text:
using Rubin
using Tests
using Elliptic
using HypergeometricFunctions
using Polylogarithms
using SpecialFunctions

@test integrate(cos(c+d*x)^7*(a+a*sin(c+d*x)), x) == :(-2*a^-5*d^-1*(a+a*sin(c+d*x))^6+-1//8*a^-7*d^-1*(a+a*sin(c+d*x))^8+6//7*a^-6*d^-1*(a+a*sin(c+d*x))^7+8//5*a^-4*d^-1*(a+a*sin(c+d*x))^5)
@test integrate(cos(c+d*x)^6*(a+a*sin(c+d*x)), x) == :(5//16*a*x+-1//7*a*d^-1*cos(c+d*x)^7+1//6*a*d^-1*cos(c+d*x)^5*sin(c+d*x)+5//16*a*d^-1*cos(c+d*x)*sin(c+d*x)+5//24*a*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate(cos(c+d*x)^5*(a+a*sin(c+d*x)), x) == :(a^-3*d^-1*(a+a*sin(c+d*x))^4+-4//5*a^-4*d^-1*(a+a*sin(c+d*x))^5+1//6*a^-5*d^-1*(a+a*sin(c+d*x))^6)
@test integrate(cos(c+d*x)^4*(a+a*sin(c+d*x)), x) == :(3//8*a*x+-1//5*a*d^-1*cos(c+d*x)^5+1//4*a*d^-1*cos(c+d*x)^3*sin(c+d*x)+3//8*a*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate(cos(c+d*x)^3*(a+a*sin(c+d*x)), x) == :(-1//4*a^-3*d^-1*(a+a*sin(c+d*x))^4+2//3*a^-2*d^-1*(a+a*sin(c+d*x))^3)
@test integrate(cos(c+d*x)^2*(a+a*sin(c+d*x)), x) == :((1/2)*a*x+-1//3*a*d^-1*cos(c+d*x)^3+(1/2)*a*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))*sec(c+d*x), x) == :(-1*a*d^-1*log(1+-1*sin(c+d*x)))
@test integrate(sec(c+d*x)^2*(a+a*sin(c+d*x)), x) == :(a*d^-1*sec(c+d*x)+a*d^-1*tan(c+d*x))
@test integrate(sec(c+d*x)^3*(a+a*sin(c+d*x)), x) == :((1/2)*a*d^-1*arctanh(sin(c+d*x))+(1/2)*a^2*d^-1*(a+-1*a*sin(c+d*x))^-1)
@test integrate(sec(c+d*x)^4*(a+a*sin(c+d*x)), x) == :(a*d^-1*tan(c+d*x)+1//3*a*d^-1*sec(c+d*x)^3+1//3*a*d^-1*tan(c+d*x)^3)
@test integrate(sec(c+d*x)^5*(a+a*sin(c+d*x)), x) == :(-1//8*a^2*d^-1*(a+a*sin(c+d*x))^-1+1//4*a^2*d^-1*(a+-1*a*sin(c+d*x))^-1+1//8*a^3*d^-1*(a+-1*a*sin(c+d*x))^-2+3//8*a*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x)^6, x) == :(45//128*x*a^2+-9//56*a^2*d^-1*cos(c+d*x)^7+-1//8*d^-1*cos(c+d*x)^7*(a^2+a^2*sin(c+d*x))+3//16*a^2*d^-1*cos(c+d*x)^5*sin(c+d*x)+15//64*a^2*d^-1*cos(c+d*x)^3*sin(c+d*x)+45//128*a^2*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x)^5, x) == :(-2//3*a^-4*d^-1*(a+a*sin(c+d*x))^6+1//7*a^-5*d^-1*(a+a*sin(c+d*x))^7+4//5*a^-3*d^-1*(a+a*sin(c+d*x))^5)
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x)^4, x) == :(7//16*x*a^2+-7//30*a^2*d^-1*cos(c+d*x)^5+-1//6*d^-1*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))+7//16*a^2*d^-1*cos(c+d*x)*sin(c+d*x)+7//24*a^2*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x)^3, x) == :((1/2)*a^-2*d^-1*(a+a*sin(c+d*x))^4+-1//5*a^-3*d^-1*(a+a*sin(c+d*x))^5)
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x)^2, x) == :(5//8*x*a^2+-5//12*a^2*d^-1*cos(c+d*x)^3+-1//4*d^-1*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))+5//8*a^2*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*cos(c+d*x), x) == :(1//3*a^-1*d^-1*(a+a*sin(c+d*x))^3)
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x), x) == :(-1*a^2*d^-1*sin(c+d*x)+-2*a^2*d^-1*log(1+-1*sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^2, x) == :(-1*x*a^2+2*a^4*d^-1*(a^2+-1*a^2*sin(c+d*x))^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^3, x) == :(a^3*d^-1*(a+-1*a*sin(c+d*x))^-1)
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^4, x) == :(1//3*a^4*d^-1*(a+-1*a*sin(c+d*x))^-2*cos(c+d*x)+1//3*a^4*d^-1*(a^2+-1*a^2*sin(c+d*x))^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^5, x) == :(1//4*a^2*d^-1*arctanh(sin(c+d*x))+1//4*a^3*d^-1*(a+-1*a*sin(c+d*x))^-1+1//4*a^4*d^-1*(a+-1*a*sin(c+d*x))^-2)
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^6, x) == :(1//5*a^2*d^-1*tan(c+d*x)^3+2//5*d^-1*sec(c+d*x)^5*(a^2+a^2*sin(c+d*x))+3//5*a^2*d^-1*tan(c+d*x))
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^7, x) == :(-1//16*a^3*d^-1*(a+a*sin(c+d*x))^-1+1//4*a^2*d^-1*arctanh(sin(c+d*x))+1//8*a^4*d^-1*(a+-1*a*sin(c+d*x))^-2+1//12*a^5*d^-1*(a+-1*a*sin(c+d*x))^-3+3//16*a^3*d^-1*(a+-1*a*sin(c+d*x))^-1)
@test integrate((a+a*sin(c+d*x))^2*sec(c+d*x)^8, x) == :(1//7*a^2*d^-1*tan(c+d*x)^5+2//7*d^-1*sec(c+d*x)^7*(a^2+a^2*sin(c+d*x))+5//7*a^2*d^-1*tan(c+d*x)+10//21*a^2*d^-1*tan(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x)^6, x) == :(55//128*x*a^3+-11//56*a^3*d^-1*cos(c+d*x)^7+-11//72*d^-1*cos(c+d*x)^7*(a^3+a^3*sin(c+d*x))+-1//9*a*d^-1*(a+a*sin(c+d*x))^2*cos(c+d*x)^7+11//48*a^3*d^-1*cos(c+d*x)^5*sin(c+d*x)+55//128*a^3*d^-1*cos(c+d*x)*sin(c+d*x)+55//192*a^3*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x)^5, x) == :(-4//7*a^-4*d^-1*(a+a*sin(c+d*x))^7+1//8*a^-5*d^-1*(a+a*sin(c+d*x))^8+2//3*a^-3*d^-1*(a+a*sin(c+d*x))^6)
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x)^4, x) == :(9//16*x*a^3+-3//10*a^3*d^-1*cos(c+d*x)^5+-3//14*d^-1*cos(c+d*x)^5*(a^3+a^3*sin(c+d*x))+-1//7*a*d^-1*(a+a*sin(c+d*x))^2*cos(c+d*x)^5+3//8*a^3*d^-1*cos(c+d*x)^3*sin(c+d*x)+9//16*a^3*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x)^3, x) == :(-1//6*a^-3*d^-1*(a+a*sin(c+d*x))^6+2//5*a^-2*d^-1*(a+a*sin(c+d*x))^5)
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x)^2, x) == :(7//8*x*a^3+-7//12*a^3*d^-1*cos(c+d*x)^3+-7//20*d^-1*cos(c+d*x)^3*(a^3+a^3*sin(c+d*x))+-1//5*a*d^-1*(a+a*sin(c+d*x))^2*cos(c+d*x)^3+7//8*a^3*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^3*cos(c+d*x), x) == :(1//4*a^-1*d^-1*(a+a*sin(c+d*x))^4)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x), x) == :(-4*a^3*d^-1*log(1+-1*sin(c+d*x))+-3*a^3*d^-1*sin(c+d*x)+-1//2*a^3*d^-1*sin(c+d*x)^2)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^2, x) == :(-3*x*a^3+3*a^3*d^-1*cos(c+d*x)+2*a^5*d^-1*(a+-1*a*sin(c+d*x))^-2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^3, x) == :(a^3*d^-1*log(1+-1*sin(c+d*x))+2*a^4*d^-1*(a+-1*a*sin(c+d*x))^-1)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^4, x) == :(1//3*a^6*d^-1*(a+-1*a*sin(c+d*x))^-3*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^5, x) == :((1/2)*a^5*d^-1*(a+-1*a*sin(c+d*x))^-2)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^6, x) == :(1//5*a^6*d^-1*(a+-1*a*sin(c+d*x))^-3*cos(c+d*x)+2//15*a^5*d^-1*(a+-1*a*sin(c+d*x))^-2*cos(c+d*x)+2//15*a^6*d^-1*(a^3+-1*a^3*sin(c+d*x))^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^7, x) == :(1//6*a^6*d^-1*(a+-1*a*sin(c+d*x))^-3+1//8*a^3*d^-1*arctanh(sin(c+d*x))+1//8*a^4*d^-1*(a+-1*a*sin(c+d*x))^-1+1//8*a^5*d^-1*(a+-1*a*sin(c+d*x))^-2)
@test integrate((a+a*sin(c+d*x))^3*sec(c+d*x)^8, x) == :(2//7*a^3*d^-1*tan(c+d*x)^3+3//7*a^3*d^-1*tan(c+d*x)+3//35*a^3*d^-1*sec(c+d*x)^5+3//35*a^3*d^-1*tan(c+d*x)^5+2//7*a*d^-1*(a+a*sin(c+d*x))^2*sec(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^8*cos(c+d*x)^5, x) == :(-1//3*a^-4*d^-1*(a+a*sin(c+d*x))^12+1//13*a^-5*d^-1*(a+a*sin(c+d*x))^13+4//11*a^-3*d^-1*(a+a*sin(c+d*x))^11)
@test integrate((a+a*sin(c+d*x))^8*cos(c+d*x)^4, x) == :(4199//1024*x*a^8+-4199//1920*a^8*d^-1*cos(c+d*x)^5+-4199//2688*d^-1*cos(c+d*x)^5*(a^8+a^8*sin(c+d*x))+-4199//4032*d^-1*(a^4+a^4*sin(c+d*x))^2*cos(c+d*x)^5+-323//792*d^-1*(a^2+a^2*sin(c+d*x))^4*cos(c+d*x)^5+-4199//6336*a^2*d^-1*(a^2+a^2*sin(c+d*x))^3*cos(c+d*x)^5+-323//1320*a^3*d^-1*(a+a*sin(c+d*x))^5*cos(c+d*x)^5+-19//132*a^2*d^-1*(a+a*sin(c+d*x))^6*cos(c+d*x)^5+-1//12*a*d^-1*(a+a*sin(c+d*x))^7*cos(c+d*x)^5+4199//1024*a^8*d^-1*cos(c+d*x)*sin(c+d*x)+4199//1536*a^8*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^8*cos(c+d*x)^3, x) == :(-1//11*a^-3*d^-1*(a+a*sin(c+d*x))^11+1//5*a^-2*d^-1*(a+a*sin(c+d*x))^10)
@test integrate((a+a*sin(c+d*x))^8*cos(c+d*x)^2, x) == :(2431//256*x*a^8+-2431//384*a^8*d^-1*cos(c+d*x)^3+-2431//640*d^-1*cos(c+d*x)^3*(a^8+a^8*sin(c+d*x))+-2431//1120*d^-1*(a^4+a^4*sin(c+d*x))^2*cos(c+d*x)^3+-221//336*d^-1*(a^2+a^2*sin(c+d*x))^4*cos(c+d*x)^3+-2431//2016*a^2*d^-1*(a^2+a^2*sin(c+d*x))^3*cos(c+d*x)^3+-17//48*a^3*d^-1*(a+a*sin(c+d*x))^5*cos(c+d*x)^3+-17//90*a^2*d^-1*(a+a*sin(c+d*x))^6*cos(c+d*x)^3+-1//10*a*d^-1*(a+a*sin(c+d*x))^7*cos(c+d*x)^3+2431//256*a^8*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^8*cos(c+d*x), x) == :(1//9*a^-1*d^-1*(a+a*sin(c+d*x))^9)
@test integrate((a+a*sin(c+d*x))^8*sec(c+d*x), x) == :(-16*d^-1*(a^4+a^4*sin(c+d*x))^2+-2*d^-1*(a^2+a^2*sin(c+d*x))^4+-128*a^8*d^-1*log(1+-1*sin(c+d*x))+-64*a^8*d^-1*sin(c+d*x)+-16//3*a^5*d^-1*(a+a*sin(c+d*x))^3+-4//5*a^3*d^-1*(a+a*sin(c+d*x))^5+-1//3*a^2*d^-1*(a+a*sin(c+d*x))^6+-1//7*a*d^-1*(a+a*sin(c+d*x))^7)
@test integrate((a+a*sin(c+d*x))^8*sec(c+d*x)^2, x) == :(-3003//16*x*a^8+1001//10*a^8*d^-1*cos(c+d*x)^5+2*a^15*d^-1*(a+-1*a*sin(c+d*x))^-7*cos(c+d*x)^13+26*a^13*d^-1*(a+-1*a*sin(c+d*x))^-5*cos(c+d*x)^11+-3003//16*a^8*d^-1*cos(c+d*x)*sin(c+d*x)+-1001//8*a^8*d^-1*cos(c+d*x)^3*sin(c+d*x)+143//2*a^16*d^-1*(a^8+-1*a^8*sin(c+d*x))^-1*cos(c+d*x)^7+286//3*a^14*d^-1*(a^2+-1*a^2*sin(c+d*x))^-3*cos(c+d*x)^9)
@test integrate((a+a*sin(c+d*x))^8*sec(c+d*x)^3, x) == :(2*a^8*d^-1*sin(c+d*x)^4+10*a^8*d^-1*sin(c+d*x)^3+36*a^8*d^-1*sin(c+d*x)^2+64*a^9*d^-1*(a+-1*a*sin(c+d*x))^-1+129*a^8*d^-1*sin(c+d*x)+192*a^8*d^-1*log(1+-1*sin(c+d*x))+1//5*a^8*d^-1*sin(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^8*sec(c+d*x)^4, x) == :(1155//8*x*a^8+-385//4*a^8*d^-1*cos(c+d*x)^3+-66*a^14*d^-1*(a^2+-1*a^2*sin(c+d*x))^-3*cos(c+d*x)^7+-231//4*a^16*d^-1*(a^8+-1*a^8*sin(c+d*x))^-1*cos(c+d*x)^5+-22//3*a^13*d^-1*(a+-1*a*sin(c+d*x))^-5*cos(c+d*x)^9+2//3*a^15*d^-1*(a+-1*a*sin(c+d*x))^-7*cos(c+d*x)^11+1155//8*a^8*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^8*sec(c+d*x)^5, x) == :(-80*a^8*d^-1*log(1+-1*sin(c+d*x))+-80*a^9*d^-1*(a+-1*a*sin(c+d*x))^-1+-31*a^8*d^-1*sin(c+d*x)+-4*a^8*d^-1*sin(c+d*x)^2+16*a^10*d^-1*(a+-1*a*sin(c+d*x))^-2+-1//3*a^8*d^-1*sin(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x)^6, x) == :(3//8*x*a^-1+1//5*a^-1*d^-1*cos(c+d*x)^5+1//4*a^-1*d^-1*cos(c+d*x)^3*sin(c+d*x)+3//8*a^-1*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x)^5, x) == :(-2//3*a^-4*d^-1*(a+-1*a*sin(c+d*x))^3+1//4*a^-5*d^-1*(a+-1*a*sin(c+d*x))^4)
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x)^4, x) == :((1/2)*x*a^-1+1//3*a^-1*d^-1*cos(c+d*x)^3+(1/2)*a^-1*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x)^3, x) == :(a^-1*d^-1*sin(c+d*x)+-1//2*a^-1*d^-1*sin(c+d*x)^2)
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x)^2, x) == :(x*a^-1+a^-1*d^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1*cos(c+d*x), x) == :(a^-1*d^-1*log(1+sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1*sec(c+d*x), x) == :(-1//2*d^-1*(a+a*sin(c+d*x))^-1+(1/2)*a^-1*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1*sec(c+d*x)^2, x) == :(-1//3*d^-1*(a+a*sin(c+d*x))^-1*sec(c+d*x)+2//3*a^-1*d^-1*tan(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+a*sin(c+d*x))^-1+1//8*d^-1*(a+-1*a*sin(c+d*x))^-1+-1//8*a*d^-1*(a+a*sin(c+d*x))^-2+3//8*a^-1*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1*sec(c+d*x)^4, x) == :(-1//5*d^-1*(a+a*sin(c+d*x))^-1*sec(c+d*x)^3+4//5*a^-1*d^-1*tan(c+d*x)+4//15*a^-1*d^-1*tan(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-1*sec(c+d*x)^5, x) == :(-3//16*d^-1*(a+a*sin(c+d*x))^-1+1//8*d^-1*(a+-1*a*sin(c+d*x))^-1+-3//32*a*d^-1*(a+a*sin(c+d*x))^-2+-1//24*a^2*d^-1*(a+a*sin(c+d*x))^-3+1//32*a*d^-1*(a+-1*a*sin(c+d*x))^-2+5//16*a^-1*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^8, x) == :(7//16*x*a^-2+1//6*d^-1*(a^2+a^2*sin(c+d*x))^-1*cos(c+d*x)^7+7//30*a^-2*d^-1*cos(c+d*x)^5+7//16*a^-2*d^-1*cos(c+d*x)*sin(c+d*x)+7//24*a^-2*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^7, x) == :(-1//2*a^-6*d^-1*(a+-1*a*sin(c+d*x))^4+1//5*a^-7*d^-1*(a+-1*a*sin(c+d*x))^5)
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^6, x) == :(5//8*x*a^-2+1//4*d^-1*(a^2+a^2*sin(c+d*x))^-1*cos(c+d*x)^5+5//12*a^-2*d^-1*cos(c+d*x)^3+5//8*a^-2*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^5, x) == :(-1//3*a^-5*d^-1*(a+-1*a*sin(c+d*x))^3)
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^4, x) == :(3//2*x*a^-2+(1/2)*d^-1*(a^2+a^2*sin(c+d*x))^-1*cos(c+d*x)^3+3//2*a^-2*d^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^3, x) == :(-1*a^-2*d^-1*sin(c+d*x)+2*a^-2*d^-1*log(1+sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x)^2, x) == :(-1*x*a^-2+-2*d^-1*(a^2+a^2*sin(c+d*x))^-1*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-2*cos(c+d*x), x) == :(-1*d^-1*(a^2+a^2*sin(c+d*x))^-1)
@test integrate((a+a*sin(c+d*x))^-2*sec(c+d*x), x) == :(-1//4*d^-1*(a+a*sin(c+d*x))^-2+-1//4*d^-1*(a^2+a^2*sin(c+d*x))^-1+1//4*a^-2*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-2*sec(c+d*x)^2, x) == :(-1//5*d^-1*(a+a*sin(c+d*x))^-2*sec(c+d*x)+-1//5*d^-1*(a^2+a^2*sin(c+d*x))^-1*sec(c+d*x)+2//5*a^-2*d^-1*tan(c+d*x))
@test integrate((a+a*sin(c+d*x))^-2*sec(c+d*x)^3, x) == :(-3//16*d^-1*(a^2+a^2*sin(c+d*x))^-1+-1//8*d^-1*(a+a*sin(c+d*x))^-2+1//16*d^-1*(a^2+-1*a^2*sin(c+d*x))^-1+-1//12*a*d^-1*(a+a*sin(c+d*x))^-3+1//4*a^-2*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-2*sec(c+d*x)^4, x) == :(-1//7*d^-1*(a+a*sin(c+d*x))^-2*sec(c+d*x)^3+-1//7*d^-1*(a^2+a^2*sin(c+d*x))^-1*sec(c+d*x)^3+4//7*a^-2*d^-1*tan(c+d*x)+4//21*a^-2*d^-1*tan(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-2*sec(c+d*x)^5, x) == :(-5//32*d^-1*(a^2+a^2*sin(c+d*x))^-1+-3//32*d^-1*(a+a*sin(c+d*x))^-2+1//64*d^-1*(a+-1*a*sin(c+d*x))^-2+5//64*d^-1*(a^2+-1*a^2*sin(c+d*x))^-1+-1//16*a*d^-1*(a+a*sin(c+d*x))^-3+-1//32*a^2*d^-1*(a+a*sin(c+d*x))^-4+15//64*a^-2*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^8, x) == :(7//8*x*a^-3+7//15*a^-3*d^-1*cos(c+d*x)^5+2//3*a^-1*d^-1*(a+a*sin(c+d*x))^-2*cos(c+d*x)^7+7//8*a^-3*d^-1*cos(c+d*x)*sin(c+d*x)+7//12*a^-3*d^-1*cos(c+d*x)^3*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^7, x) == :(-1//4*a^-7*d^-1*(a+-1*a*sin(c+d*x))^4)
