt(d*sin(e + f*x))) + (2*b^2*(a^2 - b^2)*g*(g*cos(e + f*x))^(3/2))/(a^5*d^5*f*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^3*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^5*d^5*f*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^3*sqrt(-a + b)*sqrt(a + b)*g^(5/2)*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^5*d^5*f*sqrt(d*sin(e + f*x))) - (8*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(15*a*d^6*f*sqrt(sin(2*e + 2*f*x))) + (4*(a^2 - b^2)*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(5*a^3*d^6*f*sqrt(sin(2*e + 2*f*x))) + (2*b^2*(a^2 - b^2)*g^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^5*d^6*f*sqrt(sin(2*e + 2*f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(5/2), x, 19, (a*d^(5/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^2*f*sqrt(g)) - (a*d^(5/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b^2*f*sqrt(g)) - (2*sqrt(2)*a^2*d^(5/2)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^2*sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x))) + (2*sqrt(2)*a^2*d^(5/2)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b^2*sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x))) - (a*d^(5/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^2*f*sqrt(g)) + (a*d^(5/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b^2*f*sqrt(g)) - (d^2*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))/(b*f*g) + (d^3*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(2*b*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(3/2), x, 15, -((d^(3/2)*atan(1 - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b*f*sqrt(g))) + (d^(3/2)*atan(1 + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/(sqrt(d)*sqrt(g*cos(e + f*x)))))/(sqrt(2)*b*f*sqrt(g)) + (2*sqrt(2)*a*d^(3/2)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b*sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*a*d^(3/2)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(b*sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x))) + (d^(3/2)*log(sqrt(d) - (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b*f*sqrt(g)) - (d^(3/2)*log(sqrt(d) + (sqrt(2)*sqrt(g)*sqrt(d*sin(e + f*x)))/sqrt(g*cos(e + f*x)) + sqrt(d)*tan(e + f*x)))/(2*sqrt(2)*b*f*sqrt(g))]
    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(1/2), x, 4, -((2*sqrt(2)*sqrt(d)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x)))) + (2*sqrt(2)*sqrt(d)*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(sqrt(-a^2 + b^2)*f*sqrt(g*cos(e + f*x)))]
    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(1/2), x, 7, (2*sqrt(2)*b*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a*sqrt(-a^2 + b^2)*sqrt(d)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*b*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a*sqrt(-a^2 + b^2)*sqrt(d)*f*sqrt(g*cos(e + f*x))) + (Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(3/2), x, 9, -((2*sqrt(2)*b^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*sqrt(-a^2 + b^2)*d^(3/2)*f*sqrt(g*cos(e + f*x)))) + (2*sqrt(2)*b^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*sqrt(-a^2 + b^2)*d^(3/2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(g*cos(e + f*x)))/(a*d*f*g*sqrt(d*sin(e + f*x))) - (b*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^2*d*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]
    @test_int [1/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(5/2), x, 13, (2*sqrt(2)*b^3*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^3*sqrt(-a^2 + b^2)*d^(5/2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(2)*b^3*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^3*sqrt(-a^2 + b^2)*d^(5/2)*f*sqrt(g*cos(e + f*x))) - (2*sqrt(g*cos(e + f*x)))/(3*a*d*f*g*(d*sin(e + f*x))^(3/2)) + (2*b*sqrt(g*cos(e + f*x)))/(a^2*d^2*f*g*sqrt(d*sin(e + f*x))) + (2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(3*a*d^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) + (b^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(a^3*d^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]