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^6, x) == :(5//2*x*a^-3+5//3*a^-3*d^-1*cos(c+d*x)^3+2*a^-1*d^-1*(a+a*sin(c+d*x))^-2*cos(c+d*x)^5+5//2*a^-3*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^5, x) == :((1/2)*a^-3*d^-1*sin(c+d*x)^2+-3*a^-3*d^-1*sin(c+d*x)+4*a^-3*d^-1*log(1+sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^4, x) == :(-3*x*a^-3+-3*a^-3*d^-1*cos(c+d*x)+-2*a^-1*d^-1*(a+a*sin(c+d*x))^-2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^3, x) == :(-2*d^-1*(a^3+a^3*sin(c+d*x))^-1+-1*a^-3*d^-1*log(1+sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x)^2, x) == :(-1//3*d^-1*(a+a*sin(c+d*x))^-3*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-3*cos(c+d*x), x) == :(-1//2*a^-1*d^-1*(a+a*sin(c+d*x))^-2)
@test integrate((a+a*sin(c+d*x))^-3*sec(c+d*x), x) == :(-1//6*d^-1*(a+a*sin(c+d*x))^-3+-1//8*d^-1*(a^3+a^3*sin(c+d*x))^-1+-1//8*a^-1*d^-1*(a+a*sin(c+d*x))^-2+1//8*a^-3*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3*sec(c+d*x)^2, x) == :(-4//35*d^-1*(a^3+a^3*sin(c+d*x))^-1*sec(c+d*x)+-1//7*d^-1*(a+a*sin(c+d*x))^-3*sec(c+d*x)+8//35*a^-3*d^-1*tan(c+d*x)+-4//35*a^-1*d^-1*(a+a*sin(c+d*x))^-2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3*sec(c+d*x)^3, x) == :(-1//8*d^-1*(a^3+a^3*sin(c+d*x))^-1+-1//12*d^-1*(a+a*sin(c+d*x))^-3+1//32*d^-1*(a^3+-1*a^3*sin(c+d*x))^-1+-3//32*a^-1*d^-1*(a+a*sin(c+d*x))^-2+-1//16*a*d^-1*(a+a*sin(c+d*x))^-4+5//32*a^-3*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3*sec(c+d*x)^4, x) == :(-2//21*d^-1*(a^3+a^3*sin(c+d*x))^-1*sec(c+d*x)^3+-1//9*d^-1*(a+a*sin(c+d*x))^-3*sec(c+d*x)^3+8//21*a^-3*d^-1*tan(c+d*x)+8//63*a^-3*d^-1*tan(c+d*x)^3+-2//21*a^-1*d^-1*(a+a*sin(c+d*x))^-2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-3*sec(c+d*x)^5, x) == :(-15//128*d^-1*(a^3+a^3*sin(c+d*x))^-1+-1//16*d^-1*(a+a*sin(c+d*x))^-3+3//64*d^-1*(a^3+-1*a^3*sin(c+d*x))^-1+-5//64*a^-1*d^-1*(a+a*sin(c+d*x))^-2+-3//64*a*d^-1*(a+a*sin(c+d*x))^-4+-1//40*a^2*d^-1*(a+a*sin(c+d*x))^-5+1//128*a^-1*d^-1*(a+-1*a*sin(c+d*x))^-2+21//128*a^-3*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^8, x) == :(x*a^-8+2*d^-1*(a^8+a^8*sin(c+d*x))^-1*cos(c+d*x)+-2//3*a^-2*d^-1*(a^2+a^2*sin(c+d*x))^-3*cos(c+d*x)^3+-2//7*a^-1*d^-1*(a+a*sin(c+d*x))^-7*cos(c+d*x)^7+2//5*a^-3*d^-1*(a+a*sin(c+d*x))^-5*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^7, x) == :(-1//8*d^-1*(a+-1*a*sin(c+d*x))^4*(a^3+a^3*sin(c+d*x))^-4)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^6, x) == :(-1//9*d^-1*(a+a*sin(c+d*x))^-8*cos(c+d*x)^7+-1//63*a^-1*d^-1*(a+a*sin(c+d*x))^-7*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^5, x) == :(d^-1*(a^2+a^2*sin(c+d*x))^-4+-4//5*a^-3*d^-1*(a+a*sin(c+d*x))^-5+-1//3*a^-5*d^-1*(a+a*sin(c+d*x))^-3)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^4, x) == :(-1//11*d^-1*(a+a*sin(c+d*x))^-8*cos(c+d*x)^5+-2//231*a^-2*d^-1*(a+a*sin(c+d*x))^-6*cos(c+d*x)^5+-2//1155*a^-3*d^-1*(a+a*sin(c+d*x))^-5*cos(c+d*x)^5+-1//33*a^-1*d^-1*(a+a*sin(c+d*x))^-7*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^3, x) == :(-1//3*a^-2*d^-1*(a+a*sin(c+d*x))^-6+1//5*a^-3*d^-1*(a+a*sin(c+d*x))^-5)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x)^2, x) == :(-8//3003*d^-1*(a^2+a^2*sin(c+d*x))^-4*cos(c+d*x)^3+-1//13*d^-1*(a+a*sin(c+d*x))^-8*cos(c+d*x)^3+-20//1287*a^-2*d^-1*(a+a*sin(c+d*x))^-6*cos(c+d*x)^3+-20//3003*a^-3*d^-1*(a+a*sin(c+d*x))^-5*cos(c+d*x)^3+-8//9009*a^-2*d^-1*(a^2+a^2*sin(c+d*x))^-3*cos(c+d*x)^3+-5//143*a^-1*d^-1*(a+a*sin(c+d*x))^-7*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-8*cos(c+d*x), x) == :(-1//7*a^-1*d^-1*(a+a*sin(c+d*x))^-7)
@test integrate((a+a*sin(c+d*x))^-8*sec(c+d*x), x) == :(-1//16*d^-1*(a+a*sin(c+d*x))^-8+-1//128*d^-1*(a^2+a^2*sin(c+d*x))^-4+-1//256*d^-1*(a^4+a^4*sin(c+d*x))^-2+-1//256*d^-1*(a^8+a^8*sin(c+d*x))^-1+-1//28*a^-1*d^-1*(a+a*sin(c+d*x))^-7+-1//48*a^-2*d^-1*(a+a*sin(c+d*x))^-6+-1//80*a^-3*d^-1*(a+a*sin(c+d*x))^-5+-1//192*a^-5*d^-1*(a+a*sin(c+d*x))^-3+1//256*a^-8*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-8*sec(c+d*x)^2, x) == :(-112//12155*d^-1*(a^2+a^2*sin(c+d*x))^-4*sec(c+d*x)+-64//12155*d^-1*(a^4+a^4*sin(c+d*x))^-2*sec(c+d*x)+-64//12155*d^-1*(a^8+a^8*sin(c+d*x))^-1*sec(c+d*x)+-1//17*d^-1*(a+a*sin(c+d*x))^-8*sec(c+d*x)+128//12155*a^-8*d^-1*tan(c+d*x)+-168//12155*a^-3*d^-1*(a+a*sin(c+d*x))^-5*sec(c+d*x)+-24//1105*a^-2*d^-1*(a+a*sin(c+d*x))^-6*sec(c+d*x)+-16//2431*a^-2*d^-1*(a^2+a^2*sin(c+d*x))^-3*sec(c+d*x)+-3//85*a^-1*d^-1*(a+a*sin(c+d*x))^-7*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-8*sec(c+d*x)^3, x) == :(-9//1024*d^-1*(a^8+a^8*sin(c+d*x))^-1+-3//256*d^-1*(a^2+a^2*sin(c+d*x))^-4+-1//32*d^-1*(a+a*sin(c+d*x))^-8+-1//128*d^-1*(a^4+a^4*sin(c+d*x))^-2+1//1024*d^-1*(a^8+-1*a^8*sin(c+d*x))^-1+-7//768*a^-5*d^-1*(a+a*sin(c+d*x))^-3+-3//112*a^-1*d^-1*(a+a*sin(c+d*x))^-7+-1//36*a*d^-1*(a+a*sin(c+d*x))^-9+-1//48*a^-2*d^-1*(a+a*sin(c+d*x))^-6+-1//64*a^-3*d^-1*(a+a*sin(c+d*x))^-5+5//512*a^-8*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-8*sec(c+d*x)^4, x) == :(-48//4199*d^-1*(a^2+a^2*sin(c+d*x))^-4*sec(c+d*x)^3+-32//4199*d^-1*(a^4+a^4*sin(c+d*x))^-2*sec(c+d*x)^3+-32//4199*d^-1*(a^8+a^8*sin(c+d*x))^-1*sec(c+d*x)^3+-1//19*d^-1*(a+a*sin(c+d*x))^-8*sec(c+d*x)^3+128//4199*a^-8*d^-1*tan(c+d*x)+128//12597*a^-8*d^-1*tan(c+d*x)^3+-112//12597*a^-2*d^-1*(a^2+a^2*sin(c+d*x))^-3*sec(c+d*x)^3+-66//4199*a^-3*d^-1*(a+a*sin(c+d*x))^-5*sec(c+d*x)^3+-22//969*a^-2*d^-1*(a+a*sin(c+d*x))^-6*sec(c+d*x)^3+-11//323*a^-1*d^-1*(a+a*sin(c+d*x))^-7*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-8*sec(c+d*x)^5, x) == :(-55//4096*d^-1*(a^8+a^8*sin(c+d*x))^-1+-45//4096*d^-1*(a^4+a^4*sin(c+d*x))^-2+-7//512*d^-1*(a^2+a^2*sin(c+d*x))^-4+-3//128*d^-1*(a+a*sin(c+d*x))^-8+1//4096*d^-1*(a^4+-1*a^4*sin(c+d*x))^-2+11//4096*d^-1*(a^8+-1*a^8*sin(c+d*x))^-1+-21//1280*a^-3*d^-1*(a+a*sin(c+d*x))^-5+-5//224*a^-1*d^-1*(a+a*sin(c+d*x))^-7+-5//256*a^-2*d^-1*(a+a*sin(c+d*x))^-6+-3//256*a^-5*d^-1*(a+a*sin(c+d*x))^-3+-1//48*a*d^-1*(a+a*sin(c+d*x))^-9+-1//80*a^2*d^-1*(a+a*sin(c+d*x))^-10+33//2048*a^-8*d^-1*arctanh(sin(c+d*x)))
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^7, x) == :(-24//11*a^-5*d^-1*(a+a*sin(c+d*x))^11//2+-2//15*a^-7*d^-1*(a+a*sin(c+d*x))^15//2+12//13*a^-6*d^-1*(a+a*sin(c+d*x))^13//2+16//9*a^-4*d^-1*(a+a*sin(c+d*x))^9//2)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^6, x) == :(-256//3003*a^4*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7+-64//429*a^3*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7+-24//143*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^7+-2//13*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^5, x) == :(-8//9*a^-4*d^-1*(a+a*sin(c+d*x))^9//2+2//11*a^-5*d^-1*(a+a*sin(c+d*x))^11//2+8//7*a^-3*d^-1*(a+a*sin(c+d*x))^7//2)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^4, x) == :(-64//315*a^3*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5+-16//63*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5+-2//9*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^3, x) == :(-2//7*a^-3*d^-1*(a+a*sin(c+d*x))^7//2+4//5*a^-2*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^2, x) == :(-8//15*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3+-2//5*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^(1/2)*cos(c+d*x), x) == :(2//3*a^-1*d^-1*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x), x) == :(2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^2, x) == :(d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)+-1//2*2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^3, x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^2+-3//4*a*d^-1*(a+a*sin(c+d*x))^-1//2+3//8*2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^3+-5//8*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-5//16*2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+5//6*a*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^5, x) == :(-35//64*a*d^-1*(a+a*sin(c+d*x))^-1//2+-35//96*a^2*d^-1*(a+a*sin(c+d*x))^-3//2+1//4*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^4+7//16*a*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+35//128*2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^5+-63//128*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-63//256*2^(1/2)*a^(1/2)*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-21//80*a^2*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+3//10*a*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+21//32*a*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^7, x) == :(-24//13*a^-5*d^-1*(a+a*sin(c+d*x))^13//2+-2//17*a^-7*d^-1*(a+a*sin(c+d*x))^17//2+4//5*a^-6*d^-1*(a+a*sin(c+d*x))^15//2+16//11*a^-4*d^-1*(a+a*sin(c+d*x))^11//2)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^6, x) == :(-4096//45045*a^5*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7+-1024//6435*a^4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7+-128//715*a^3*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^7+-32//195*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^7+-2//15*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^5, x) == :(-8//11*a^-4*d^-1*(a+a*sin(c+d*x))^11//2+2//13*a^-5*d^-1*(a+a*sin(c+d*x))^13//2+8//9*a^-3*d^-1*(a+a*sin(c+d*x))^9//2)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^4, x) == :(-256//1155*a^4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5+-64//231*a^3*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5+-8//33*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^5+-2//11*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^3, x) == :(-2//9*a^-3*d^-1*(a+a*sin(c+d*x))^9//2+4//7*a^-2*d^-1*(a+a*sin(c+d*x))^7//2)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x)^2, x) == :(-64//105*a^3*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3+-16//35*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^3+-2//7*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^3//2*cos(c+d*x), x) == :(2//5*a^-1*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x), x) == :(-2*a*d^-1*(a+a*sin(c+d*x))^(1/2)+2*2^(1/2)*a^3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x)^2, x) == :(2*a*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x)^3, x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^2+1//4*2^(1/2)*a^3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^3+(1/2)*a*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)+-1//4*2^(1/2)*a^3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x)^5, x) == :(-15//32*a^2*d^-1*(a+a*sin(c+d*x))^-1//2+1//4*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^4+5//16*a*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^2+15//64*2^(1/2)*a^3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^3//2*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^5+-7//16*a^3*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-7//32*2^(1/2)*a^3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+7//12*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)+7//30*a*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^5//2*cos(c+d*x)^5, x) == :(-8//13*a^-4*d^-1*(a+a*sin(c+d*x))^13//2+2//15*a^-5*d^-1*(a+a*sin(c+d*x))^15//2+8//11*a^-3*d^-1*(a+a*sin(c+d*x))^11//2)
@test integrate((a+a*sin(c+d*x))^5//2*cos(c+d*x)^4, x) == :(-4096//15015*a^5*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5+-1024//3003*a^4*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5+-128//429*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^5+-32//143*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^5+-2//13*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^5//2*cos(c+d*x)^3, x) == :(-2//11*a^-3*d^-1*(a+a*sin(c+d*x))^11//2+4//9*a^-2*d^-1*(a+a*sin(c+d*x))^9//2)
@test integrate((a+a*sin(c+d*x))^5//2*cos(c+d*x)^2, x) == :(-256//315*a^4*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3+-64//105*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^3+-8//21*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^3+-2//9*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^5//2*cos(c+d*x), x) == :(2//7*a^-1*d^-1*(a+a*sin(c+d*x))^7//2)
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x), x) == :(-4*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)+-2//3*a*d^-1*(a+a*sin(c+d*x))^3//2+4*2^(1/2)*a^5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^2, x) == :(-2*a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)+8*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^3, x) == :(a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^2+-1//2*2^(1/2)*a^5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^4, x) == :(2//3*a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^5, x) == :(1//4*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^4+3//16*a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^2+3//32*2^(1/2)*a^5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^5+-1//8*2^(1/2)*a^5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)+1//6*a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^5//2*sec(c+d*x)^7, x) == :(-35//128*a^3*d^-1*(a+a*sin(c+d*x))^-1//2+1//6*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^6+7//48*a*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^4+35//192*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^2+35//256*2^(1/2)*a^5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^7, x) == :(-24//17*a^-5*d^-1*(a+a*sin(c+d*x))^17//2+-2//21*a^-7*d^-1*(a+a*sin(c+d*x))^21//2+12//19*a^-6*d^-1*(a+a*sin(c+d*x))^19//2+16//15*a^-4*d^-1*(a+a*sin(c+d*x))^15//2)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^6, x) == :(-131072//969969*a^7*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7+-32768//138567*a^6*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7+-12288//46189*a^5*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^7+-1024//4199*a^4*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^7+-64//323*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^7+-48//323*a^2*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)^7+-2//19*a*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^5, x) == :(-8//15*a^-4*d^-1*(a+a*sin(c+d*x))^15//2+2//17*a^-5*d^-1*(a+a*sin(c+d*x))^17//2+8//13*a^-3*d^-1*(a+a*sin(c+d*x))^13//2)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^4, x) == :(-16384//45045*a^6*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5+-4096//9009*a^5*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5+-512//1287*a^4*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^5+-128//429*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^5+-8//39*a^2*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)^5+-2//15*a*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^3, x) == :(-2//13*a^-3*d^-1*(a+a*sin(c+d*x))^13//2+4//11*a^-2*d^-1*(a+a*sin(c+d*x))^11//2)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x)^2, x) == :(-4096//3465*a^5*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3+-1024//1155*a^4*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)^3+-128//231*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)^3+-32//99*a^2*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)^3+-2//11*a*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^7//2*cos(c+d*x), x) == :(2//9*a^-1*d^-1*(a+a*sin(c+d*x))^9//2)
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x), x) == :(-8*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)+-4//3*a^2*d^-1*(a+a*sin(c+d*x))^3//2+-2//5*a*d^-1*(a+a*sin(c+d*x))^5//2+8*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^2, x) == :(-16//3*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)+-2//3*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)+64//3*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^3, x) == :(3*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)+a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^2+-3*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^4, x) == :(2*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^3+-8//3*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^5, x) == :((1/2)*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^4+-1//8*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^2+-1//16*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^6, x) == :(2//5*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^7, x) == :(1//6*d^-1*(a+a*sin(c+d*x))^7//2*sec(c+d*x)^6+5//48*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^4+5//64*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^2+5//128*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^8, x) == :(1//7*d^-1*(a+a*sin(c+d*x))^7//2*sec(c+d*x)^7+-1//16*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//8*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)+1//10*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^5+1//12*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^9, x) == :(-315//2048*a^4*d^-1*(a+a*sin(c+d*x))^-1//2+1//8*d^-1*(a+a*sin(c+d*x))^7//2*sec(c+d*x)^8+3//32*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^6+21//256*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^4+105//1024*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^2+315//4096*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^7//2*sec(c+d*x)^10, x) == :(1//9*d^-1*(a+a*sin(c+d*x))^7//2*sec(c+d*x)^9+-11//64*a^5*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-11//128*2^(1/2)*a^7//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+11//48*a^4*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)+11//120*a^3*d^-1*(a+a*sin(c+d*x))^(1/2)*sec(c+d*x)^3+11//126*a*d^-1*(a+a*sin(c+d*x))^5//2*sec(c+d*x)^7+11//140*a^2*d^-1*(a+a*sin(c+d*x))^3//2*sec(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^7, x) == :(-8//3*a^-5*d^-1*(a+a*sin(c+d*x))^9//2+-2//13*a^-7*d^-1*(a+a*sin(c+d*x))^13//2+12//11*a^-6*d^-1*(a+a*sin(c+d*x))^11//2+16//7*a^-4*d^-1*(a+a*sin(c+d*x))^7//2)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^6, x) == :(-64//693*a^3*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7+-16//99*a^2*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7+-2//11*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^5, x) == :(-8//7*a^-4*d^-1*(a+a*sin(c+d*x))^7//2+2//9*a^-5*d^-1*(a+a*sin(c+d*x))^9//2+8//5*a^-3*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^4, x) == :(-8//35*a^2*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5+-2//7*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^3, x) == :(-2//5*a^-3*d^-1*(a+a*sin(c+d*x))^5//2+4//3*a^-2*d^-1*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x)^2, x) == :(-2//3*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-1//2*cos(c+d*x), x) == :(2*a^-1*d^-1*(a+a*sin(c+d*x))^(1/2))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x), x) == :(-1*d^-1*(a+a*sin(c+d*x))^-1//2+(1/2)*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2, x) == :(d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)+-3//4*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-3//8*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3, x) == :(-5//8*d^-1*(a+a*sin(c+d*x))^-1//2+(1/2)*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+-5//12*a*d^-1*(a+a*sin(c+d*x))^-3//2+5//16*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+35//48*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)+-35//64*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-35//128*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-7//24*a*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x)^5, x) == :(-63//128*d^-1*(a+a*sin(c+d*x))^-1//2+-21//64*a*d^-1*(a+a*sin(c+d*x))^-3//2+1//4*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^4+63//160*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+-9//40*a*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^2+63//256*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-1//2*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^5+11//40*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+77//128*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)+-231//512*a*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-231//1024*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-77//320*a*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+-11//60*a*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^7, x) == :(-24//7*a^-5*d^-1*(a+a*sin(c+d*x))^7//2+-2//11*a^-7*d^-1*(a+a*sin(c+d*x))^11//2+4//3*a^-6*d^-1*(a+a*sin(c+d*x))^9//2+16//5*a^-4*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^6, x) == :(-8//63*a^2*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7+-2//9*a*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^5, x) == :(-8//5*a^-4*d^-1*(a+a*sin(c+d*x))^5//2+2//7*a^-5*d^-1*(a+a*sin(c+d*x))^7//2+8//3*a^-3*d^-1*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^4, x) == :(-2//5*a*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3, x) == :(4*a^-2*d^-1*(a+a*sin(c+d*x))^(1/2)+-2//3*a^-3*d^-1*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x)^2, x) == :(-2*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+2*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*cos(c+d*x), x) == :(-2*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2)
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x), x) == :(-1//3*d^-1*(a+a*sin(c+d*x))^-3//2+-1//2*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2+1//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x)^2, x) == :(-15//32*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-1//4*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+-15//64*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+5//8*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x)^3, x) == :(-7//24*d^-1*(a+a*sin(c+d*x))^-3//2+-7//16*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2+-1//5*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^2+7//20*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+7//32*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x)^4, x) == :(-105//256*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-7//32*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+-1//6*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^3+-105//512*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+35//64*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x)^5, x) == :(-33//128*d^-1*(a+a*sin(c+d*x))^-3//2+-99//256*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2+-99//560*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^2+-1//7*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^4+11//56*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^4+99//320*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+99//512*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-3//2*sec(c+d*x)^6, x) == :(-3003//8192*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-1001//5120*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+-143//960*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^3+-1//8*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^5+-3003//16384*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+13//80*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^5+143//640*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+1001//2048*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^10, x) == :(-64//2145*a^3*d^-1*(a+a*sin(c+d*x))^-11//2*cos(c+d*x)^11+-16//195*a^2*d^-1*(a+a*sin(c+d*x))^-9//2*cos(c+d*x)^11+-2//15*a*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^11)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^9, x) == :(-64//7*a^-6*d^-1*(a+a*sin(c+d*x))^7//2+-16//11*a^-8*d^-1*(a+a*sin(c+d*x))^11//2+2//13*a^-9*d^-1*(a+a*sin(c+d*x))^13//2+16//3*a^-7*d^-1*(a+a*sin(c+d*x))^9//2+32//5*a^-5*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^8, x) == :(-8//99*a^2*d^-1*(a+a*sin(c+d*x))^-9//2*cos(c+d*x)^9+-2//11*a*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^9)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^7, x) == :(-24//5*a^-5*d^-1*(a+a*sin(c+d*x))^5//2+-2//9*a^-7*d^-1*(a+a*sin(c+d*x))^9//2+12//7*a^-6*d^-1*(a+a*sin(c+d*x))^7//2+16//3*a^-4*d^-1*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^6, x) == :(-2//7*a*d^-1*(a+a*sin(c+d*x))^-7//2*cos(c+d*x)^7)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^5, x) == :(8*a^-3*d^-1*(a+a*sin(c+d*x))^(1/2)+-8//3*a^-4*d^-1*(a+a*sin(c+d*x))^3//2+2//5*a^-5*d^-1*(a+a*sin(c+d*x))^5//2)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^4, x) == :(-4*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+4*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+2//3*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)^3)
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^3, x) == :(-4*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2+-2*a^-3*d^-1*(a+a*sin(c+d*x))^(1/2))
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x)^2, x) == :((1/2)*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*cos(c+d*x), x) == :(-2//3*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2)
@test integrate((a+a*sin(c+d*x))^-5//2*sec(c+d*x), x) == :(-1//5*d^-1*(a+a*sin(c+d*x))^-5//2+-1//4*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2+-1//6*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2+1//8*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-5//2*sec(c+d*x)^2, x) == :(-1//6*d^-1*(a+a*sin(c+d*x))^-5//2*sec(c+d*x)+-35//128*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-35//256*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-7//48*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+35//96*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*sec(c+d*x)^3, x) == :(-9//32*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2+-3//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2+-1//7*d^-1*(a+a*sin(c+d*x))^-5//2*sec(c+d*x)^2+-9//70*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^2+9//40*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^2+9//64*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^-1//2*(a+a*sin(c+d*x))^(1/2)))