    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(5/2), x, 31, -((a^2*d^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*(a^2 - b^2)*f*g^(3/2))) + (b*d^(5/2)*atan(1 - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*atan(1 + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/(sqrt(g)*sqrt(d*sin(e + f*x)))))/(sqrt(2)*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) - (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*(a^2 - b^2)*f*g^(3/2)) - (a^2*d^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*b*(a^2 - b^2)*f*g^(3/2)) + (b*d^(5/2)*log(sqrt(g) + sqrt(g)*cot(e + f*x) + (sqrt(2)*sqrt(d)*sqrt(g*cos(e + f*x)))/sqrt(d*sin(e + f*x))))/(2*sqrt(2)*(a^2 - b^2)*f*g^(3/2)) - (2*sqrt(2)*a^3*d^3*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*a^3*d^3*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*b*d^2*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) + (2*a*d*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) - (2*a*d^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g^2*sqrt(sin(2*e + 2*f*x)))]
    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(3/2), x, 10, (2*sqrt(2)*a^2*d^2*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*a^2*d^2*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) + (2*a*d*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) - (2*b*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) + (2*b*d*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g^2*sqrt(sin(2*e + 2*f*x)))]
    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))*(d*sin(e + f*x))^(1/2), x, 11, -((2*sqrt(2)*a*b*d*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x)))) + (2*sqrt(2)*a*b*d*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*b*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g*sqrt(g*cos(e + f*x))) + (2*a*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*d*f*g*sqrt(g*cos(e + f*x))) - (2*a*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*f*g^2*sqrt(sin(2*e + 2*f*x)))]
    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(1/2), x, 11, (2*sqrt(2)*b^2*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^2*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e + f*x))) + (2*a*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*d*f*g*sqrt(g*cos(e + f*x))) - (2*b*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*d^2*f*g*sqrt(g*cos(e + f*x))) + (2*b*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*d*f*g^2*sqrt(sin(2*e + 2*f*x)))]
    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(3/2), x, 16, -((2*a)/((a^2 - b^2)*d*f*g*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))) + (2*b^2*(g*cos(e + f*x))^(3/2))/(a*(a^2 - b^2)*d*f*g^3*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^3*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^3*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*b*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*d^2*f*g*sqrt(g*cos(e + f*x))) + (4*a*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*d^3*f*g*sqrt(g*cos(e + f*x))) - (4*a*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*d^2*f*g^2*sqrt(sin(2*e + 2*f*x))) + (2*b^2*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a*(a^2 - b^2)*d^2*f*g^2*sqrt(sin(2*e + 2*f*x)))]
    @test_int [1/((g*cos(e + f*x))^(3/2)*(a + b*sin(e + f*x)))/(d*sin(e + f*x))^(5/2), x, 19, -((2*a)/(3*(a^2 - b^2)*d*f*g*sqrt(g*cos(e + f*x))*(d*sin(e + f*x))^(3/2))) + (2*b^2*(g*cos(e + f*x))^(3/2))/(3*a*(a^2 - b^2)*d*f*g^3*(d*sin(e + f*x))^(3/2)) + (2*b)/((a^2 - b^2)*d^2*f*g*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x))) - (2*b^3*(g*cos(e + f*x))^(3/2))/(a^2*(a^2 - b^2)*d^2*f*g^3*sqrt(d*sin(e + f*x))) + (2*sqrt(2)*b^4*Elliptic.Pi(-(sqrt(-a + b)/sqrt(a + b)), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*sqrt(d*sin(e + f*x))) - (2*sqrt(2)*b^4*Elliptic.Pi(sqrt(-a + b)/sqrt(a + b), asin(sqrt(g*cos(e + f*x))/(sqrt(g)*sqrt(1 + sin(e + f*x)))), -1)*sqrt(sin(e + f*x)))/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*sqrt(d*sin(e + f*x))) + (8*a*sqrt(d*sin(e + f*x)))/(3*(a^2 - b^2)*d^3*f*g*sqrt(g*cos(e + f*x))) - (4*b*(d*sin(e + f*x))^(3/2))/((a^2 - b^2)*d^4*f*g*sqrt(g*cos(e + f*x))) + (4*b*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/((a^2 - b^2)*d^3*f*g^2*sqrt(sin(2*e + 2*f*x))) - (2*b^3*sqrt(g*cos(e + f*x))*Elliptic.E(e - pi/4 + f*x, 2)*sqrt(d*sin(e + f*x)))/(a^2*(a^2 - b^2)*d^3*f*g^2*sqrt(sin(2*e + 2*f*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(d*sin(e+f*x))^n / (a+b*sin(e+f*x))^2=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(p/2)*(d*sin(e+f*x))^(n/2) / (a+b*sin(e+f*x))^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(g*cos(e + f*x))^(3/2)/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^2), x, 8, (sqrt(2)*b*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b - sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*sqrt(-a^2 + b^2)*sqrt(d)*f*sqrt(g*cos(e + f*x))) - (sqrt(2)*b*g^2*sqrt(cos(e + f*x))*Elliptic.Pi(-(a/(b + sqrt(-a^2 + b^2))), asin(sqrt(d*sin(e + f*x))/(sqrt(d)*sqrt(1 + cos(e + f*x)))), -1))/(a^2*sqrt(-a^2 + b^2)*sqrt(d)*f*sqrt(g*cos(e + f*x))) + (g*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))/(a*d*f*(a + b*sin(e + f*x))) + (g^2*Elliptic.F(e - pi/4 + f*x, 2)*sqrt(sin(2*e + 2*f*x)))/(2*a^2*f*sqrt(g*cos(e + f*x))*sqrt(d*sin(e + f*x)))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section:: =#
    #=Integrands*of*the*form*sec(e+f*x)^1*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^2*sin(c + d*x)^4*(a + b*sin(c + d*x)), x, 8, -((3*a*x)/2) + (2*b*cos(c + d*x))/d - (b*cos(c + d*x)^3)/(3*d) + (b*sec(c + d*x))/d + (3*a*tan(c + d*x))/(2*d) - (a*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + b*sin(c + d*x)), x, 8, -((3*b*x)/2) + (a*cos(c + d*x))/d + (a*sec(c + d*x))/d + (3*b*tan(c + d*x))/(2*d) - (b*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x)), x, 7, (-a)*x + (b*cos(c + d*x))/d + (b*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x)), x, 5, (-b)*x + (a*sec(c + d*x))/d + (b*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x)), x, 6, -((a*atanh(cos(c + d*x)))/d) + (a*sec(c + d*x))/d + (b*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x)), x, 7, -((b*atanh(cos(c + d*x)))/d) - (a*cot(c + d*x))/d + (b*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x)), x, 8, -((3*a*atanh(cos(c + d*x)))/(2*d)) - (b*cot(c + d*x))/d + (3*a*sec(c + d*x))/(2*d) - (a*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (b*tan(c + d*x))/d]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 8, -3*a*b*x + ((a^2 + 2*b^2)*cos(c + d*x))/d - (b^2*cos(c + d*x)^3)/(3*d) + ((a^2 + b^2)*sec(c + d*x))/d + (3*a*b*tan(c + d*x))/d - (a*b*sin(c + d*x)^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 11, (-a^2)*x - (3*b^2*x)/2 + (2*a*b*cos(c + d*x))/d + (2*a*b*sec(c + d*x))/d + (a^2*tan(c + d*x))/d + (3*b^2*tan(c + d*x))/(2*d) - (b^2*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 4, -2*a*b*x + (2*b^2*cos(c + d*x))/d + (sec(c + d*x)*(a + b*sin(c + d*x))^2)/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 8, ((a^2 + b^2)*sec(c + d*x))/d - (a^2*atanh(cos(c + d*x)))/d + (2*a*b*tan(c + d*x))/d, ((a^2 + b^2)*sec(c + d*x))/d - (a^2*atanh(sqrt(cos(c + d*x)^2))*sqrt(cos(c + d*x)^2)*sec(c + d*x))/d + (2*a*b*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 7, -((2*a*b*atanh(cos(c + d*x)))/d) - (a^2*cot(c + d*x))/d + (2*a*b*sec(c + d*x))/d + ((a^2 + b^2)*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 10, -(((3*a^2 + 2*b^2)*atanh(cos(c + d*x)))/(2*d)) - (2*a*b*cot(c + d*x))/d + ((3*a^2 + 2*b^2)*sec(c + d*x))/(2*d) - (a^2*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (2*a*b*tan(c + d*x))/d, -((2*a*b*cot(c + d*x))/d) + ((3*a^2 + 2*b^2)*sec(c + d*x))/(2*d) - ((3*a^2 + 2*b^2)*atanh(sqrt(cos(c + d*x)^2))*sqrt(cos(c + d*x)^2)*sec(c + d*x))/(2*d) - (a^2*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (2*a*b*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 8, -((3*a*b*atanh(cos(c + d*x)))/d) - ((2*a^2 + b^2)*cot(c + d*x))/d - (a^2*cot(c + d*x)^3)/(3*d) + (3*a*b*sec(c + d*x))/d - (a*b*csc(c + d*x)^2*sec(c + d*x))/d + ((a^2 + b^2)*tan(c + d*x))/d]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 17, (-(9/2))*a^2*b*x - (15*b^3*x)/8 + (a^3*cos(c + d*x))/d + (6*a*b^2*cos(c + d*x))/d - (a*b^2*cos(c + d*x)^3)/d + (a^3*sec(c + d*x))/d + (3*a*b^2*sec(c + d*x))/d + (9*a^2*b*tan(c + d*x))/(2*d) + (15*b^3*tan(c + d*x))/(8*d) - (3*a^2*b*sin(c + d*x)^2*tan(c + d*x))/(2*d) - (5*b^3*sin(c + d*x)^2*tan(c + d*x))/(8*d) - (b^3*sin(c + d*x)^4*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 14, (-a^3)*x - (9/2)*a*b^2*x + (3*a^2*b*cos(c + d*x))/d + (2*b^3*cos(c + d*x))/d - (b^3*cos(c + d*x)^3)/(3*d) + (3*a^2*b*sec(c + d*x))/d + (b^3*sec(c + d*x))/d + (a^3*tan(c + d*x))/d + (9*a*b^2*tan(c + d*x))/(2*d) - (3*a*b^2*sin(c + d*x)^2*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 3, (-(3/2))*b*(2*a^2 + b^2)*x + (6*a*b^2*cos(c + d*x))/d + (3*b^3*cos(c + d*x)*sin(c + d*x))/(2*d) + (sec(c + d*x)*(a + b*sin(c + d*x))^3)/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 11, (-b^3)*x - (a^3*atanh(cos(c + d*x)))/d + (a^3*sec(c + d*x))/d + (3*a*b^2*sec(c + d*x))/d + (3*a^2*b*tan(c + d*x))/d + (b^3*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 12, -((3*a^2*b*atanh(cos(c + d*x)))/d) - (a^3*cot(c + d*x))/d + (3*a^2*b*sec(c + d*x))/d + (b^3*sec(c + d*x))/d + (a^3*tan(c + d*x))/d + (3*a*b^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 14, -((3*a^3*atanh(cos(c + d*x)))/(2*d)) - (3*a*b^2*atanh(cos(c + d*x)))/d - (3*a^2*b*cot(c + d*x))/d + (3*a^3*sec(c + d*x))/(2*d) + (3*a*b^2*sec(c + d*x))/d - (a^3*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (3*a^2*b*tan(c + d*x))/d + (b^3*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 15, -((9*a^2*b*atanh(cos(c + d*x)))/(2*d)) - (b^3*atanh(cos(c + d*x)))/d - (2*a^3*cot(c + d*x))/d - (3*a*b^2*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) + (9*a^2*b*sec(c + d*x))/(2*d) + (b^3*sec(c + d*x))/d - (3*a^2*b*csc(c + d*x)^2*sec(c + d*x))/(2*d) + (a^3*tan(c + d*x))/d + (3*a*b^2*tan(c + d*x))/d]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 12, -(x/b^2) - (2*a^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(5/2)*d) + (4*a^3*(a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(5/2)*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) - (a^4*cos(c + d*x))/(b*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 12, (2*a^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(5/2)*d) - (2*a^2*(a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(5/2)*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) + (a^3*cos(c + d*x))/((a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 12, -((2*a^3*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) - (4*a*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) - (a^2*b*cos(c + d*x))/((a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 6, (2*b*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (a*sec(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x))) + (sec(c + d*x)*(2*a^2 + b^2 - 3*a*b*sin(c + d*x)))/((a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 13, (2*b^3*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) + (2*b^3*(3*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(5/2)*d) - atanh(cos(c + d*x))/(a^2*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) + (b^4*cos(c + d*x))/(a*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 15, -((2*b^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*(a^2 - b^2)^(5/2)*d)) - (4*b^4*(2*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d) + (2*b*atanh(cos(c + d*x)))/(a^3*d) - cot(c + d*x)/(a^2*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) - (b^5*cos(c + d*x))/(a^2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 17, (2*b^5*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(5/2)*d) + (2*b^5*(5*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - atanh(cos(c + d*x))/(2*a^2*d) - ((a^2 + 3*b^2)*atanh(cos(c + d*x)))/(a^4*d) + (2*b*cot(c + d*x))/(a^3*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^2*d) + cos(c + d*x)/(2*(a + b)^2*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^2*d*(1 + sin(c + d*x))) + (b^6*cos(c + d*x))/(a^3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]