@test integrate((a+a*sin(c+d*x))^-5//2*sec(c+d*x)^4, x) == :(-1//8*d^-1*(a+a*sin(c+d*x))^-5//2*sec(c+d*x)^3+-1155//4096*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-1155//8192*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-77//512*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)+-11//96*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sec(c+d*x)^3+11//64*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x)^3+385//1024*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*sec(c+d*x))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x)), x) == :(-2//9*a*d^-1*e^-1*(e*cos(c+d*x))^9//2+2//7*a*e*d^-1*(e*cos(c+d*x))^5//2*sin(c+d*x)+10//21*a*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+10//21*a*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x)), x) == :(-2//7*a*d^-1*e^-1*(e*cos(c+d*x))^7//2+2//5*a*e*d^-1*(e*cos(c+d*x))^3//2*sin(c+d*x)+6//5*a*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x)), x) == :(-2//5*a*d^-1*e^-1*(e*cos(c+d*x))^5//2+2//3*a*e*d^-1*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+2//3*a*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x)), x) == :(-2//3*a*d^-1*e^-1*(e*cos(c+d*x))^3//2+2*a*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x)), x) == :(-2*a*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+2*a*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x)), x) == :(2*a*d^-1*e^-1*(e*cos(c+d*x))^-1//2+2*a*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-2*a*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x)), x) == :(2//3*a*d^-1*e^-1*(e*cos(c+d*x))^-3//2+2//3*a*d^-1*e^-1*(e*cos(c+d*x))^-3//2*sin(c+d*x)+2//3*a*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x)), x) == :(2//5*a*d^-1*e^-1*(e*cos(c+d*x))^-5//2+2//5*a*d^-1*e^-1*(e*cos(c+d*x))^-5//2*sin(c+d*x)+6//5*a*d^-1*e^-3*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-6//5*a*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^2, x) == :(-26//99*a^2*d^-1*e^-1*(e*cos(c+d*x))^9//2+-2//11*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a^2+a^2*sin(c+d*x))+26//77*e*a^2*d^-1*(e*cos(c+d*x))^5//2*sin(c+d*x)+130//231*a^2*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+130//231*a^2*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^2, x) == :(-22//63*a^2*d^-1*e^-1*(e*cos(c+d*x))^7//2+-2//9*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a^2+a^2*sin(c+d*x))+22//45*e*a^2*d^-1*(e*cos(c+d*x))^3//2*sin(c+d*x)+22//15*a^2*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^2, x) == :(-18//35*a^2*d^-1*e^-1*(e*cos(c+d*x))^5//2+-2//7*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a^2+a^2*sin(c+d*x))+6//7*e*a^2*d^-1*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+6//7*a^2*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^2, x) == :(-14//15*a^2*d^-1*e^-1*(e*cos(c+d*x))^3//2+-2//5*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))+14//5*a^2*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^2, x) == :(-10//3*a^2*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+-2//3*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))+10//3*a^2*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^2, x) == :(4*a^4*d^-1*e^-3*(e*cos(c+d*x))^3//2*(a^2+-1*a^2*sin(c+d*x))^-1+-6*a^2*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^2, x) == :(4//3*a^4*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a^2+-1*a^2*sin(c+d*x))^-1+-2//3*a^2*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^2, x) == :(2//5*a^4*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a+-1*a*sin(c+d*x))^-2+2//5*a^4*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a^2+-1*a^2*sin(c+d*x))^-1+-2//5*a^2*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^2, x) == :(1//7*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(4*a^2+4*a^2*sin(c+d*x))+2//7*a^2*d^-1*e^-3*(e*cos(c+d*x))^-3//2*sin(c+d*x)+2//7*a^2*d^-1*e^-4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^2, x) == :(1//9*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(4*a^2+4*a^2*sin(c+d*x))+2//3*a^2*d^-1*e^-5*(e*cos(c+d*x))^-1//2*sin(c+d*x)+2//9*a^2*d^-1*e^-3*(e*cos(c+d*x))^-5//2*sin(c+d*x)+-2//3*a^2*d^-1*e^-6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^3, x) == :(-34//99*a^3*d^-1*e^-1*(e*cos(c+d*x))^9//2+-34//143*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a^3+a^3*sin(c+d*x))+-2//13*a*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^2+34//77*e*a^3*d^-1*(e*cos(c+d*x))^5//2*sin(c+d*x)+170//231*a^3*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+170//231*a^3*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^3, x) == :(-10//21*a^3*d^-1*e^-1*(e*cos(c+d*x))^7//2+-10//33*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a^3+a^3*sin(c+d*x))+-2//11*a*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^2+2//3*e*a^3*d^-1*(e*cos(c+d*x))^3//2*sin(c+d*x)+2*a^3*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^3, x) == :(-26//35*a^3*d^-1*e^-1*(e*cos(c+d*x))^5//2+-26//63*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a^3+a^3*sin(c+d*x))+-2//9*a*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^2+26//21*e*a^3*d^-1*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+26//21*a^3*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^3, x) == :(-22//15*a^3*d^-1*e^-1*(e*cos(c+d*x))^3//2+-22//35*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^3+a^3*sin(c+d*x))+-2//7*a*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^2+22//5*a^3*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^3, x) == :(-6*a^3*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+-6//5*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^3+a^3*sin(c+d*x))+6*a^3*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2)+-2//5*a*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^2)
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3, x) == :(14//3*a^3*d^-1*e^-3*(e*cos(c+d*x))^3//2+4*a^5*d^-1*e^-5*(e*cos(c+d*x))^7//2*(a+-1*a*sin(c+d*x))^-2+-14*a^3*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3, x) == :(10//3*a^3*d^-1*e^-3*(e*cos(c+d*x))^(1/2)+4//3*a^5*d^-1*e^-5*(e*cos(c+d*x))^5//2*(a+-1*a*sin(c+d*x))^-2+-10//3*a^3*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^3, x) == :(-6//5*a^6*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a^3+-1*a^3*sin(c+d*x))^-1+4//5*a^5*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a+-1*a*sin(c+d*x))^-2+6//5*a^3*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^3, x) == :(-2//21*a^6*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(a^3+-1*a^3*sin(c+d*x))^-1+4//7*a^5*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(a+-1*a*sin(c+d*x))^-2+-2//21*a^3*d^-1*e^-4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^3, x) == :(2//9*a^6*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a+-1*a*sin(c+d*x))^-3+2//15*a^5*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a+-1*a*sin(c+d*x))^-2+2//15*a^6*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a^3+-1*a^3*sin(c+d*x))^-1+-2//15*a^3*d^-1*e^-6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^4, x) == :(-442//385*a^4*d^-1*e^-1*(e*cos(c+d*x))^5//2+-442//693*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a^4+a^4*sin(c+d*x))+-34//99*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a^2+a^2*sin(c+d*x))^2+-2//11*a*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^3+442//231*e*a^4*d^-1*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+442//231*a^4*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^4, x) == :(-22//9*a^4*d^-1*e^-1*(e*cos(c+d*x))^3//2+-22//21*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^4+a^4*sin(c+d*x))+-10//21*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))^2+-2//9*a*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^3+22//3*a^4*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^4, x) == :(-78//7*a^4*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+-78//35*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^4+a^4*sin(c+d*x))+-26//35*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))^2+-2//7*a*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^3+78//7*a^4*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^4, x) == :(4*a^7*d^-1*e^-7*(e*cos(c+d*x))^11//2*(a+-1*a*sin(c+d*x))^-3+-154//15*a^4*d^-1*e^-3*(e*cos(c+d*x))^3//2*sin(c+d*x)+44//3*a^8*d^-1*e^-5*(e*cos(c+d*x))^7//2*(a^4+-1*a^4*sin(c+d*x))^-1+-154//5*a^4*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^4, x) == :(-10*a^4*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+12*a^8*d^-1*e^-5*(e*cos(c+d*x))^5//2*(a^4+-1*a^4*sin(c+d*x))^-1+4//3*a^7*d^-1*e^-7*(e*cos(c+d*x))^9//2*(a+-1*a*sin(c+d*x))^-3+-10*a^4*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^4, x) == :(-28//5*a^8*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a^4+-1*a^4*sin(c+d*x))^-1+4//5*a^7*d^-1*e^-7*(e*cos(c+d*x))^7//2*(a+-1*a*sin(c+d*x))^-3+42//5*a^4*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^4, x) == :(-20//21*a^8*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(a^4+-1*a^4*sin(c+d*x))^-1+4//7*a^7*d^-1*e^-7*(e*cos(c+d*x))^5//2*(a+-1*a*sin(c+d*x))^-3+10//21*a^4*d^-1*e^-4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^4, x) == :(-2//15*a^8*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a^2+-1*a^2*sin(c+d*x))^-2+-2//15*a^8*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a^4+-1*a^4*sin(c+d*x))^-1+4//9*a^7*d^-1*e^-7*(e*cos(c+d*x))^3//2*(a+-1*a*sin(c+d*x))^-3+2//15*a^4*d^-1*e^-6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-13//2*(a+a*sin(c+d*x))^4, x) == :(-2//77*a^8*d^-1*e^-7*(e*cos(c+d*x))^(1/2)*(a^2+-1*a^2*sin(c+d*x))^-2+-2//77*a^8*d^-1*e^-7*(e*cos(c+d*x))^(1/2)*(a^4+-1*a^4*sin(c+d*x))^-1+4//11*a^7*d^-1*e^-7*(e*cos(c+d*x))^(1/2)*(a+-1*a*sin(c+d*x))^-3+-2//77*a^4*d^-1*e^-6*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-1, x) == :(2//9*e*a^-1*d^-1*(e*cos(c+d*x))^9//2+2//7*a^-1*d^-1*e^3*(e*cos(c+d*x))^5//2*sin(c+d*x)+10//21*a^-1*d^-1*e^5*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+10//21*a^-1*d^-1*e^6*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-1, x) == :(2//7*e*a^-1*d^-1*(e*cos(c+d*x))^7//2+2//5*a^-1*d^-1*e^3*(e*cos(c+d*x))^3//2*sin(c+d*x)+6//5*a^-1*d^-1*e^4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-1, x) == :(2//5*e*a^-1*d^-1*(e*cos(c+d*x))^5//2+2//3*a^-1*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+2//3*a^-1*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1, x) == :(2//3*e*a^-1*d^-1*(e*cos(c+d*x))^3//2+2*a^-1*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1, x) == :(2*e*a^-1*d^-1*(e*cos(c+d*x))^(1/2)+2*a^-1*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1, x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1+-2*a^-1*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1, x) == :(-2//3*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1+2//3*a^-1*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1, x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1+6//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-6//5*a^-1*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-1, x) == :(-2//7*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1+10//21*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-3//2*sin(c+d*x)+10//21*a^-1*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^-1, x) == :(-2//9*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-1+14//15*a^-1*d^-1*e^-3*(e*cos(c+d*x))^-1//2*sin(c+d*x)+14//45*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-5//2*sin(c+d*x)+-14//15*a^-1*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-2, x) == :(4//5*e*d^-1*(e*cos(c+d*x))^9//2*(a^2+a^2*sin(c+d*x))^-1+6//7*a^-2*d^-1*e^5*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+18//35*a^-2*d^-1*e^3*(e*cos(c+d*x))^5//2*sin(c+d*x)+6//7*a^-2*d^-1*e^6*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-2, x) == :(4//3*e*d^-1*(e*cos(c+d*x))^7//2*(a^2+a^2*sin(c+d*x))^-1+14//15*a^-2*d^-1*e^3*(e*cos(c+d*x))^3//2*sin(c+d*x)+14//5*a^-2*d^-1*e^4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-2, x) == :(4*e*d^-1*(e*cos(c+d*x))^5//2*(a^2+a^2*sin(c+d*x))^-1+10//3*a^-2*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+10//3*a^-2*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-2, x) == :(-4*e*d^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))^-1+-6*a^-2*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-2, x) == :(-4//3*e*d^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))^-1+-2//3*a^-2*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-2, x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-2+-2//5*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))^-1+-2//5*a^-2*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-2, x) == :(-2//7*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-2+-2//7*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))^-1+2//7*a^-2*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-2, x) == :(-2//9*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-2+-2//9*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a^2+a^2*sin(c+d*x))^-1+2//3*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-2//3*a^-2*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-2, x) == :(-2//11*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-2+-2//11*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a^2+a^2*sin(c+d*x))^-1+10//33*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-3//2*sin(c+d*x)+10//33*a^-2*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^-2, x) == :(-2//13*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-2+-2//13*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a^2+a^2*sin(c+d*x))^-1+14//65*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-5//2*sin(c+d*x)+42//65*a^-2*d^-1*e^-3*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-42//65*a^-2*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^15//2*(a+a*sin(c+d*x))^-3, x) == :(26//45*a^-3*d^-1*e^3*(e*cos(c+d*x))^9//2+4//5*e*a^-1*d^-1*(e*cos(c+d*x))^13//2*(a+a*sin(c+d*x))^-2+26//21*a^-3*d^-1*e^7*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+26//35*a^-3*d^-1*e^5*(e*cos(c+d*x))^5//2*sin(c+d*x)+26//21*a^-3*d^-1*e^8*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^13//2*(a+a*sin(c+d*x))^-3, x) == :(22//21*a^-3*d^-1*e^3*(e*cos(c+d*x))^7//2+4//3*e*a^-1*d^-1*(e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-2+22//15*a^-3*d^-1*e^5*(e*cos(c+d*x))^3//2*sin(c+d*x)+22//5*a^-3*d^-1*e^6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-3, x) == :(18//5*a^-3*d^-1*e^3*(e*cos(c+d*x))^5//2+4*e*a^-1*d^-1*(e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-2+6*a^-3*d^-1*e^5*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+6*a^-3*d^-1*e^6*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-3, x) == :(-14//3*a^-3*d^-1*e^3*(e*cos(c+d*x))^3//2+-4*e*a^-1*d^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-2+-14*a^-3*d^-1*e^4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3, x) == :(-10//3*a^-3*d^-1*e^3*(e*cos(c+d*x))^(1/2)+-4//3*e*a^-1*d^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-2+-10//3*a^-3*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-3, x) == :(6//5*e*d^-1*(e*cos(c+d*x))^3//2*(a^3+a^3*sin(c+d*x))^-1+-4//5*e*a^-1*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-2+6//5*a^-3*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3, x) == :(2//21*e*d^-1*(e*cos(c+d*x))^(1/2)*(a^3+a^3*sin(c+d*x))^-1+-4//7*e*a^-1*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-2+-2//21*a^-3*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3, x) == :(-2//9*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3+-2//15*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^3+a^3*sin(c+d*x))^-1+-2//15*a^-1*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-2+-2//15*a^-3*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3, x) == :(-10//77*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^3+a^3*sin(c+d*x))^-1+-2//11*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3+-10//77*a^-1*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-2+10//77*a^-3*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-3, x) == :(-14//117*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a^3+a^3*sin(c+d*x))^-1+-2//13*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3+-14//117*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-2+14//39*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-14//39*a^-3*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^15//2*(a+a*sin(c+d*x))^-4, x) == :(52//5*d^-1*e^3*(e*cos(c+d*x))^9//2*(a^4+a^4*sin(c+d*x))^-1+4*e*a^-1*d^-1*(e*cos(c+d*x))^13//2*(a+a*sin(c+d*x))^-3+78//7*a^-4*d^-1*e^7*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+234//35*a^-4*d^-1*e^5*(e*cos(c+d*x))^5//2*sin(c+d*x)+78//7*a^-4*d^-1*e^8*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^13//2*(a+a*sin(c+d*x))^-4, x) == :(-44//3*d^-1*e^3*(e*cos(c+d*x))^7//2*(a^4+a^4*sin(c+d*x))^-1+-4*e*a^-1*d^-1*(e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-3+-154//15*a^-4*d^-1*e^5*(e*cos(c+d*x))^3//2*sin(c+d*x)+-154//5*a^-4*d^-1*e^6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^11//2*(a+a*sin(c+d*x))^-4, x) == :(-12*d^-1*e^3*(e*cos(c+d*x))^5//2*(a^4+a^4*sin(c+d*x))^-1+-10*a^-4*d^-1*e^5*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+-4//3*e*a^-1*d^-1*(e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-3+-10*a^-4*d^-1*e^6*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-4, x) == :(28//5*d^-1*e^3*(e*cos(c+d*x))^3//2*(a^4+a^4*sin(c+d*x))^-1+-4//5*e*a^-1*d^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3+42//5*a^-4*d^-1*e^4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-4, x) == :(20//21*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(a^4+a^4*sin(c+d*x))^-1+-4//7*e*a^-1*d^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-3+10//21*a^-4*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-4, x) == :(2//15*e*d^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))^-2+2//15*e*d^-1*(e*cos(c+d*x))^3//2*(a^4+a^4*sin(c+d*x))^-1+-4//9*e*a^-1*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3+2//15*a^-4*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-4, x) == :(2//77*e*d^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))^-2+2//77*e*d^-1*(e*cos(c+d*x))^(1/2)*(a^4+a^4*sin(c+d*x))^-1+-4//11*e*a^-1*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3+-2//77*a^-4*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-4, x) == :(-2//13*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-4+-2//39*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^2+a^2*sin(c+d*x))^-2+-2//39*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a^4+a^4*sin(c+d*x))^-1+-10//117*a^-1*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3+-2//39*a^-4*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-4, x) == :(-2//15*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-4+-2//33*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^2+a^2*sin(c+d*x))^-2+-2//33*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a^4+a^4*sin(c+d*x))^-1+-14//165*a^-1*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3+2//33*a^-4*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-4, x) == :(-14//221*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a^2+a^2*sin(c+d*x))^-2+-14//221*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a^4+a^4*sin(c+d*x))^-1+-2//17*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-4+-18//221*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3+42//221*a^-4*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-42//221*a^-4*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^(1/2), x) == :(3//4*e*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-1//2*a*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1//2+-3//4*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+3//4*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2), x) == :(-1*a*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2), x) == :(-2*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+2*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^(1/2), x) == :(2*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^(1/2), x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^(1/2)+4//3*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^(1/2), x) == :(-2//3*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^(1/2)+-16//15*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2+8//3*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^(1/2), x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^(1/2)+-32//35*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^7//2+-12//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^3//2+16//5*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^5//2)
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^3//2, x) == :(-15//32*a^3*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3//2+-3//8*a^2*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-1//2+-1//4*a*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^(1/2)+15//64*e*a^2*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+45//64*a*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+45//64*a*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^3//2, x) == :(-7//12*a^2*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1//2+-1//3*a*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^(1/2)+7//8*a*e*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-7//8*a*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+7//8*a*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^3//2, x) == :(-5//4*a^2*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+-1//2*a*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^(1/2)+5//4*a*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+5//4*a*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^3//2, x) == :(-1*a*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-3*a*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+3*a*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2, x) == :(4*a*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2)+-2*a^2*d^-1*e^-3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+-2*a^2*d^-1*e^-3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3//2, x) == :(2//3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^3//2, x) == :(2*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3//2+-4//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2)
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^3//2, x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^3//2+-16//21*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^7//2+8//3*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^5//2)
@test integrate((e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^3//2, x) == :(-2//3*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^3//2+4*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^5//2+-16//5*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^7//2+32//45*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^9//2)
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^5//2, x) == :(-77//96*a^3*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1//2+-11//24*a^2*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^(1/2)+-1//4*a*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^3//2+77//64*e*a^2*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-77//64*a^2*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+77//64*a^2*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^5//2, x) == :(-15//8*a^3*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+-3//4*a^2*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^(1/2)+-1//3*a*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^3//2+15//8*a^2*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+15//8*a^2*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^5//2, x) == :(-7//4*a^2*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-1//2*a*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^3//2+-21//4*a^2*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+21//4*a^2*d^-1*e^-1//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^5//2, x) == :(4*a*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^3//2+5*a^3*d^-1*e^-3*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+-5*a^2*d^-1*e^-3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+-5*a^2*d^-1*e^-3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2, x) == :(4//3*a*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2+-2*a^2*d^-1*e^-5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x))+2*a^2*d^-1*e^-5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2)))
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^5//2, x) == :(2//5*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2)
@test integrate((e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^5//2, x) == :(2//3*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^5//2+-4//21*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^7//2)
@test integrate((e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^5//2, x) == :(2*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^5//2+-8//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^7//2+16//45*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+a*sin(c+d*x))^9//2)
@test integrate((e*cos(c+d*x))^-13//2*(a+a*sin(c+d*x))^5//2, x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^5//2+4*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^7//2+-16//7*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^9//2+32//77*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-11//2*(a+a*sin(c+d*x))^11//2)
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1//2, x) == :(1//4*e*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+-1//2*a*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3//2+3//4*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+3//4*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2, x) == :(e*a^-1*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+a^-1*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x))+-1*a^-1*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2)))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1//2, x) == :(2*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+2*d^-1*e^(1/2)*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a+a*cos(c+d*x)+a*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1//2, x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1//2)