    @test_int [sec(c + d*x)^2*sin(c + d*x)^4/(a + b*sin(c + d*x))^3, x, 18, (4*a^4*(a^2 - 2*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(7/2)*d) - (a^4*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(7/2)*d) - (2*a^2*(a^4 - 3*a^2*b^2 + 6*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(7/2)*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) - (a^4*cos(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) - (3*a^5*cos(c + d*x))/(2*b*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) + (2*a^3*(a^2 - 2*b^2)*cos(c + d*x))/(b*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 18, -((2*a^3*(a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(7/2)*d)) + (a^3*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(7/2)*d) + (2*a*b*(a^2 + 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) + (a^3*cos(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) + (3*a^4*cos(c + d*x))/(2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) - (a^2*(a^2 - 3*b^2)*cos(c + d*x))/((a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 18, -((4*a^2*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - (a^2*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - (2*b^2*(3*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) - (a^2*b*cos(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) - (3*a^3*b*cos(c + d*x))/(2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) - (2*a*b^3*cos(c + d*x))/((a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sin(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 7, (3*a*b*(2*a^2 + 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - (a*sec(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - ((3*a^2 + 2*b^2)*sec(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) + (sec(c + d*x)*(3*a*(2*a^2 + 3*b^2) - b*(11*a^2 + 4*b^2)*sin(c + d*x)))/(2*(a^2 - b^2)^3*d)]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 19, (2*b^3*(3*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*(a^2 - b^2)^(7/2)*d) + (b^3*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*(a^2 - b^2)^(7/2)*d) + (2*b^3*(6*a^4 - 3*a^2*b^2 + b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(7/2)*d) - atanh(cos(c + d*x))/(a^3*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) + (b^4*cos(c + d*x))/(2*a*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) + (3*b^4*cos(c + d*x))/(2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) + (b^4*(3*a^2 - b^2)*cos(c + d*x))/(a^2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 21, -((4*b^4*(2*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(7/2)*d)) - (b^4*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(7/2)*d) - (2*b^4*(10*a^4 - 9*a^2*b^2 + 3*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d) + (3*b*atanh(cos(c + d*x)))/(a^4*d) - cot(c + d*x)/(a^3*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) - cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) - (b^5*cos(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) - (3*b^5*cos(c + d*x))/(2*a*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) - (2*b^5*(2*a^2 - b^2)*cos(c + d*x))/(a^3*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*csc(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 23, (2*b^5*(5*a^2 - 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^2*b^2 + 6*b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*d) - atanh(cos(c + d*x))/(2*a^3*d) - ((a^2 + 6*b^2)*atanh(cos(c + d*x)))/(a^5*d) + (3*b*cot(c + d*x))/(a^4*d) - (cot(c + d*x)*csc(c + d*x))/(2*a^3*d) + cos(c + d*x)/(2*(a + b)^3*d*(1 - sin(c + d*x))) + cos(c + d*x)/(2*(a - b)^3*d*(1 + sin(c + d*x))) + (b^6*cos(c + d*x))/(2*a^3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) + (3*b^6*cos(c + d*x))/(2*a^2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) + (b^6*(5*a^2 - 3*b^2)*cos(c + d*x))/(a^4*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^2*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [sec(e + f*x)^2*(a + b*sin(e + f*x))^(1/2)/sqrt(d*sin(e + f*x)), x, 2, (sec(e + f*x)*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))/(d*f) - (sqrt(a + b)*sqrt((a*(1 - csc(e + f*x)))/(a + b))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(sqrt(d)*f)]

    @test_int [sec(e + f*x)^2*(a + b*sin(e + f*x))^(3/2)/sqrt(d*sin(e + f*x)), x, -1, (sec(e + f*x)*(b + a*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))/(f*sqrt(d*sin(e + f*x))) - ((a + b)^(3/2)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(sqrt(d)*f) - (b*(a + b)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*sqrt((b + a*csc(e + f*x))/(-a + b))*Elliptic.E(asin(sqrt(-((b + a*csc(e + f*x))/(a - b)))), (-a + b)/(a + b))*(1 + sin(e + f*x))*tan(e + f*x))/(f*sqrt((a*(1 + csc(e + f*x)))/(a - b))*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))]

    #= ::Section:: =#
    #=Integrands*of*the*form*sec(e+f*x)^3*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^4*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [(sec(e + f*x)^4*(a + b*sin(e + f*x))^(5/2))/sqrt(d*sin(e + f*x)), x, -1, (5*a*sec(e + f*x)*(b + a*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))/(6*f*sqrt(d*sin(e + f*x))) + (sec(e + f*x)^3*sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^(5/2))/(3*d*f) - (5*a*(a + b)^(3/2)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(6*sqrt(d)*f) - (5*a*b*(a + b)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*sqrt((b + a*csc(e + f*x))/(-a + b))*Elliptic.E(asin(sqrt(-((b + a*csc(e + f*x))/(a - b)))), (-a + b)/(a + b))*(1 + sin(e + f*x))*tan(e + f*x))/(6*f*sqrt((a*(1 + csc(e + f*x)))/(a - b))*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(c + d*x)^5*sin(c + d*x)^7*(a + b*sin(c + d*x)), x, 11, (35*b*atanh(sin(c + d*x)))/(8*d) + (a*cos(c + d*x)^2)/(2*d) - (3*a*log(cos(c + d*x)))/d - (3*a*sec(c + d*x)^2)/(2*d) + (a*sec(c + d*x)^4)/(4*d) - (35*b*sin(c + d*x))/(8*d) - (35*b*sin(c + d*x)^3)/(24*d) - (7*b*sin(c + d*x)^3*tan(c + d*x)^2)/(8*d) + (b*sin(c + d*x)^3*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^6*(a + b*sin(c + d*x)), x, 10, (15*a*atanh(sin(c + d*x)))/(8*d) + (b*cos(c + d*x)^2)/(2*d) - (3*b*log(cos(c + d*x)))/d - (3*b*sec(c + d*x)^2)/(2*d) + (b*sec(c + d*x)^4)/(4*d) - (15*a*sin(c + d*x))/(8*d) - (5*a*sin(c + d*x)*tan(c + d*x)^2)/(8*d) + (a*sin(c + d*x)*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + b*sin(c + d*x)), x, 7, -(((8*a + 15*b)*log(1 - sin(c + d*x)))/(16*d)) - ((8*a - 15*b)*log(1 + sin(c + d*x)))/(16*d) - (15*b*sin(c + d*x))/(8*d) - ((4*a + 5*b*sin(c + d*x))*tan(c + d*x)^2)/(8*d) + ((a + b*sin(c + d*x))*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + b*sin(c + d*x)), x, 7, (3*a*atanh(sin(c + d*x)))/(8*d) - (b*log(cos(c + d*x)))/d - (3*a*sec(c + d*x)*tan(c + d*x))/(8*d) - (b*tan(c + d*x)^2)/(2*d) + (a*sec(c + d*x)*tan(c + d*x)^3)/(4*d) + (b*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x)), x, 6, (3*b*atanh(sin(c + d*x)))/(8*d) - (3*b*sec(c + d*x)*tan(c + d*x))/(8*d) + (b*sec(c + d*x)*tan(c + d*x)^3)/(4*d) + (a*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + b*sin(c + d*x)), x, 6, -((a*atanh(sin(c + d*x)))/(8*d)) - (a*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (b*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + b*sin(c + d*x)), x, 6, -((b*atanh(sin(c + d*x)))/(8*d)) + (a*sec(c + d*x)^4)/(4*d) - (b*sec(c + d*x)*tan(c + d*x))/(8*d) + (b*sec(c + d*x)^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + b*sin(c + d*x)), x, 8, (3*b*atanh(sin(c + d*x)))/(8*d) + (a*log(tan(c + d*x)))/d + (3*b*sec(c + d*x)*tan(c + d*x))/(8*d) + (b*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (a*tan(c + d*x)^2)/d + (a*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + b*sin(c + d*x)), x, 10, (15*a*atanh(sin(c + d*x)))/(8*d) - (15*a*csc(c + d*x))/(8*d) + (b*log(tan(c + d*x)))/d + (5*a*csc(c + d*x)*sec(c + d*x)^2)/(8*d) + (a*csc(c + d*x)*sec(c + d*x)^4)/(4*d) + (b*tan(c + d*x)^2)/d + (b*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + b*sin(c + d*x)), x, 10, (15*b*atanh(sin(c + d*x)))/(8*d) - (a*cot(c + d*x)^2)/(2*d) - (15*b*csc(c + d*x))/(8*d) + (3*a*log(tan(c + d*x)))/d + (5*b*csc(c + d*x)*sec(c + d*x)^2)/(8*d) + (b*csc(c + d*x)*sec(c + d*x)^4)/(4*d) + (3*a*tan(c + d*x)^2)/(2*d) + (a*tan(c + d*x)^4)/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^4*(a + b*sin(c + d*x)), x, 11, (35*a*atanh(sin(c + d*x)))/(8*d) - (b*cot(c + d*x)^2)/(2*d) - (35*a*csc(c + d*x))/(8*d) - (35*a*csc(c + d*x)^3)/(24*d) + (3*b*log(tan(c + d*x)))/d + (7*a*csc(c + d*x)^3*sec(c + d*x)^2)/(8*d) + (a*csc(c + d*x)^3*sec(c + d*x)^4)/(4*d) + (3*b*tan(c + d*x)^2)/(2*d) + (b*tan(c + d*x)^4)/(4*d)]