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1//2, x) == :(-2//3*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1//2+4//3*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-1//2, x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1//2+-8//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^(1/2)+16//15*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^-1//2, x) == :(-2//7*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-1//2+-32//35*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2+-4//7*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^(1/2)+16//7*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3//2, x) == :((1/2)*e*a^-1*d^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-1//2+5//4*a^-2*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-5//4*a^-2*d^-1*e^7//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+5//4*a^-2*d^-1*e^7//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-3//2, x) == :(e*a^-1*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+3*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^2+a^2*cos(c+d*x)+a^2*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+3*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^2+a^2*cos(c+d*x)+a^2*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3//2, x) == :(-2*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-3//2+-2*e*a^-2*d^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-2*a^-2*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x))+2*a^-2*d^-1*e^3//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2)))
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3//2, x) == :(-2//3*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3//2)
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3//2, x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3//2+-4//5*a^-1*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1//2)
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-3//2, x) == :(-2//7*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3//2+-8//21*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1//2+16//21*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-3//2, x) == :(-2//9*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-3//2+-16//15*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^(1/2)+-4//15*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1//2+32//45*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^-7//2*(a+a*sin(c+d*x))^-3//2, x) == :(-2//11*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-3//2+-256//385*a^-4*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^5//2+-32//77*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^(1/2)+-16//77*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-1//2+128//77*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^3//2)
@test integrate((e*cos(c+d*x))^9//2*(a+a*sin(c+d*x))^-5//2, x) == :((1/2)*e*a^-1*d^-1*(e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-3//2+7//4*a^-2*d^-1*e^3*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-1//2+21//4*d^-1*e^9//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+21//4*d^-1*e^9//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^7//2*(a+a*sin(c+d*x))^-5//2, x) == :(-5*a^-3*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)+-4*e*a^-1*d^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-3//2+-5*a^-3*d^-1*e^7//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x))+5*a^-3*d^-1*e^7//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(1+cos(c+d*x)+sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2)))
@test integrate((e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-5//2, x) == :(-4//3*e*a^-1*d^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3//2+-2*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x))^-1*arcsinh(e^-1//2*(e*cos(c+d*x))^(1/2))+-2*d^-1*e^5//2*(1+cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^(1/2)*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x))^-1*arctan(e^(1/2)*(e*cos(c+d*x))^-1//2*(1+cos(c+d*x))^-1//2*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-5//2, x) == :(-2//5*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+a*sin(c+d*x))^-5//2)
@test integrate((e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-5//2, x) == :(-2//7*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-5//2+-4//21*a^-1*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+a*sin(c+d*x))^-3//2)
@test integrate((e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-5//2, x) == :(-2//9*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-5//2+-16//45*a^-2*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-1//2+-8//45*a^-1*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^-3//2)
@test integrate((e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-5//2, x) == :(-2//11*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-5//2+-16//77*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-1//2+-12//77*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^-3//2+32//77*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+a*sin(c+d*x))^(1/2))
@test integrate((e*cos(c+d*x))^-5//2*(a+a*sin(c+d*x))^-5//2, x) == :(-2//13*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-5//2+-128//195*a^-3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^(1/2)+-32//195*a^-2*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-1//2+-16//117*a^-1*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^-3//2+256//585*a^-4*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+a*sin(c+d*x))^3//2)
@test integrate((a+a*sin(c+d*x))^m*cos(c+d*x)^7, x) == :(-1*a^-7*d^-1*(7+m)^-1*(a+a*sin(c+d*x))^(7+m)+-12*a^-5*d^-1*(5+m)^-1*(a+a*sin(c+d*x))^(5+m)+6*a^-6*d^-1*(6+m)^-1*(a+a*sin(c+d*x))^(6+m)+8*a^-4*d^-1*(4+m)^-1*(a+a*sin(c+d*x))^(4+m))
@test integrate((a+a*sin(c+d*x))^m*cos(c+d*x)^5, x) == :(a^-5*d^-1*(5+m)^-1*(a+a*sin(c+d*x))^(5+m)+-4*a^-4*d^-1*(4+m)^-1*(a+a*sin(c+d*x))^(4+m)+4*a^-3*d^-1*(3+m)^-1*(a+a*sin(c+d*x))^(3+m))
@test integrate((a+a*sin(c+d*x))^m*cos(c+d*x)^3, x) == :(-1*a^-3*d^-1*(3+m)^-1*(a+a*sin(c+d*x))^(3+m)+2*a^-2*d^-1*(2+m)^-1*(a+a*sin(c+d*x))^(2+m))
@test integrate((a+a*sin(c+d*x))^m*cos(c+d*x), x) == :(a^-1*d^-1*(1+m)^-1*(a+a*sin(c+d*x))^(1+m))
@test integrate((e*cos(c+d*x))^(-4+-1m)*(a+a*sin(c+d*x))^m, x) == :(-1*d^-1*e^-1*(e*cos(c+d*x))^(-3+-1m)*(3+-1m)^-1*(a+a*sin(c+d*x))^m+-6*a^-3*d^-1*e^-1*(e*cos(c+d*x))^(-3+-1m)*(a+a*sin(c+d*x))^(3+m)*(9+m^4+-10*m^2)^-1+-3*a^-1*d^-1*e^-1*(e*cos(c+d*x))^(-3+-1m)*(1+-1m)^-1*(3+-1m)^-1*(a+a*sin(c+d*x))^(1+m)+6*a^-2*d^-1*e^-1*(e*cos(c+d*x))^(-3+-1m)*(1+-1*m^2)^-1*(3+-1m)^-1*(a+a*sin(c+d*x))^(2+m))
@test integrate((e*cos(c+d*x))^(-3+-1m)*(a+a*sin(c+d*x))^m, x) == :(-1*d^-1*e^-1*(e*cos(c+d*x))^(-2+-1m)*(2+-1m)^-1*(a+a*sin(c+d*x))^m+-2*a^-2*d^-1*e^-1*m^-1*(e*cos(c+d*x))^(-2+-1m)*(4+-1*m^2)^-1*(a+a*sin(c+d*x))^(2+m)+2*a^-1*d^-1*e^-1*m^-1*(e*cos(c+d*x))^(-2+-1m)*(2+-1m)^-1*(a+a*sin(c+d*x))^(1+m))
@test integrate((e*cos(c+d*x))^(-2+-1m)*(a+a*sin(c+d*x))^m, x) == :(-1*d^-1*e^-1*(e*cos(c+d*x))^(-1+-1m)*(1+-1m)^-1*(a+a*sin(c+d*x))^m+a^-1*d^-1*e^-1*(e*cos(c+d*x))^(-1+-1m)*(1+-1*m^2)^-1*(a+a*sin(c+d*x))^(1+m))
@test integrate((e*cos(c+d*x))^(-1+-1m)*(a+a*sin(c+d*x))^m, x) == :(d^-1*e^-1*m^-1*(e*cos(c+d*x))^(-1m)*(a+a*sin(c+d*x))^m)
@test integrate((e*cos(c+d*x))^(1+-2m)*(a+a*sin(c+d*x))^m, x) == :(-1*a*d^-1*e^-1*(e*cos(c+d*x))^(2+-2m)*(1+-1m)^-1*(a+a*sin(c+d*x))^(-1+m))
@test integrate(cos(c+d*x)^5*(a+b*sin(c+d*x)), x) == :(a*d^-1*sin(c+d*x)+-2//3*a*d^-1*sin(c+d*x)^3+-1//6*b*d^-1*cos(c+d*x)^6+1//5*a*d^-1*sin(c+d*x)^5)
@test integrate(cos(c+d*x)^3*(a+b*sin(c+d*x)), x) == :(a*d^-1*sin(c+d*x)+-1//3*a*d^-1*sin(c+d*x)^3+-1//4*b*d^-1*cos(c+d*x)^4)
@test integrate((a+b*sin(c+d*x))*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)*log(1+sin(c+d*x))+-1//2*d^-1*(a+b)*log(1+-1*sin(c+d*x)))
@test integrate(sec(c+d*x)^3*(a+b*sin(c+d*x)), x) == :((1/2)*a*d^-1*arctanh(sin(c+d*x))+(1/2)*d^-1*sec(c+d*x)^2*(b+a*sin(c+d*x)))
@test integrate(sec(c+d*x)^5*(a+b*sin(c+d*x)), x) == :(1//4*d^-1*sec(c+d*x)^4*(b+a*sin(c+d*x))+3//8*a*d^-1*arctanh(sin(c+d*x))+3//8*a*d^-1*sec(c+d*x)*tan(c+d*x))
@test integrate(cos(c+d*x)^4*(a+b*sin(c+d*x)), x) == :(3//8*a*x+-1//5*b*d^-1*cos(c+d*x)^5+1//4*a*d^-1*cos(c+d*x)^3*sin(c+d*x)+3//8*a*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate(cos(c+d*x)^2*(a+b*sin(c+d*x)), x) == :((1/2)*a*x+-1//3*b*d^-1*cos(c+d*x)^3+(1/2)*a*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate(sec(c+d*x)^2*(a+b*sin(c+d*x)), x) == :(a*d^-1*tan(c+d*x)+b*d^-1*sec(c+d*x))
@test integrate(sec(c+d*x)^4*(a+b*sin(c+d*x)), x) == :(a*d^-1*tan(c+d*x)+1//3*a*d^-1*tan(c+d*x)^3+1//3*b*d^-1*sec(c+d*x)^3)
@test integrate(sec(c+d*x)^6*(a+b*sin(c+d*x)), x) == :(a*d^-1*tan(c+d*x)+1//5*a*d^-1*tan(c+d*x)^5+1//5*b*d^-1*sec(c+d*x)^5+2//3*a*d^-1*tan(c+d*x)^3)
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x)^5, x) == :(a^2*d^-1*sin(c+d*x)+-1//3*d^-1*sin(c+d*x)^3*(-1*b^2+2*a^2)+1//5*d^-1*sin(c+d*x)^5*(a^2+-2*b^2)+1//7*b^2*d^-1*sin(c+d*x)^7+-1//3*a*b*d^-1*cos(c+d*x)^6)
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x)^3, x) == :(-1//5*b^-3*d^-1*(a+b*sin(c+d*x))^5+(1/2)*a*b^-3*d^-1*(a+b*sin(c+d*x))^4+-1//3*b^-3*d^-1*(a+b*sin(c+d*x))^3*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x), x) == :(1//3*b^-1*d^-1*(a+b*sin(c+d*x))^3)
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)^2*log(1+sin(c+d*x))+-1*b^2*d^-1*sin(c+d*x)+-1//2*d^-1*(a+b)^2*log(1+-1*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^3, x) == :((1/2)*d^-1*(a^2+-1*b^2)*arctanh(sin(c+d*x))+(1/2)*d^-1*sec(c+d*x)^2*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^5, x) == :(1//8*d^-1*sec(c+d*x)^2*((-1*b^2+3*a^2)*sin(c+d*x)+2*a*b)+1//8*d^-1*(-1*b^2+3*a^2)*arctanh(sin(c+d*x))+1//4*d^-1*sec(c+d*x)^4*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x)^6, x) == :(x*(5//16*a^2+5//128*b^2)+-9//56*a*b*d^-1*cos(c+d*x)^7+-1//8*b*d^-1*cos(c+d*x)^7*(a+b*sin(c+d*x))+1//48*d^-1*cos(c+d*x)^5*(b^2+8*a^2)*sin(c+d*x)+1//128*d^-1*(5*b^2+40*a^2)*cos(c+d*x)*sin(c+d*x)+1//192*d^-1*cos(c+d*x)^3*(5*b^2+40*a^2)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x)^4, x) == :(x*(1//16*b^2+3//8*a^2)+-7//30*a*b*d^-1*cos(c+d*x)^5+-1//6*b*d^-1*cos(c+d*x)^5*(a+b*sin(c+d*x))+1//16*d^-1*(b^2+6*a^2)*cos(c+d*x)*sin(c+d*x)+1//24*d^-1*cos(c+d*x)^3*(b^2+6*a^2)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^2*cos(c+d*x)^2, x) == :(x*((1/2)*a^2+1//8*b^2)+-5//12*a*b*d^-1*cos(c+d*x)^3+-1//4*b*d^-1*cos(c+d*x)^3*(a+b*sin(c+d*x))+1//8*d^-1*(b^2+4*a^2)*cos(c+d*x)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^2, x) == :(-1*x*b^2+a*b*d^-1*cos(c+d*x)+d^-1*(a+b*sin(c+d*x))*(b+a*sin(c+d*x))*sec(c+d*x))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^4, x) == :(1//3*d^-1*(-1*b^2+2*a^2)*tan(c+d*x)+1//3*a*b*d^-1*sec(c+d*x)+1//3*d^-1*sec(c+d*x)^3*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^6, x) == :(1//5*d^-1*(-1*b^2+4*a^2)*tan(c+d*x)+1//15*d^-1*tan(c+d*x)^3*(-1*b^2+4*a^2)+1//5*a*b*d^-1*sec(c+d*x)^3+1//5*d^-1*sec(c+d*x)^5*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^2*sec(c+d*x)^8, x) == :(1//7*d^-1*(-1*b^2+6*a^2)*tan(c+d*x)+1//21*d^-1*tan(c+d*x)^3*(-2*b^2+12*a^2)+1//35*d^-1*tan(c+d*x)^5*(-1*b^2+6*a^2)+1//7*a*b*d^-1*sec(c+d*x)^5+1//7*d^-1*sec(c+d*x)^7*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^3*cos(c+d*x)^5, x) == :(1//8*b^-5*d^-1*(a+b*sin(c+d*x))^8+-4//7*a*b^-5*d^-1*(a+b*sin(c+d*x))^7+1//3*b^-5*d^-1*(a+b*sin(c+d*x))^6*(-1*b^2+3*a^2)+1//4*b^-5*d^-1*(a+b*sin(c+d*x))^4*(a^2+-1*b^2)^2+-4//5*a*b^-5*d^-1*(a+b*sin(c+d*x))^5*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^3*cos(c+d*x)^3, x) == :(-1//6*b^-3*d^-1*(a+b*sin(c+d*x))^6+-1//4*b^-3*d^-1*(a+b*sin(c+d*x))^4*(a^2+-1*b^2)+2//5*a*b^-3*d^-1*(a+b*sin(c+d*x))^5)
@test integrate((a+b*sin(c+d*x))^3*cos(c+d*x), x) == :(1//4*b^-1*d^-1*(a+b*sin(c+d*x))^4)
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)^3*log(1+sin(c+d*x))+-1//2*b^3*d^-1*sin(c+d*x)^2+-1//2*d^-1*(a+b)^3*log(1+-1*sin(c+d*x))+-3*a*b^2*d^-1*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^3, x) == :((1/2)*a*b^2*d^-1*sin(c+d*x)+(1/2)*d^-1*(a+b*sin(c+d*x))^2*sec(c+d*x)^2*(b+a*sin(c+d*x))+-1//4*d^-1*(a+b)^2*(a+-2b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^2*(a+2b)*log(1+sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^5, x) == :(1//4*d^-1*(a+b*sin(c+d*x))^3*sec(c+d*x)^3*tan(c+d*x)+3//8*a*d^-1*(a^2+-1*b^2)*arctanh(sin(c+d*x))+3//8*a*d^-1*sec(c+d*x)^2*(a+b*sin(c+d*x))*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^3*cos(c+d*x)^4, x) == :(3//16*a*x*(b^2+2*a^2)+-1//7*b*d^-1*(a+b*sin(c+d*x))^2*cos(c+d*x)^5+-1//70*b*d^-1*cos(c+d*x)^5*(4*b^2+17*a^2)+-3//14*a*b*d^-1*cos(c+d*x)^5*(a+b*sin(c+d*x))+1//8*a*d^-1*cos(c+d*x)^3*(b^2+2*a^2)*sin(c+d*x)+3//16*a*d^-1*(b^2+2*a^2)*cos(c+d*x)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*cos(c+d*x)^2, x) == :(1//8*a*x*(3*b^2+4*a^2)+-1//5*b*d^-1*(a+b*sin(c+d*x))^2*cos(c+d*x)^3+-1//60*b*d^-1*cos(c+d*x)^3*(8*b^2+27*a^2)+-7//20*a*b*d^-1*cos(c+d*x)^3*(a+b*sin(c+d*x))+1//8*a*d^-1*(3*b^2+4*a^2)*cos(c+d*x)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^2, x) == :(-3*a*x*b^2+d^-1*(a+b*sin(c+d*x))^2*(b+a*sin(c+d*x))*sec(c+d*x)+2*b*d^-1*(a^2+b^2)*cos(c+d*x)+a*b^2*d^-1*cos(c+d*x)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+b*sin(c+d*x))^2*sec(c+d*x)^3*(b+a*sin(c+d*x))+2//3*a*d^-1*(a^2+-1*b^2)*tan(c+d*x)+2//3*b*d^-1*(a^2+-1*b^2)*sec(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+b*sin(c+d*x))^2*sec(c+d*x)^5*(b+a*sin(c+d*x))+2//15*a*d^-1*(-3*b^2+4*a^2)*tan(c+d*x)+2//15*b*d^-1*(-1*b^2+2*a^2)*sec(c+d*x)+2//15*d^-1*sec(c+d*x)^3*(a+b*sin(c+d*x))*(a*b+(-1*b^2+2*a^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^8, x) == :(1//7*d^-1*(a+b*sin(c+d*x))^2*sec(c+d*x)^7*(b+a*sin(c+d*x))+2//35*b*d^-1*sec(c+d*x)^3*(-1*b^2+3*a^2)+2//35*d^-1*sec(c+d*x)^5*(a+b*sin(c+d*x))*((-1*b^2+3*a^2)*sin(c+d*x)+2*a*b)+4//35*a*d^-1*tan(c+d*x)^3*(-1*b^2+2*a^2)+12//35*a*d^-1*(-1*b^2+2*a^2)*tan(c+d*x))