    @test_int [sec(c + d*x)^5*sin(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 9, -(((15*a^2 + 48*a*b + 35*b^2)*log(1 - sin(c + d*x)))/(16*d)) + ((15*a^2 - 48*a*b + 35*b^2)*log(1 + sin(c + d*x)))/(16*d) - ((a^2 + 3*b^2)*sin(c + d*x))/d - (a*b*sin(c + d*x)^2)/d - (b^2*sin(c + d*x)^3)/(3*d) - (sec(c + d*x)^2*(11*b + 9*a*sin(c + d*x))*(a + b*sin(c + d*x)))/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^2*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 8, -(((4*a^2 + 15*a*b + 12*b^2)*log(1 - sin(c + d*x)))/(8*d)) + ((15*a*b - 4*(a^2 + 3*b^2))*log(1 + sin(c + d*x)))/(8*d) - (2*a*b*sin(c + d*x))/d - (b^2*sin(c + d*x)^2)/(2*d) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^2)/(4*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))*(4*a + 5*b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 9, -(((3*a^2 + 16*a*b + 15*b^2)*log(1 - sin(c + d*x)))/(16*d)) + ((3*a^2 - 16*a*b + 15*b^2)*log(1 + sin(c + d*x)))/(16*d) - (b^2*sin(c + d*x))/d - (sec(c + d*x)^2*(7*b + 5*a*sin(c + d*x))*(a + b*sin(c + d*x)))/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^2*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 7, -((b*(3*a + 4*b)*log(1 - sin(c + d*x)))/(8*d)) + ((3*a - 4*b)*b*log(1 + sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^2)/(4*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))*(2*a + 3*b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 5, -(((a^2 - 3*b^2)*atanh(sin(c + d*x)))/(8*d)) - (sec(c + d*x)^2*(4*a*b + (a^2 + 3*b^2)*sin(c + d*x)))/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^2*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, -((a*b*atanh(sin(c + d*x)))/(4*d)) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^2)/(4*d) - (sec(c + d*x)^2*(b^2 + a*b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, -((a*(4*a + 3*b)*log(1 - sin(c + d*x)))/(8*d)) + (a^2*log(sin(c + d*x)))/d - (a*(4*a - 3*b)*log(1 + sin(c + d*x)))/(8*d) + (a*sec(c + d*x)^2*(2*a + 3*b*sin(c + d*x)))/(4*d) + (sec(c + d*x)^4*(a^2 + b^2 + 2*a*b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 6, -((a^2*csc(c + d*x))/d) - ((15*a^2 + 16*a*b + 3*b^2)*log(1 - sin(c + d*x)))/(16*d) + (2*a*b*log(sin(c + d*x)))/d + ((15*a^2 - 16*a*b + 3*b^2)*log(1 + sin(c + d*x)))/(16*d) + (b*sec(c + d*x)^2*(8*a + (3 + (7*a^2)/b^2)*b*sin(c + d*x)))/(8*d) + (b*sec(c + d*x)^4*(2*a + ((a^2 + b^2)*sin(c + d*x))/b))/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 6, -((2*a*b*csc(c + d*x))/d) - (a^2*csc(c + d*x)^2)/(2*d) - ((12*a^2 + 15*a*b + 4*b^2)*log(1 - sin(c + d*x)))/(8*d) + ((3*a^2 + b^2)*log(sin(c + d*x)))/d - ((12*a^2 - 15*a*b + 4*b^2)*log(1 + sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(a^2 + b^2 + 2*a*b*sin(c + d*x)))/(4*d) + (sec(c + d*x)^2*(2*(2*a^2 + b^2) + 7*a*b*sin(c + d*x)))/(4*d)]

    @test_int [sec(c + d*x)^5*sin(c + d*x)^5*(a + b*sin(c + d*x))^3, x, 8, -(((a + b)*(8*a^2 + 37*a*b + 35*b^2)*log(1 - sin(c + d*x)))/(16*d)) - ((a - b)*(8*a^2 - 37*a*b + 35*b^2)*log(1 + sin(c + d*x)))/(16*d) - (b*(24*a^2 + 35*b^2)*sin(c + d*x))/(8*d) - (3*a*b^2*sin(c + d*x)^2)/(2*d) - (b^3*sin(c + d*x)^3)/(3*d) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^3)/(4*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))^2*(8*a + 11*b*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 9, -((3*(a + b)*(a^2 + 7*a*b + 8*b^2)*log(1 - sin(c + d*x)))/(16*d)) + (3*(a - b)*(a^2 - 7*a*b + 8*b^2)*log(1 + sin(c + d*x)))/(16*d) - (29*a*b^2*sin(c + d*x))/(8*d) - (b^3*sin(c + d*x)^2)/(2*d) - (sec(c + d*x)^2*(8*b + 5*a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 8, -((3*b*(a + b)*(3*a + 5*b)*log(1 - sin(c + d*x)))/(16*d)) + (3*(3*a - 5*b)*(a - b)*b*log(1 + sin(c + d*x)))/(16*d) - (15*b^3*sin(c + d*x))/(8*d) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^3)/(4*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))^2*(4*a + 7*b*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 7, ((a^3 - 9*a*b^2 - 8*b^3)*log(1 - sin(c + d*x)))/(16*d) - ((a^3 - 9*a*b^2 + 8*b^3)*log(1 + sin(c + d*x)))/(16*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))*(5*a*b + (a^2 + 4*b^2)*sin(c + d*x)))/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 5, -((3*b*(a^2 - b^2)*atanh(sin(c + d*x)))/(8*d)) + (sec(c + d*x)^4*(a + b*sin(c + d*x))^3)/(4*d) - (3*sec(c + d*x)^2*(a + b*sin(c + d*x))*(b^2 + a*b*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 6, -(((8*a^3 + 9*a^2*b - b^3)*log(1 - sin(c + d*x)))/(16*d)) + (a^3*log(sin(c + d*x)))/d - ((8*a^3 - 9*a^2*b + b^3)*log(1 + sin(c + d*x)))/(16*d) + (sec(c + d*x)^2*(4*a^3 + b*(9*a^2 - b^2)*sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 6, -((a^3*csc(c + d*x))/d) - (3*a*(a + b)*(5*a + 3*b)*log(1 - sin(c + d*x)))/(16*d) + (3*a^2*b*log(sin(c + d*x)))/d + (3*a*(5*a - 3*b)*(a - b)*log(1 + sin(c + d*x)))/(16*d) + (b*sec(c + d*x)^4*(3*a^2 + b^2 + a*(3 + a^2/b^2)*b*sin(c + d*x)))/(4*d) + (a*b*sec(c + d*x)^2*(12*a + (9 + (7*a^2)/b^2)*b*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^5*csc(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 6, -((3*a^2*b*csc(c + d*x))/d) - (a^3*csc(c + d*x)^2)/(2*d) - (3*(a + b)*(8*a^2 + 7*a*b + b^2)*log(1 - sin(c + d*x)))/(16*d) + (3*a*(a^2 + b^2)*log(sin(c + d*x)))/d - (3*(a - b)*(8*a^2 - 7*a*b + b^2)*log(1 + sin(c + d*x)))/(16*d) + (b^2*sec(c + d*x)^4*(a*(3 + a^2/b^2) + (1 + (3*a^2)/b^2)*b*sin(c + d*x)))/(4*d) + (b^2*sec(c + d*x)^2*(4*a*(3 + (2*a^2)/b^2) + 3*(1 + (7*a^2)/b^2)*b*sin(c + d*x)))/(8*d)]

    #= ::Subsubsection:: =#
    #=m<0=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [sec(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^4, x, 6, -(((6*a^2*b^2*(1 - n^2) - a^4*(3 - 4*n + n^2) - b^4*(3 + 4*n + n^2))*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(8*d*(1 + n))) - (a*b*n*(a^2*(2 - n) - b^2*(2 + n))*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(2*d*(2 + n)) + (sec(c + d*x)^4*sin(c + d*x)^(1 + n)*(a^4 + 6*a^2*b^2 + b^4 + 4*a*b*(a^2 + b^2)*sin(c + d*x)))/(4*d) + (sec(c + d*x)^2*sin(c + d*x)^(1 + n)*(a^4*(3 - n) - 6*a^2*b^2*(1 + n) - b^4*(5 + n) + 4*a*b*(a^2*(2 - n) - b^2*(2 + n))*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^3, x, 5, (a*(a^2*(3 - n) - 3*b^2*(1 + n))*HypergeometricFunctions._₂F₁(2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(4*d*(1 + n)) + (b*(3*a^2*(2 - n) - b^2*(2 + n))*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(4*d*(2 + n)) + (sec(c + d*x)^4*sin(c + d*x)^(1 + n)*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^2, x, 5, ((a^2*(3 - n) - b^2*(1 + n))*HypergeometricFunctions._₂F₁(2, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(4*d*(1 + n)) + (a*b*(2 - n)*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(2*d*(2 + n)) + (sec(c + d*x)^4*sin(c + d*x)^(1 + n)*(a^2 + b^2 + 2*a*b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^1, x, 4, (a*HypergeometricFunctions._₂F₁(3, (1 + n)/2, (3 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(1 + n))/(d*(1 + n)) + (b*HypergeometricFunctions._₂F₁(3, (2 + n)/2, (4 + n)/2, sin(c + d*x)^2)*sin(c + d*x)^(2 + n))/(d*(2 + n))]
    @test_int [sec(c + d*x)^5*sin(c + d*x)^n/(a + b*sin(c + d*x))^1, x, 10, ((3*a^2 - 9*a*b + 8*b^2)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(16*(a - b)^3*d*(1 + n)) + ((3*a^2 + 9*a*b + 8*b^2)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, sin(c + d*x))*sin(c + d*x)^(1 + n))/(16*(a + b)^3*d*(1 + n)) - (b^6*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -((b*sin(c + d*x))/a))*sin(c + d*x)^(1 + n))/(a*(a^2 - b^2)^3*d*(1 + n)) + ((3*a - 5*b)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(16*(a - b)^2*d*(1 + n)) + ((3*a + 5*b)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, sin(c + d*x))*sin(c + d*x)^(1 + n))/(16*(a + b)^2*d*(1 + n)) + (HypergeometricFunctions._₂F₁(3, 1 + n, 2 + n, -sin(c + d*x))*sin(c + d*x)^(1 + n))/(8*(a - b)*d*(1 + n)) + (HypergeometricFunctions._₂F₁(3, 1 + n, 2 + n, sin(c + d*x))*sin(c + d*x)^(1 + n))/(8*(a + b)*d*(1 + n))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^5*sin(e+f*x)^n*(a+b*sin(e+f*x))^m*with*m*symbolic=#

    @test_int [sec(c + d*x)^5*sin(c + d*x)^n*(a + b*sin(c + d*x))^p, x, 17, (1/(16*d*(1 + n)))*((3*AppellF1(1 + n, -p, 1, 2 + n, -((b*sin(c + d*x))/a), -sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1(1 + n, -p, 1, 2 + n, -((b*sin(c + d*x))/a), sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1(1 + n, -p, 2, 2 + n, -((b*sin(c + d*x))/a), -sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1(1 + n, -p, 2, 2 + n, -((b*sin(c + d*x))/a), sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p) + (1/(8*d*(1 + n)))*((AppellF1(1 + n, -p, 3, 2 + n, -((b*sin(c + d*x))/a), -sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p) + (1/(8*d*(1 + n)))*((AppellF1(1 + n, -p, 3, 2 + n, -((b*sin(c + d*x))/a), sin(c + d*x))*sin(c + d*x)^(1 + n)*(a + b*sin(c + d*x))^p)/(1 + (b*sin(c + d*x))/a)^p)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^6*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^6*sin(e+f*x)^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*sec(e+f*x)^6*sin(e+f*x)^n*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^6*(d*sin(e+f*x))^(n/2)*(a+b*sin(e+f*x))^(m/2)=#

    @test_int [(sec(e + f*x)^6*(a + b*sin(e + f*x))^(9/2))/sqrt(d*sin(e + f*x)), x, -1, -((3*a*b*(-2*a^2 + b^2)*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(5*f*sqrt(d*sin(e + f*x)))) + (sec(e + f*x)^5*sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))^(9/2))/(5*d*f) - (3*a*sec(e + f*x)^3*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*((-a)*(7*a^2 + b^2) + 2*b*(-7*a^2 + b^2)*sin(e + f*x) + 5*a*(a^2 - b^2)*sin(e + f*x)^2 + (8*a^2*b - 4*b^3)*sin(e + f*x)^3))/(20*d*f) - (3*a*(a + b)^(3/2)*(5*a^2 + 3*a*b - 4*b^2)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*sqrt((a*(1 + csc(e + f*x)))/(a - b))*Elliptic.F(asin((sqrt(d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(d*sin(e + f*x)))), -((a + b)/(a - b)))*tan(e + f*x))/(20*sqrt(d)*f) - (3*b*(2*a^4 - 3*a^2*b^2 + b^4)*sqrt(-((a*(-1 + csc(e + f*x)))/(a + b)))*Elliptic.E(asin(sqrt(-((b + a*csc(e + f*x))/(a - b)))), 1 - (2*a)/(a + b))*sqrt(d*sin(e + f*x))*sqrt(-((a*csc(e + f*x)^2*(1 + sin(e + f*x))*(a + b*sin(e + f*x)))/(a - b)^2))*tan(e + f*x))/(5*d*f*sqrt(a + b*sin(e + f*x)))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(n/3)=#