@test integrate((a+b*sin(c+d*x))^3*sec(c+d*x)^10, x) == :(1//9*d^-1*(a+b*sin(c+d*x))^2*sec(c+d*x)^9*(b+a*sin(c+d*x))+2//21*a*d^-1*(-3*b^2+8*a^2)*tan(c+d*x)+2//63*b*d^-1*sec(c+d*x)^5*(-1*b^2+4*a^2)+2//63*d^-1*sec(c+d*x)^7*(a+b*sin(c+d*x))*((-1*b^2+4*a^2)*sin(c+d*x)+3*a*b)+2//105*a*d^-1*tan(c+d*x)^5*(-3*b^2+8*a^2)+4//63*a*d^-1*tan(c+d*x)^3*(-3*b^2+8*a^2))
@test integrate((a+b*sin(c+d*x))^8*cos(c+d*x)^5, x) == :(1//13*b^-5*d^-1*(a+b*sin(c+d*x))^13+-1//3*a*b^-5*d^-1*(a+b*sin(c+d*x))^12+1//9*b^-5*d^-1*(a+b*sin(c+d*x))^9*(a^2+-1*b^2)^2+1//11*b^-5*d^-1*(a+b*sin(c+d*x))^11*(-2*b^2+6*a^2)+-2//5*a*b^-5*d^-1*(a+b*sin(c+d*x))^10*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^8*cos(c+d*x)^3, x) == :(-1//11*b^-3*d^-1*(a+b*sin(c+d*x))^11+-1//9*b^-3*d^-1*(a+b*sin(c+d*x))^9*(a^2+-1*b^2)+1//5*a*b^-3*d^-1*(a+b*sin(c+d*x))^10)
@test integrate((a+b*sin(c+d*x))^8*cos(c+d*x), x) == :(1//9*b^-1*d^-1*(a+b*sin(c+d*x))^9)
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)^8*log(1+sin(c+d*x))+-1//2*d^-1*(a+b)^8*log(1+-1*sin(c+d*x))+-1//7*b^8*d^-1*sin(c+d*x)^7+-1*b^2*d^-1*(b^6+28*a^6+28*a^2*b^4+70*a^4*b^2)*sin(c+d*x)+-4//3*a*b^7*d^-1*sin(c+d*x)^6+-1//3*b^4*d^-1*sin(c+d*x)^3*(b^4+70*a^4+28*a^2*b^2)+-1//5*b^6*d^-1*sin(c+d*x)^5*(b^2+28*a^2)+-4*a*b^3*d^-1*sin(c+d*x)^2*(b^4+7*a^4+7*a^2*b^2)+-2*a*b^5*d^-1*sin(c+d*x)^4*(b^2+7*a^2))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^3, x) == :((1/2)*a*b^7*d^-1*sin(c+d*x)^6+(1/2)*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^2*(b+a*sin(c+d*x))+-1//4*d^-1*(a+b)^7*(a+-7b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^7*(a+7b)*log(1+sin(c+d*x))+7//2*b^2*d^-1*(b^6+3*a^6+20*a^2*b^4+30*a^4*b^2)*sin(c+d*x)+7//6*b^4*d^-1*sin(c+d*x)^3*(b^4+15*a^4+20*a^2*b^2)+7//10*b^6*d^-1*sin(c+d*x)^5*(b^2+5*a^2)+(1/2)*a*b^3*d^-1*sin(c+d*x)^2*(24*b^4+35*a^4+112*a^2*b^2)+3//2*a*b^5*d^-1*sin(c+d*x)^4*(4*b^2+7*a^2))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^5, x) == :(-1//8*d^-1*(a+b*sin(c+d*x))^5*sec(c+d*x)^2*(b*(a^2+7*b^2)+-1*a*(-11*b^2+3*a^2)*sin(c+d*x))+-1//16*d^-1*(a+b)^6*(3*a^2+35*b^2+-18*a*b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^4*(b+a*sin(c+d*x))+1//16*d^-1*(a+-1b)^6*(3*a^2+35*b^2+18*a*b)*log(1+sin(c+d*x))+5//8*b^2*d^-1*(-7*b^6+6*a^6+-84*a^2*b^4+-35*a^4*b^2)*sin(c+d*x)+5//24*b^4*d^-1*sin(c+d*x)^3*(-7*b^4+9*a^4+-42*a^2*b^2)+-1//8*a*b^7*d^-1*sin(c+d*x)^4*(13+-3*a^2*b^-2)+1//4*a*b^3*d^-1*sin(c+d*x)^2*(-48*b^4+15*a^4+-77*a^2*b^2))
@test integrate((a+b*sin(c+d*x))^8*cos(c+d*x)^2, x) == :(x*((1/2)*a^8+7//256*b^8+7//2*a^6*b^2+35//8*a^4*b^4+35//32*a^2*b^6)+-1//10*b*d^-1*(a+b*sin(c+d*x))^7*cos(c+d*x)^3+1//256*d^-1*(7*b^8+128*a^8+280*a^2*b^6+896*a^6*b^2+1120*a^4*b^4)*cos(c+d*x)*sin(c+d*x)+-17//90*a*b*d^-1*(a+b*sin(c+d*x))^6*cos(c+d*x)^3+-11//40320*a*b*d^-1*cos(c+d*x)^3*(1289*b^6+1792*a^6+9588*a^2*b^4+10536*a^4*b^2)+-1//240*b*d^-1*(a+b*sin(c+d*x))^5*cos(c+d*x)^3*(21*b^2+64*a^2)+-1//2016*b*d^-1*(a+b*sin(c+d*x))^3*cos(c+d*x)^3*(147*b^4+784*a^4+1500*a^2*b^2)+-1//13440*b*d^-1*cos(c+d*x)^3*(a+b*sin(c+d*x))*(735*b^6+6272*a^6+15956*a^2*b^4+28088*a^4*b^2)+-13//3360*a*b*d^-1*(a+b*sin(c+d*x))^2*cos(c+d*x)^3*(101*b^4+112*a^4+348*a^2*b^2)+-1//336*a*b*d^-1*(a+b*sin(c+d*x))^4*cos(c+d*x)^3*(109*b^2+112*a^2))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^2, x) == :(-7//16*x*b^2*(5*b^6+64*a^6+120*a^2*b^4+240*a^4*b^2)+d^-1*(a+b*sin(c+d*x))^7*(b+a*sin(c+d*x))*sec(c+d*x)+a*b*d^-1*(a+b*sin(c+d*x))^6*cos(c+d*x)+1//6*b*d^-1*(a+b*sin(c+d*x))^5*(6*a^2+7*b^2)*cos(c+d*x)+1//20*a*b*d^-1*(40*a^6+512*b^6+1664*a^4*b^2+2789*a^2*b^4)*cos(c+d*x)+1//80*b^2*d^-1*(80*a^6+175*b^6+2248*a^4*b^2+2502*a^2*b^4)*cos(c+d*x)*sin(c+d*x)+1//120*b*d^-1*(a+b*sin(c+d*x))^3*(120*a^4+175*b^4+992*a^2*b^2)*cos(c+d*x)+1//30*a*b*d^-1*(a+b*sin(c+d*x))^4*(30*a^2+113*b^2)*cos(c+d*x)+1//40*a*b*d^-1*(a+b*sin(c+d*x))^2*(40*a^4+337*b^4+624*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^4, x) == :(35//8*x*b^4*(b^4+16*a^4+16*a^2*b^2)+-1//3*d^-1*(a+b*sin(c+d*x))^6*(-1*(-7*b^2+2*a^2)*sin(c+d*x)+5*a*b)*sec(c+d*x)+1//3*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^3*(b+a*sin(c+d*x))+1//3*b*d^-1*(a+b*sin(c+d*x))^5*(-7*b^2+2*a^2)*cos(c+d*x)+1//6*a*b*d^-1*(-256*b^6+8*a^6+-803*a^2*b^4+-104*a^4*b^2)*cos(c+d*x)+1//12*b*d^-1*(a+b*sin(c+d*x))^3*(-35*b^4+8*a^4+-72*a^2*b^2)*cos(c+d*x)+1//24*b^2*d^-1*(-105*b^6+16*a^6+-866*a^2*b^4+-200*a^4*b^2)*cos(c+d*x)*sin(c+d*x)+1//3*a*b*d^-1*(a+b*sin(c+d*x))^4*(-13*b^2+2*a^2)*cos(c+d*x)+1//12*a*b*d^-1*(a+b*sin(c+d*x))^2*(-151*b^4+8*a^4+-88*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^6, x) == :(-7//2*x*b^6*(b^2+8*a^2)+-4//15*d^-1*(a+b*sin(c+d*x))^5*(b*(-7*b^2+4*a^2)+-1*a*(b^2+2*a^2)*sin(c+d*x))*sec(c+d*x)+-1//15*d^-1*(a+b*sin(c+d*x))^6*sec(c+d*x)^3*(-1*(-7*b^2+4*a^2)*sin(c+d*x)+3*a*b)+1//5*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^5*(b+a*sin(c+d*x))+1//15*b*d^-1*(a+b*sin(c+d*x))^3*(8*a^4+35*b^4+-16*a^2*b^2)*cos(c+d*x)+1//30*b^2*d^-1*(16*a^6+105*b^6+-88*a^4*b^2+282*a^2*b^4)*cos(c+d*x)*sin(c+d*x)+2//15*a*b*d^-1*(8*a^6+192*b^6+-48*a^4*b^2+163*a^2*b^4)*cos(c+d*x)+1//15*a*b*d^-1*(a+b*sin(c+d*x))^2*(8*a^4+87*b^4+-32*a^2*b^2)*cos(c+d*x)+4//15*a*b*d^-1*(a+b*sin(c+d*x))^4*(b^2+2*a^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^8, x) == :(x*b^8+-2//105*d^-1*(a+b*sin(c+d*x))^4*(-1*(-35*b^4+24*a^4+8*a^2*b^2)*sin(c+d*x)+3*a*b*(-11*b^2+12*a^2))*sec(c+d*x)+-2//105*d^-1*(a+b*sin(c+d*x))^5*sec(c+d*x)^3*(b*(-7*b^2+6*a^2)+-1*a*(-11*b^2+12*a^2)*sin(c+d*x))+-1//35*d^-1*(a+b*sin(c+d*x))^6*sec(c+d*x)^5*(a*b+-1*(-7*b^2+6*a^2)*sin(c+d*x))+1//7*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^7*(b+a*sin(c+d*x))+1//105*b^2*d^-1*(-105*b^6+48*a^6+-152*a^4*b^2+174*a^2*b^4)*cos(c+d*x)*sin(c+d*x)+2//105*b*d^-1*(a+b*sin(c+d*x))^3*(-35*b^4+24*a^4+8*a^2*b^2)*cos(c+d*x)+4//105*a*b*d^-1*(-96*b^6+24*a^6+-88*a^4*b^2+125*a^2*b^4)*cos(c+d*x)+2//105*a*b*d^-1*(a+b*sin(c+d*x))^2*(9*b^4+24*a^4+-40*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^8*sec(c+d*x)^10, x) == :(1//9*d^-1*(a+b*sin(c+d*x))^7*sec(c+d*x)^9*(b+a*sin(c+d*x))+1//63*d^-1*(a+b*sin(c+d*x))^6*sec(c+d*x)^7*(a*b+(-7*b^2+8*a^2)*sin(c+d*x))+128//315*a^2*d^-1*(a^2+-1*b^2)^3*tan(c+d*x)+128//315*a*b*d^-1*(a^2+-1*b^2)^3*sec(c+d*x)+16//105*a*d^-1*(a+b*sin(c+d*x))^4*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a^2+-1*b^2)+64//315*a*d^-1*(a+b*sin(c+d*x))^2*(a^2+-1*b^2)^2*sec(c+d*x)^3*(b+a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x)^5, x) == :(1//4*b^-1*d^-1*sin(c+d*x)^4+b^-5*d^-1*(a^2+-1*b^2)^2*log(a+b*sin(c+d*x))+(1/2)*b^-3*d^-1*sin(c+d*x)^2*(a^2+-2*b^2)+-1//3*a*b^-2*d^-1*sin(c+d*x)^3+-1*a*b^-4*d^-1*(a^2+-2*b^2)*sin(c+d*x))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x)^3, x) == :(-1//2*b^-1*d^-1*sin(c+d*x)^2+a*b^-2*d^-1*sin(c+d*x)+-1*b^-3*d^-1*(a^2+-1*b^2)*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x), x) == :(b^-1*d^-1*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x), x) == :(d^-1*(-2b+2a)^-1*log(1+sin(c+d*x))+-1*d^-1*(2a+2b)^-1*log(1+-1*sin(c+d*x))+-1*b*d^-1*(a^2+-1*b^2)^-1*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x)^3, x) == :(b^3*d^-1*(a^2+-1*b^2)^-2*log(a+b*sin(c+d*x))+-1*d^-1*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+-1//4*d^-1*(a+b)^-2*(a+2b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^-2*(a+-2b)*log(1+sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x)^5, x) == :(-1*b^5*d^-1*(a^2+-1*b^2)^-3*log(a+b*sin(c+d*x))+-1*d^-1*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-1//16*d^-1*(a+b)^-3*(3*a^2+8*b^2+9*a*b)*log(1+-1*sin(c+d*x))+1//8*d^-1*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(4*b^3+a*(-7*b^2+3*a^2)*sin(c+d*x))+1//16*d^-1*(a+-1b)^-3*(3*a^2+8*b^2+-9*a*b)*log(1+sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x)^6, x) == :(1//5*b^-1*d^-1*cos(c+d*x)^5+-2*b^-6*d^-1*(a^2+-1*b^2)^5//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//12*b^-3*d^-1*cos(c+d*x)^3*(-4*b^2+4*a^2+-3*a*b*sin(c+d*x))+1//8*a*x*b^-6*(8*a^4+15*b^4+-20*a^2*b^2)+1//8*b^-5*d^-1*(8*(a^2+-1*b^2)^2+-1*a*b*(-7*b^2+4*a^2)*sin(c+d*x))*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x)^4, x) == :(1//3*b^-1*d^-1*cos(c+d*x)^3+2*b^-4*d^-1*(a^2+-1*b^2)^3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//2*a*x*b^-4*(-3*b^2+2*a^2)+-1//2*b^-3*d^-1*(-2*b^2+2*a^2+-1*a*b*sin(c+d*x))*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-1*cos(c+d*x)^2, x) == :(a*x*b^-2+b^-1*d^-1*cos(c+d*x)+-2*b^-2*d^-1*(a^2+-1*b^2)^(1/2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x)^2, x) == :(-1*d^-1*(a^2+-1*b^2)^-1*(b+-1*a*sin(c+d*x))*sec(c+d*x)+-2*b^2*d^-1*(a^2+-1*b^2)^-3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x)^4, x) == :(-1*d^-1*(-3*b^2+3*a^2)^-1*sec(c+d*x)^3*(b+-1*a*sin(c+d*x))+2*b^4*d^-1*(a^2+-1*b^2)^-5//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+1//3*d^-1*(a^2+-1*b^2)^-2*(3*b^3+a*(-5*b^2+2*a^2)*sin(c+d*x))*sec(c+d*x))
@test integrate((a+b*sin(c+d*x))^-1*sec(c+d*x)^6, x) == :(-1*d^-1*(-5*b^2+5*a^2)^-1*sec(c+d*x)^5*(b+-1*a*sin(c+d*x))+-2*b^6*d^-1*(a^2+-1*b^2)^-7//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//15*d^-1*(a^2+-1*b^2)^-3*(15*b^5+-1*a*(8*a^4+33*b^4+-26*a^2*b^2)*sin(c+d*x))*sec(c+d*x)+1//15*d^-1*(a^2+-1*b^2)^-2*sec(c+d*x)^3*(5*b^3+a*(-9*b^2+4*a^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^7, x) == :(-1//5*b^-2*d^-1*sin(c+d*x)^5+b^-7*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^3+(1/2)*a*b^-3*d^-1*sin(c+d*x)^4+-1*b^-6*d^-1*(3*b^4+5*a^4+-9*a^2*b^2)*sin(c+d*x)+-1*b^-4*d^-1*sin(c+d*x)^3*(a^2+-1*b^2)+a*b^-5*d^-1*sin(c+d*x)^2*(-3*b^2+2*a^2)+6*a*b^-7*d^-1*(a^2+-1*b^2)^2*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^5, x) == :(1//3*b^-2*d^-1*sin(c+d*x)^3+b^-4*d^-1*(-2*b^2+3*a^2)*sin(c+d*x)+-1*a*b^-3*d^-1*sin(c+d*x)^2+-1*b^-5*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^2+-4*a*b^-5*d^-1*(a^2+-1*b^2)*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^3, x) == :(-1*b^-2*d^-1*sin(c+d*x)+b^-3*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)+2*a*b^-3*d^-1*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x), x) == :(-1*b^-1*d^-1*(a+b*sin(c+d*x))^-1)
@test integrate((a+b*sin(c+d*x))^-2*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)^-2*log(1+sin(c+d*x))+-1//2*d^-1*(a+b)^-2*log(1+-1*sin(c+d*x))+b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-1+-2*a*b*d^-1*(a^2+-1*b^2)^-2*log(a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+b)^-3*(a+3b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^-3*(a+-3b)*log(1+sin(c+d*x))+-1*d^-1*(a+b*sin(c+d*x))^-1*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+4*a*b^3*d^-1*(a^2+-1*b^2)^-3*log(a+b*sin(c+d*x))+-1//2*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-2*(a^2+3*b^2))
@test integrate((a+b*sin(c+d*x))^-2*sec(c+d*x)^5, x) == :(1//16*d^-1*(a+b)^-4*(-15*b^2+-3*a^2+-12*a*b)*log(1+-1*sin(c+d*x))+1//16*d^-1*(a+-1b)^-4*(3*a^2+15*b^2+-12*a*b)*log(1+sin(c+d*x))+-1*d^-1*(a+b*sin(c+d*x))^-1*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-6*a*b^5*d^-1*(a^2+-1*b^2)^-4*log(a+b*sin(c+d*x))+-3//8*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-3*(a^4+-5*b^4+-4*a^2*b^2)+1//8*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(b*(a^2+5*b^2)+3*a*(a^2+-3*b^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^6, x) == :(-1//8*x*b^-6*(15*b^4+40*a^4+-60*a^2*b^2)+-1*b^-1*d^-1*(a+b*sin(c+d*x))^-1*cos(c+d*x)^5+-5//8*b^-5*d^-1*(8*a*(a^2+-1*b^2)+-1*b*(-3*b^2+4*a^2)*sin(c+d*x))*cos(c+d*x)+5//12*b^-3*d^-1*cos(c+d*x)^3*(4a+-3*b*sin(c+d*x))+10*a*b^-6*d^-1*(a^2+-1*b^2)^3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^4, x) == :((1/2)*x*b^-4*(-3*b^2+6*a^2)+-1*b^-1*d^-1*(a+b*sin(c+d*x))^-1*cos(c+d*x)^3+3//2*b^-3*d^-1*(2a+-1*b*sin(c+d*x))*cos(c+d*x)+-6*a*b^-4*d^-1*(a^2+-1*b^2)^(1/2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-2*cos(c+d*x)^2, x) == :(-1*x*b^-2+-1*b^-1*d^-1*(a+b*sin(c+d*x))^-1*cos(c+d*x)+2*a*b^-2*d^-1*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-2*sec(c+d*x)^2, x) == :(-1*d^-1*(a^2+-1*b^2)^-2*(-1*(a^2+2*b^2)*sin(c+d*x)+3*a*b)*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-1*sec(c+d*x)+-6*a*b^2*d^-1*(a^2+-1*b^2)^-5//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-2*sec(c+d*x)^4, x) == :(-1//3*d^-1*(a^2+-1*b^2)^-2*sec(c+d*x)^3*(-1*(a^2+4*b^2)*sin(c+d*x)+5*a*b)+1//3*d^-1*(a^2+-1*b^2)^-3*((-8*b^4+2*a^4+-9*a^2*b^2)*sin(c+d*x)+15*a*b^3)*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-1*sec(c+d*x)^3+10*a*b^4*d^-1*(a^2+-1*b^2)^-7//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x))))
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^7, x) == :(-1//4*b^-3*d^-1*sin(c+d*x)^4+a*b^-4*d^-1*sin(c+d*x)^3+(1/2)*b^-7*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^3+-1*b^-7*d^-1*(3*b^4+15*a^4+-18*a^2*b^2)*log(a+b*sin(c+d*x))+-1//2*b^-5*d^-1*sin(c+d*x)^2*(-3*b^2+6*a^2)+a*b^-6*d^-1*(-9*b^2+10*a^2)*sin(c+d*x)+-6*a*b^-7*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^2)
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^5, x) == :((1/2)*b^-3*d^-1*sin(c+d*x)^2+b^-5*d^-1*(-2*b^2+6*a^2)*log(a+b*sin(c+d*x))+-3*a*b^-4*d^-1*sin(c+d*x)+-1//2*b^-5*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^2+4*a*b^-5*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^3, x) == :(-1*b^-3*d^-1*log(a+b*sin(c+d*x))+(1/2)*b^-3*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)+-2*a*b^-3*d^-1*(a+b*sin(c+d*x))^-1)