    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(4/3), x, 11, -((9*(64*a*b*c*d - 26*a^2*d^2 - b^2*(18*c^2 - 13*d^2))*cos(e + f*x)*(c + d*sin(e + f*x))^(7/3))/(2080*d^3*f)) - (9*b*(3*b*c - 2*a*d)*cos(e + f*x)*sin(e + f*x)*(c + d*sin(e + f*x))^(7/3))/(208*d^2*f) + (3*cos(e + f*x)*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(7/3))/(16*d*f) - (3*(c + d)^2*(208*a^2*c*d^2 - 64*a*b*d*(3*c^2 - 5*d^2) + b^2*c*(54*c^2 + d^2))*AppellF1(1/2, 1/2, -(7/3), 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1/3))/(1040*sqrt(2)*d^4*f*sqrt(1 + sin(e + f*x))*((c + d*sin(e + f*x))/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(192*a*b*c*d - 208*a^2*d^2 - b^2*(54*c^2 + 91*d^2))*AppellF1(1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1/3))/(1040*sqrt(2)*d^4*f*sqrt(1 + sin(e + f*x))*((c + d*sin(e + f*x))/(c + d))^(1/3))]
    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^(4/3), x, 10, -((3*(6*b*c - 13*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(7/3))/(130*d^2*f)) + (3*b*cos(e + f*x)*sin(e + f*x)*(c + d*sin(e + f*x))^(7/3))/(13*d*f) + (3*(c + d)^2*(6*b*c^2 - 13*a*c*d - 10*b*d^2)*AppellF1(1/2, 1/2, -(7/3), 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1/3))/(65*sqrt(2)*d^3*f*sqrt(1 + sin(e + f*x))*((c + d*sin(e + f*x))/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(6*b*c - 13*a*d)*AppellF1(1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1/3))/(65*sqrt(2)*d^3*f*sqrt(1 + sin(e + f*x))*((c + d*sin(e + f*x))/(c + d))^(1/3))]
    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^0*(c + d*sin(e + f*x))^(4/3), x, 2, (3*AppellF1(7/3, -(1/2), -(1/2), 10/3, (c + d*sin(e + f*x))/(c - d), (c + d*sin(e + f*x))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(7/3))/(7*d*f*sqrt(1 - (c + d*sin(e + f*x))/(c - d))*sqrt(1 - (c + d*sin(e + f*x))/(c + d)))]
    @test_int [cos(e + f*x)^2/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^(4/3), x, 0, Unintegrable((cos(e + f*x)^2*(c + d*sin(e + f*x))^(4/3))/(a + b*sin(e + f*x)), x)]
    @test_int [cos(e + f*x)^2/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(4/3), x, 0, Unintegrable((cos(e + f*x)^2*(c + d*sin(e + f*x))^(4/3))/(a + b*sin(e + f*x))^2, x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 0, Unintegrable(cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x)]

    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(4/3), x, 0, Unintegrable(cos(e + f*x)^2*(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(4/3), x)]

    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 11, (((2*a^2*d^2*(3 + n) - 4*a*b*c*d*(4 + n) + b^2*(6*c^2 - d^2*(3 + n)))*cos(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d^3*f*(2 + n)*(3 + n)*(4 + n)) - (b*(3*b*c - 2*a*d)*cos(e + f*x)*sin(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d^2*f*(3 + n)*(4 + n)) + (cos(e + f*x)*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(1 + n))/(d*f*(4 + n)) - (sqrt(2)*(c + d)*(a^2*c*d^2*(12 + 7*n + n^2) - 2*a*b*d*(4 + n)*(2*c^2 - d^2*(2 + n)) + b^2*c*(6*c^2 - d^2*(3 - n - n^2)))*AppellF1(1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d^4*f*(2 + n)*(3 + n)*(4 + n)*sqrt(1 + sin(e + f*x)))) - (sqrt(2)*(c^2 - d^2)*(4*a*b*c*d*(4 + n) - a^2*d^2*(12 + 7*n + n^2) - b^2*(6*c^2 + d^2*(3 + 4*n + n^2)))*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d^4*f*(2 + n)*(3 + n)*(4 + n)*sqrt(1 + sin(e + f*x)))))]
    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 10, -(((2*b*c - a*d*(3 + n))*cos(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d^2*f*(2 + n)*(3 + n))) + (b*cos(e + f*x)*sin(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d*f*(3 + n)) - (sqrt(2)*(c + d)*(a*c*d*(3 + n) - b*(2*c^2 - d^2*(2 + n)))*AppellF1(1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d^3*f*(2 + n)*(3 + n)*sqrt(1 + sin(e + f*x)))) - (sqrt(2)*(c^2 - d^2)*(2*b*c - a*d*(3 + n))*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d^3*f*(2 + n)*(3 + n)*sqrt(1 + sin(e + f*x))))]
    @test_int [cos(e + f*x)^2*(a + b*sin(e + f*x))^0*(c + d*sin(e + f*x))^n, x, 2, (AppellF1(1 + n, -(1/2), -(1/2), 2 + n, (c + d*sin(e + f*x))/(c - d), (c + d*sin(e + f*x))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d*f*(1 + n)*sqrt(1 - (c + d*sin(e + f*x))/(c - d))*sqrt(1 - (c + d*sin(e + f*x))/(c + d)))]
    @test_int [cos(e + f*x)^2/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 0, Unintegrable((cos(e + f*x)^2*(c + d*sin(e + f*x))^n)/(a + b*sin(e + f*x)), x)]
    @test_int [cos(e + f*x)^2/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 0, Unintegrable((cos(e + f*x)^2*(c + d*sin(e + f*x))^n)/(a + b*sin(e + f*x))^2, x)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^m*(a+b*sin(e+f*x))^n*(A+B*sin(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [cos(c + d*x)^7*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (a*A*sin(c + d*x))/d + ((A*b + a*B)*sin(c + d*x)^2)/(2*d) - ((3*a*A - b*B)*sin(c + d*x)^3)/(3*d) - (3*(A*b + a*B)*sin(c + d*x)^4)/(4*d) + (3*(a*A - b*B)*sin(c + d*x)^5)/(5*d) + ((A*b + a*B)*sin(c + d*x)^6)/(2*d) - ((a*A - 3*b*B)*sin(c + d*x)^7)/(7*d) - ((A*b + a*B)*sin(c + d*x)^8)/(8*d) - (b*B*sin(c + d*x)^9)/(9*d)]
    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (a*A*sin(c + d*x))/d + ((A*b + a*B)*sin(c + d*x)^2)/(2*d) - ((2*a*A - b*B)*sin(c + d*x)^3)/(3*d) - ((A*b + a*B)*sin(c + d*x)^4)/(2*d) + ((a*A - 2*b*B)*sin(c + d*x)^5)/(5*d) + ((A*b + a*B)*sin(c + d*x)^6)/(6*d) + (b*B*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (a*A*sin(c + d*x))/d + ((A*b + a*B)*sin(c + d*x)^2)/(2*d) - ((a*A - b*B)*sin(c + d*x)^3)/(3*d) - ((A*b + a*B)*sin(c + d*x)^4)/(4*d) - (b*B*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, (a*A*sin(c + d*x))/d + ((A*b + a*B)*sin(c + d*x)^2)/(2*d) + (b*B*sin(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 5, -((a + b)*(A + B)*log(1 - sin(c + d*x)))/(2*d) + ((a - b)*(A - B)*log(1 + sin(c + d*x)))/(2*d) - (b*B*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 3, ((a*A - b*B)*atanh(sin(c + d*x)))/(2*d) + (sec(c + d*x)^2*(A*b + a*B + (a*A + b*B)*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 4, ((3*a*A - b*B)*atanh(sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(A*b + a*B + (a*A + b*B)*sin(c + d*x)))/(4*d) + ((3*a*A - b*B)*sec(c + d*x)*tan(c + d*x))/(8*d)]
    @test_int [sec(c + d*x)^7*(a + b*sin(c + d*x))*(A + B*sin(c + d*x)), x, 5, ((5*a*A - b*B)*atanh(sin(c + d*x)))/(16*d) + (sec(c + d*x)^6*(A*b + a*B + (a*A + b*B)*sin(c + d*x)))/(6*d) + ((5*a*A - b*B)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((5*a*A - b*B)*sec(c + d*x)^3*tan(c + d*x))/(24*d)]

    @test_int [cos(c + d*x)^7*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, -(((a^2 - b^2)^3*(A*b - a*B)*(a + b*sin(c + d*x))^3)/(3*b^8*d)) + ((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*(a + b*sin(c + d*x))^4)/(4*b^8*d) - (3*(a^2 - b^2)*(5*a^2*A*b - A*b^3 - 7*a^3*B + 3*a*b^2*B)*(a + b*sin(c + d*x))^5)/(5*b^8*d) + ((20*a^3*A*b - 12*a*A*b^3 - 35*a^4*B + 30*a^2*b^2*B - 3*b^4*B)*(a + b*sin(c + d*x))^6)/(6*b^8*d) - ((15*a^2*A*b - 3*A*b^3 - 35*a^3*B + 15*a*b^2*B)*(a + b*sin(c + d*x))^7)/(7*b^8*d) + (3*(2*a*A*b - 7*a^2*B + b^2*B)*(a + b*sin(c + d*x))^8)/(8*b^8*d) - ((A*b - 7*a*B)*(a + b*sin(c + d*x))^9)/(9*b^8*d) - (B*(a + b*sin(c + d*x))^10)/(10*b^8*d)]
    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, ((a^2 - b^2)^2*(A*b - a*B)*(a + b*sin(c + d*x))^3)/(3*b^6*d) - ((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*(a + b*sin(c + d*x))^4)/(4*b^6*d) + (2*(3*a^2*A*b - A*b^3 - 5*a^3*B + 3*a*b^2*B)*(a + b*sin(c + d*x))^5)/(5*b^6*d) - ((2*a*A*b - 5*a^2*B + b^2*B)*(a + b*sin(c + d*x))^6)/(3*b^6*d) + ((A*b - 5*a*B)*(a + b*sin(c + d*x))^7)/(7*b^6*d) + (B*(a + b*sin(c + d*x))^8)/(8*b^6*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, -((a^2 - b^2)*(A*b - a*B)*(a + b*sin(c + d*x))^3)/(3*b^4*d) + ((2*a*A*b - 3*a^2*B + b^2*B)*(a + b*sin(c + d*x))^4)/(4*b^4*d) - ((A*b - 3*a*B)*(a + b*sin(c + d*x))^5)/(5*b^4*d) - (B*(a + b*sin(c + d*x))^6)/(6*b^4*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 3, ((A*b - a*B)*(a + b*sin(c + d*x))^3)/(3*b^2*d) + (B*(a + b*sin(c + d*x))^4)/(4*b^2*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 6, -((a + b)^2*(A + B)*log(1 - sin(c + d*x)))/(2*d) + ((a - b)^2*(A - B)*log(1 + sin(c + d*x)))/(2*d) - (b*(A*b + 2*a*B)*sin(c + d*x))/d - (b^2*B*sin(c + d*x)^2)/(2*d)]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 5, -(((a + b)*(a*A - b*(A + 2*B))*log(1 - sin(c + d*x)))/(4*d)) + ((a - b)*(a*A + b*(A - 2*B))*log(1 + sin(c + d*x)))/(4*d) + (sec(c + d*x)^2*(a + b*sin(c + d*x))*(A*b + a*B + (a*A + b*B)*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 4, ((3*a^2*A - A*b^2 - 2*a*b*B)*atanh(sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(B + A*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(4*d) + (sec(c + d*x)^2*(2*b*(2*a*A - b*B) + (3*a^2*A + A*b^2 - 2*a*b*B)*sin(c + d*x)))/(8*d)]
    @test_int [sec(c + d*x)^7*(a + b*sin(c + d*x))^2*(A + B*sin(c + d*x)), x, 5, ((5*a^2*A - A*b^2 - 2*a*b*B)*atanh(sin(c + d*x)))/(16*d) + (sec(c + d*x)^6*(B + A*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(6*d) + (sec(c + d*x)^4*(2*b*(4*a*A - b*B) + (5*a^2*A + 3*A*b^2 - 2*a*b*B)*sin(c + d*x)))/(24*d) + ((5*a^2*A - A*b^2 - 2*a*b*B)*sec(c + d*x)*tan(c + d*x))/(16*d)]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [(cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 3, -(((a^2 - b^2)^3*(A*b - a*B)*log(a + b*sin(c + d*x)))/(b^8*d)) + ((a^5*A*b - 3*a^3*A*b^3 + 3*a*A*b^5 - a^6*B + 3*a^4*b^2*B - 3*a^2*b^4*B + b^6*B)*sin(c + d*x))/(b^7*d) - ((a^4 - 3*a^2*b^2 + 3*b^4)*(A*b - a*B)*sin(c + d*x)^2)/(2*b^6*d) + ((a^3*A*b - 3*a*A*b^3 - a^4*B + 3*a^2*b^2*B - 3*b^4*B)*sin(c + d*x)^3)/(3*b^5*d) - ((a^2 - 3*b^2)*(A*b - a*B)*sin(c + d*x)^4)/(4*b^4*d) + ((a*A*b - a^2*B + 3*b^2*B)*sin(c + d*x)^5)/(5*b^3*d) - ((A*b - a*B)*sin(c + d*x)^6)/(6*b^2*d) - (B*sin(c + d*x)^7)/(7*b*d)]
    @test_int [(cos(c + d*x)^5*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 3, ((a^2 - b^2)^2*(A*b - a*B)*log(a + b*sin(c + d*x)))/(b^6*d) - ((a^3*A*b - 2*a*A*b^3 - a^4*B + 2*a^2*b^2*B - b^4*B)*sin(c + d*x))/(b^5*d) + ((a^2 - 2*b^2)*(A*b - a*B)*sin(c + d*x)^2)/(2*b^4*d) - ((a*A*b - a^2*B + 2*b^2*B)*sin(c + d*x)^3)/(3*b^3*d) + ((A*b - a*B)*sin(c + d*x)^4)/(4*b^2*d) + (B*sin(c + d*x)^5)/(5*b*d)]
    @test_int [(cos(c + d*x)^3*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 3, -(((a^2 - b^2)*(A*b - a*B)*log(a + b*sin(c + d*x)))/(b^4*d)) + ((a*A*b - a^2*B + b^2*B)*sin(c + d*x))/(b^3*d) - ((A*b - a*B)*sin(c + d*x)^2)/(2*b^2*d) - (B*sin(c + d*x)^3)/(3*b*d)]
    @test_int [(cos(c + d*x)^1*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 3, ((A*b - a*B)*log(a + b*sin(c + d*x)))/(b^2*d) + (B*sin(c + d*x))/(b*d)]
    @test_int [(sec(c + d*x)^1*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 3, -((A + B)*log(1 - sin(c + d*x)))/(2*(a + b)*d) + ((A - B)*log(1 + sin(c + d*x)))/(2*(a - b)*d) - ((A*b - a*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [(sec(c + d*x)^3*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 4, -((a*A + b*(2*A + B))*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) + ((a*A - b*(2*A - B))*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) + (b^2*(A*b - a*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) - (sec(c + d*x)^2*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [(sec(c + d*x)^5*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 5, -((3*a^2*A + a*b*(9*A + B) + b^2*(8*A + 3*B))*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) + ((3*a^2*A + b^2*(8*A - 3*B) - a*b*(9*A - B))*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (b^4*(A*b - a*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (sec(c + d*x)^4*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(4*b^2*(A*b - a*B) + (3*a^3*A - 7*a*A*b^2 + a^2*b*B + 3*b^3*B)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]
    @test_int [(sec(c + d*x)^7*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x)), x, 6, -((5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*log(1 - sin(c + d*x)))/(32*(a + b)^4*d) + ((5*a^3*A - b^3*(16*A - 5*B) + a*b^2*(29*A - 4*B) - a^2*b*(20*A - B))*log(1 + sin(c + d*x)))/(32*(a - b)^4*d) + (b^6*(A*b - a*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^4*d) - (sec(c + d*x)^6*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(6*(a^2 - b^2)*d) + (sec(c + d*x)^4*(6*b^2*(A*b - a*B) + (5*a^3*A - 11*a*A*b^2 + a^2*b*B + 5*b^3*B)*sin(c + d*x)))/(24*(a^2 - b^2)^2*d) - (sec(c + d*x)^2*(8*b^4*(A*b - a*B) - (5*a^5*A - 16*a^3*A*b^2 + 19*a*A*b^4 + a^4*b*B - 4*a^2*b^3*B - 5*b^5*B)*sin(c + d*x)))/(16*(a^2 - b^2)^3*d)]

    @test_int [(cos(c + d*x)^7*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 3, ((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*log(a + b*sin(c + d*x)))/(b^8*d) - ((5*a^4*A*b - 9*a^2*A*b^3 + 3*A*b^5 - 6*a^5*B + 12*a^3*b^2*B - 6*a*b^4*B)*sin(c + d*x))/(b^7*d) + ((4*a^3*A*b - 6*a*A*b^3 - 5*a^4*B + 9*a^2*b^2*B - 3*b^4*B)*sin(c + d*x)^2)/(2*b^6*d) - ((3*a^2*A*b - 3*A*b^3 - 4*a^3*B + 6*a*b^2*B)*sin(c + d*x)^3)/(3*b^5*d) + ((2*a*A*b - 3*a^2*B + 3*b^2*B)*sin(c + d*x)^4)/(4*b^4*d) - ((A*b - 2*a*B)*sin(c + d*x)^5)/(5*b^3*d) - (B*sin(c + d*x)^6)/(6*b^2*d) + ((a^2 - b^2)^3*(A*b - a*B))/(b^8*d*(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^5*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 3, -(((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*log(a + b*sin(c + d*x)))/(b^6*d)) + ((3*a^2*A*b - 2*A*b^3 - 4*a^3*B + 4*a*b^2*B)*sin(c + d*x))/(b^5*d) - ((2*a*A*b - 3*a^2*B + 2*b^2*B)*sin(c + d*x)^2)/(2*b^4*d) + ((A*b - 2*a*B)*sin(c + d*x)^3)/(3*b^3*d) + (B*sin(c + d*x)^4)/(4*b^2*d) - ((a^2 - b^2)^2*(A*b - a*B))/(b^6*d*(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^3*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 3, ((2*a*A*b - 3*a^2*B + b^2*B)*log(a + b*sin(c + d*x)))/(b^4*d) - ((A*b - 2*a*B)*sin(c + d*x))/(b^3*d) - (B*sin(c + d*x)^2)/(2*b^2*d) + ((a^2 - b^2)*(A*b - a*B))/(b^4*d*(a + b*sin(c + d*x)))]
    @test_int [(cos(c + d*x)^1*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 3, (B*log(a + b*sin(c + d*x)))/(b^2*d) - (A*b - a*B)/(b^2*d*(a + b*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^1*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 3, -((A + B)*log(1 - sin(c + d*x)))/(2*(a + b)^2*d) + ((A - B)*log(1 + sin(c + d*x)))/(2*(a - b)^2*d) - ((2*a*A*b - a^2*B - b^2*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) + (A*b - a*B)/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^3*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 4, -((a*A + 3*A*b + 2*b*B)*log(1 - sin(c + d*x)))/(4*(a + b)^3*d) + ((a*A - 3*A*b + 2*b*B)*log(1 + sin(c + d*x)))/(4*(a - b)^3*d) + (b^2*(4*a*A*b - 3*a^2*B - b^2*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (b*(a^2*A + 3*A*b^2 - 4*a*b*B))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^2*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^5*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 5, -((3*a^2*A + 2*a*b*(6*A + B) + b^2*(15*A + 8*B))*log(1 - sin(c + d*x)))/(16*(a + b)^4*d) + ((3*a^2*A + b^2*(15*A - 8*B) - 2*a*b*(6*A - B))*log(1 + sin(c + d*x)))/(16*(a - b)^4*d) - (b^4*(6*a*A*b - 5*a^2*B - b^2*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^4*d) - (b*(3*a^4*A - 12*a^2*A*b^2 - 15*A*b^4 + 2*a^3*b*B + 22*a*b^3*B))/(8*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^4*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(4*(a^2 - b^2)*d*(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(b*(a^2*A + 5*A*b^2 - 6*a*b*B) + (3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [(sec(c + d*x)^7*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2, x, 6, -((5*a^3*A + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B) + b^3*(35*A + 16*B))*log(1 - sin(c + d*x)))/(32*(a + b)^5*d) + ((5*a^3*A - b^3*(35*A - 16*B) + a*b^2*(47*A - 10*B) - a^2*(25*A*b - 2*b*B))*log(1 + sin(c + d*x)))/(32*(a - b)^5*d) + (b^6*(8*a*A*b - 7*a^2*B - b^2*B)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^5*d) - (b*(5*a^6*A - 23*a^4*A*b^2 + 47*a^2*A*b^4 + 35*A*b^6 + 2*a^5*b*B - 12*a^3*b^3*B - 54*a*b^5*B))/(16*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^6*(A*b - a*B - (a*A - b*B)*sin(c + d*x)))/(6*(a^2 - b^2)*d*(a + b*sin(c + d*x))) + (sec(c + d*x)^4*(b*(a^2*A + 7*A*b^2 - 8*a*b*B) + (5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B)*sin(c + d*x)))/(24*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(b*(5*a^4*A - 18*a^2*A*b^2 - 35*A*b^4 + 2*a^3*b*B + 46*a*b^3*B) + 3*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*sin(c + d*x)))/(48*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^m*(a+b*sin(e+f*x))^(n/2)*(A+B*sin(e+f*x))=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(m/2)*(a+b*sin(e+f*x))^n*(A+B*sin(e+f*x))=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^(m/2)*(a+b*sin(e+f*x))^(n/2)*(A+B*sin(e+f*x))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^m*(a+b*sin(e+f*x))^n*(A+B*sin(e+f*x))*with*m*and/or*n*symbolic=#

    @test_int [(g*cos(e + f*x))^(-1 - m)*(a + b*sin(e + f*x))^m*(A + B*sin(e + f*x)), x, 0, Unintegrable((g*cos(e + f*x))^(-1 - m)*(a + b*sin(e + f*x))^m*(A + B*sin(e + f*x)), x)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*p*symbolic=#

    @test_int [(g*cos(e + f*x))^p/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^1), x, 4, -((g*AppellF1(1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*sin(e + f*x)), (a - b)/(a + b*sin(e + f*x)))*(g*cos(e + f*x))^(-1 + p)*(-((b*(1 - sin(e + f*x)))/(a + b*sin(e + f*x))))^((1 - p)/2)*((b*(1 + sin(e + f*x)))/(a + b*sin(e + f*x)))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))) + (g*AppellF1(1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*sin(e + f*x)), (c - d)/(c + d*sin(e + f*x)))*(g*cos(e + f*x))^(-1 + p)*(-((d*(1 - sin(e + f*x)))/(c + d*sin(e + f*x))))^((1 - p)/2)*((d*(1 + sin(e + f*x)))/(c + d*sin(e + f*x)))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))]
    @test_int [(g*cos(e + f*x))^p/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^2), x, 5, -((b*g*AppellF1(1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*sin(e + f*x)), (a - b)/(a + b*sin(e + f*x)))*(g*cos(e + f*x))^(-1 + p)*(-((b*(1 - sin(e + f*x)))/(a + b*sin(e + f*x))))^((1 - p)/2)*((b*(1 + sin(e + f*x)))/(a + b*sin(e + f*x)))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p))) + (b*g*AppellF1(1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*sin(e + f*x)), (c - d)/(c + d*sin(e + f*x)))*(g*cos(e + f*x))^(-1 + p)*(-((d*(1 - sin(e + f*x)))/(c + d*sin(e + f*x))))^((1 - p)/2)*((d*(1 + sin(e + f*x)))/(c + d*sin(e + f*x)))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p)) + (g*AppellF1(2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (c + d)/(c + d*sin(e + f*x)), (c - d)/(c + d*sin(e + f*x)))*(g*cos(e + f*x))^(-1 + p)*(-((d*(1 - sin(e + f*x)))/(c + d*sin(e + f*x))))^((1 - p)/2)*((d*(1 + sin(e + f*x)))/(c + d*sin(e + f*x)))^((1 - p)/2))/((b*c - a*d)*f*(2 - p)*(c + d*sin(e + f*x)))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*p*symbolic=#

    @test_int [(g*sec(e + f*x))^p/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x, 5, (-((AppellF1(1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (a + b)/(a + b*sin(e + f*x)), (a - b)/(a + b*sin(e + f*x)))*sec(e + f*x)*(g*sec(e + f*x))^p*(-((b*(1 - sin(e + f*x)))/(a + b*sin(e + f*x))))^((1 + p)/2)*((b*(1 + sin(e + f*x)))/(a + b*sin(e + f*x)))^((1 + p)/2))/((b*c - a*d)*f*(1 + p))) + (AppellF1(1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (c + d)/(c + d*sin(e + f*x)), (c - d)/(c + d*sin(e + f*x)))*sec(e + f*x)*(g*sec(e + f*x))^p*(-((d*(1 - sin(e + f*x)))/(c + d*sin(e + f*x))))^((1 + p)/2)*((d*(1 + sin(e + f*x)))/(c + d*sin(e + f*x)))^((1 + p)/2))/((b*c - a*d)*f*(1 + p)))]
    end

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.jl