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x), x) == :(-1//2*b^-1*d^-1*(a+b*sin(c+d*x))^-2)
@test integrate((a+b*sin(c+d*x))^-3*sec(c+d*x), x) == :((1/2)*d^-1*(a+-1b)^-3*log(1+sin(c+d*x))+-1//2*d^-1*(a+b)^-3*log(1+-1*sin(c+d*x))+b*d^-1*(a+b*sin(c+d*x))^-2*(-2*b^2+2*a^2)^-1+-1*b*d^-1*(a^2+-1*b^2)^-3*(b^2+3*a^2)*log(a+b*sin(c+d*x))+2*a*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-2)
@test integrate((a+b*sin(c+d*x))^-3*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+b)^-4*(a+4b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^-4*(a+-4b)*log(1+sin(c+d*x))+-1*d^-1*(a+b*sin(c+d*x))^-2*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+2*b^3*d^-1*(a^2+-1*b^2)^-4*(b^2+5*a^2)*log(a+b*sin(c+d*x))+-1//2*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-2*(a^2+2*b^2)+-1//2*a*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-3*(a^2+11*b^2))
@test integrate((a+b*sin(c+d*x))^-3*sec(c+d*x)^5, x) == :(1//16*d^-1*(a+b)^-5*(-24*b^2+-3*a^2+-15*a*b)*log(1+-1*sin(c+d*x))+1//16*d^-1*(a+-1b)^-5*(3*a^2+24*b^2+-15*a*b)*log(1+sin(c+d*x))+-1*d^-1*(a+b*sin(c+d*x))^-2*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-3*b^5*d^-1*(a^2+-1*b^2)^-5*(b^2+7*a^2)*log(a+b*sin(c+d*x))+-3//8*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-3*(a^4+-4*b^4+-5*a^2*b^2)+1//8*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(2*b*(a^2+3*b^2)+a*(-11*b^2+3*a^2)*sin(c+d*x))+-3//8*a*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-4*(a^4+-27*b^4+-6*a^2*b^2))
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^6, x) == :(-1//2*b^-1*d^-1*(a+b*sin(c+d*x))^-2*cos(c+d*x)^5+5//2*a*x*b^-6*(-3*b^2+4*a^2)+5//2*b^-5*d^-1*(-1*b^2+4*a^2+-2*a*b*sin(c+d*x))*cos(c+d*x)+-1*b^-6*d^-1*(a^2+-1*b^2)^-1//2*(5*b^4+20*a^4+-25*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-5//6*b^-3*d^-1*(a+b*sin(c+d*x))^-1*cos(c+d*x)^3*(4a+b*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^4, x) == :(-3*a*x*b^-4+-1//2*b^-1*d^-1*(a+b*sin(c+d*x))^-2*cos(c+d*x)^3+b^-4*d^-1*(a^2+-1*b^2)^-1//2*(-3*b^2+6*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-3//2*b^-3*d^-1*(a+b*sin(c+d*x))^-1*(2a+b*sin(c+d*x))*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-3*cos(c+d*x)^2, x) == :(d^-1*(a^2+-1*b^2)^-3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//2*b^-1*d^-1*(a+b*sin(c+d*x))^-2*cos(c+d*x)+(1/2)*a*b^-1*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-1*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-3*sec(c+d*x)^2, x) == :(-1//2*d^-1*(a^2+-1*b^2)^-3*(3*b*(b^2+4*a^2)+-1*a*(2*a^2+13*b^2)*sin(c+d*x))*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-2*(-2*b^2+2*a^2)^-1*sec(c+d*x)+-3*b^2*d^-1*(a^2+-1*b^2)^-7//2*(b^2+4*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+5//2*a*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-2*sec(c+d*x))
@test integrate((a+b*sin(c+d*x))^-3*sec(c+d*x)^4, x) == :(-1//6*d^-1*(a^2+-1*b^2)^-3*sec(c+d*x)^3*(5*b*(b^2+6*a^2)+-1*a*(2*a^2+33*b^2)*sin(c+d*x))+1//6*d^-1*(a^2+-1*b^2)^-4*(15*b^3*(b^2+6*a^2)+a*(-81*b^4+4*a^4+-28*a^2*b^2)*sin(c+d*x))*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-2*(-2*b^2+2*a^2)^-1*sec(c+d*x)^3+5*b^4*d^-1*(a^2+-1*b^2)^-9//2*(b^2+6*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+7//2*a*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-2*sec(c+d*x)^3)
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^7, x) == :(b^-7*d^-1*(a+b*sin(c+d*x))^-1+b^-7*d^-1*(a+b*sin(c+d*x))^-3*(-1*b^2+5*a^2)+-3*a*b^-7*d^-1*(a+b*sin(c+d*x))^-2+1//5*b^-7*d^-1*(a+b*sin(c+d*x))^-5*(3*b^4+15*a^4+-18*a^2*b^2)+1//7*b^-7*d^-1*(a+b*sin(c+d*x))^-7*(a^2+-1*b^2)^3+-1*a*b^-7*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^2+-1*a*b^-7*d^-1*(a+b*sin(c+d*x))^-4*(-3*b^2+5*a^2))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^5, x) == :(-1//3*b^-5*d^-1*(a+b*sin(c+d*x))^-3+a*b^-5*d^-1*(a+b*sin(c+d*x))^-4+-1//5*b^-5*d^-1*(a+b*sin(c+d*x))^-5*(-2*b^2+6*a^2)+-1//7*b^-5*d^-1*(a+b*sin(c+d*x))^-7*(a^2+-1*b^2)^2+2//3*a*b^-5*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^3, x) == :(1//5*b^-3*d^-1*(a+b*sin(c+d*x))^-5+-1//3*a*b^-3*d^-1*(a+b*sin(c+d*x))^-6+1//7*b^-3*d^-1*(a+b*sin(c+d*x))^-7*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x), x) == :(-1//7*b^-1*d^-1*(a+b*sin(c+d*x))^-7)
@test integrate((a+b*sin(c+d*x))^-8*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+b)^-9*(a+9b)*log(1+-1*sin(c+d*x))+1//4*d^-1*(a+-1b)^-9*(a+-9b)*log(1+sin(c+d*x))+-1*d^-1*(a+b*sin(c+d*x))^-7*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+-1//2*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-8*(a^8+9*b^8+196*a^6*b^2+244*a^2*b^6+574*a^4*b^4)+-1//6*b*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-6*(3*a^6+9*b^6+115*a^4*b^2+129*a^2*b^4)+-1//10*b*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-4*(5*a^4+9*b^4+50*a^2*b^2)+-1//14*b*d^-1*(a+b*sin(c+d*x))^-7*(a^2+-1*b^2)^-2*(7*a^2+9*b^2)+8*a*b^3*d^-1*(a^2+-1*b^2)^-9*(5*b^6+15*a^6+45*a^2*b^4+63*a^4*b^2)*log(a+b*sin(c+d*x))+-1//2*a*b*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-5*(a^4+11*b^4+20*a^2*b^2)+-1//2*a*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-7*(a^6+31*b^6+77*a^4*b^2+147*a^2*b^4)+-1//6*a*b*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^-3*(3*a^2+13*b^2))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^8, x) == :(x*b^-8+-1//7*b^-1*d^-1*(a+b*sin(c+d*x))^-7*cos(c+d*x)^7+-1//8*a*b^-8*d^-1*(a^2+-1*b^2)^-7//2*(-35*b^6+16*a^6+-56*a^4*b^2+70*a^2*b^4)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//24*b^-5*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-2*cos(c+d*x)^3*(8*(a^2+-1*b^2)^2+a*b*(-11*b^2+6*a^2)*sin(c+d*x))+1//6*a*b^-1*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^-1*cos(c+d*x)^7+1//16*b^-7*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-3*(16*(a^2+-1*b^2)^3+a*b*(8*a^4+19*b^4+-22*a^2*b^2)*sin(c+d*x))*cos(c+d*x)+1//30*b^-3*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-1*cos(c+d*x)^5*(-6*b^2+6*a^2+5*a*b*sin(c+d*x))+-1//24*a*b^-3*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-2*cos(c+d*x)^5*(-11*b^2+6*a^2)+1//16*a*b^-5*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-3*cos(c+d*x)^3*(8*a^4+19*b^4+-22*a^2*b^2))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^6, x) == :(-1//7*b^-1*d^-1*(a+b*sin(c+d*x))^-7*cos(c+d*x)^5+5//8*a*d^-1*(a^2+-1*b^2)^-9//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//42*b^-5*d^-1*(a+b*sin(c+d*x))^-5*(4*a^2+9*b^2+10*a*b*sin(c+d*x))*cos(c+d*x)+5//42*b^-3*d^-1*(a+b*sin(c+d*x))^-6*cos(c+d*x)^3*(2a+3*b*sin(c+d*x))+1//168*b^-5*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-2*(4*a^4+12*b^4+-9*a^2*b^2)*cos(c+d*x)+1//336*b^-5*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-4*(8*a^6+48*b^6+-38*a^4*b^2+87*a^2*b^4)*cos(c+d*x)+1//168*a*b^-5*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-1*(-1*b^2+4*a^2)*cos(c+d*x)+1//336*a*b^-5*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-3*(8*a^4+57*b^4+-30*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^4, x) == :(-1//7*b^-1*d^-1*(a+b*sin(c+d*x))^-7*cos(c+d*x)^3+1//28*b^-3*d^-1*(a+b*sin(c+d*x))^-6*(a+3*b*sin(c+d*x))*cos(c+d*x)+3//8*a*d^-1*(a^2+-1*b^2)^-11//2*(b^2+2*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+-1//140*b^-3*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-1*(a^2+-3*b^2)*cos(c+d*x)+-1//280*b^-3*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-3*(-8*b^4+2*a^4+-15*a^2*b^2)*cos(c+d*x)+-1//560*b^-3*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-5*(-32*b^6+4*a^6+-247*a^2*b^4+-40*a^4*b^2)*cos(c+d*x)+-1//280*a*b^-3*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-2*(-11*b^2+2*a^2)*cos(c+d*x)+-1//560*a*b^-3*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-4*(-73*b^4+4*a^4+-36*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-8*cos(c+d*x)^2, x) == :(-1//7*b^-1*d^-1*(a+b*sin(c+d*x))^-7*cos(c+d*x)+1//8*a*d^-1*(a^2+-1*b^2)^-13//2*(5*b^4+8*a^4+20*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+1//42*a*b^-1*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^-1*cos(c+d*x)+1//210*b^-1*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-2*(5*a^2+6*b^2)*cos(c+d*x)+1//840*b^-1*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-4*(20*a^4+32*b^4+179*a^2*b^2)*cos(c+d*x)+1//1680*b^-1*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-6*(40*a^6+128*b^6+1518*a^4*b^2+1779*a^2*b^4)*cos(c+d*x)+1//840*a*b^-1*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-3*(20*a^2+79*b^2)*cos(c+d*x)+1//1680*a*b^-1*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-5*(40*a^4+397*b^4+718*a^2*b^2)*cos(c+d*x))
@test integrate((a+b*sin(c+d*x))^-8*sec(c+d*x)^2, x) == :(-1//560*d^-1*(a^2+-1*b^2)^-8*(-1*(560*a^8+2048*b^8+42472*a^6*b^2+54511*a^2*b^6+125634*a^4*b^4)*sin(c+d*x)+315*a*b*(35*b^6+64*a^6+280*a^2*b^4+336*a^4*b^2))*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-7*(-7*b^2+7*a^2)^-1*sec(c+d*x)+-9//8*a*b^2*d^-1*(a^2+-1*b^2)^-17//2*(35*b^6+64*a^6+280*a^2*b^4+336*a^4*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+1//70*b*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-3*(16*b^2+49*a^2)*sec(c+d*x)+1//280*b*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-5*(128*b^4+700*a^4+1317*a^2*b^2)*sec(c+d*x)+1//560*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-7*(1024*b^6+9800*a^6+22767*a^2*b^4+41484*a^4*b^2)*sec(c+d*x)+5//14*a*b*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^-2*sec(c+d*x)+11//560*a*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-6*(241*b^4+280*a^4+844*a^2*b^2)*sec(c+d*x)+13//280*a*b*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-4*(27*b^2+28*a^2)*sec(c+d*x))
@test integrate((a+b*sin(c+d*x))^-8*sec(c+d*x)^4, x) == :(-1//336*d^-1*(a^2+-1*b^2)^-8*sec(c+d*x)^3*(-1*(112*a^8+2048*b^8+52528*a^6*b^2+57665*a^2*b^6+142902*a^4*b^4)*sin(c+d*x)+1155*a*b*(7*b^6+32*a^6+70*a^2*b^4+112*a^4*b^2))+1//336*d^-1*(a^2+-1*b^2)^-9*((-4096*b^10+224*a^10+-413024*a^4*b^6+-207332*a^6*b^4+-135489*a^2*b^8+-6048*a^8*b^2)*sin(c+d*x)+3465*a*b^3*(7*b^6+32*a^6+70*a^2*b^4+112*a^4*b^2))*sec(c+d*x)+b*d^-1*(a+b*sin(c+d*x))^-7*(-7*b^2+7*a^2)^-1*sec(c+d*x)^3+1//14*b*d^-1*(a+b*sin(c+d*x))^-5*(a^2+-1*b^2)^-3*sec(c+d*x)^3*(4*b^2+13*a^2)+1//112*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-7*sec(c+d*x)^3*(512*b^6+9212*a^6+12907*a^2*b^4+28420*a^4*b^2)+1//168*b*d^-1*(a+b*sin(c+d*x))^-3*(a^2+-1*b^2)^-5*sec(c+d*x)^3*(128*b^4+882*a^4+1421*a^2*b^2)+17//42*a*b*d^-1*(a+b*sin(c+d*x))^-6*(a^2+-1*b^2)^-2*sec(c+d*x)^3+165//8*a*b^4*d^-1*(a^2+-1*b^2)^-19//2*(7*b^6+32*a^6+70*a^2*b^4+112*a^4*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+1//56*a*b*d^-1*(a+b*sin(c+d*x))^-4*(a^2+-1*b^2)^-4*sec(c+d*x)^3*(103*b^2+118*a^2)+13//112*a*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-6*sec(c+d*x)^3*(85*b^4+140*a^4+336*a^2*b^2))
@test integrate((a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^5, x) == :(2//11*b^-5*d^-1*(a+b*sin(c+d*x))^11//2+-8//9*a*b^-5*d^-1*(a+b*sin(c+d*x))^9//2+1//7*b^-5*d^-1*(a+b*sin(c+d*x))^7//2*(-4*b^2+12*a^2)+2//3*b^-5*d^-1*(a+b*sin(c+d*x))^3//2*(a^2+-1*b^2)^2+-8//5*a*b^-5*d^-1*(a+b*sin(c+d*x))^5//2*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3, x) == :(-2//7*b^-3*d^-1*(a+b*sin(c+d*x))^7//2+1//3*b^-3*d^-1*(a+b*sin(c+d*x))^3//2*(-2*a^2+2*b^2)+4//5*a*b^-3*d^-1*(a+b*sin(c+d*x))^5//2)
@test integrate((a+b*sin(c+d*x))^(1/2)*cos(c+d*x), x) == :(2//3*b^-1*d^-1*(a+b*sin(c+d*x))^3//2)
@test integrate((a+b*sin(c+d*x))^(1/2)*sec(c+d*x), x) == :(d^-1*(a+b)^(1/2)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^(1/2)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^3, x) == :((1/2)*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)*tan(c+d*x)+-1//4*d^-1*(a+-1b)^-1//2*(-1b+2a)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^-1//2*(b+2a)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^5, x) == :(-1//32*d^-1*(a+-1b)^-3//2*(5*b^2+12*a^2+-18*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^3*tan(c+d*x)+1//32*d^-1*(a+b)^-3//2*(5*b^2+12*a^2+18*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-16*b^2+16*a^2)^-1*sec(c+d*x)^2*(a*b+-1*(-5*b^2+6*a^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^4, x) == :(2//9*b^-1*d^-1*(a+b*sin(c+d*x))^3//2*cos(c+d*x)^3+-4//21*a*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3+-4//315*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(4*a*(a^2+-3*b^2)+-3*b*(a^2+7*b^2)*sin(c+d*x))*cos(c+d*x)+-1//315*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-168*b^4+32*a^4+-120*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+32//315*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^4+3*b^4+-4*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^2, x) == :(2//5*b^-1*d^-1*(a+b*sin(c+d*x))^3//2*cos(c+d*x)+-4//15*a*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)+1//15*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(4*a^2+12*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-4//15*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^2, x) == :(d^-1*(a+b*sin(c+d*x))^(1/2)*tan(c+d*x)+-1*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^2*tan(c+d*x)+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-6*b^2+6*a^2)^-1*(a*b+-1*(-3*b^2+4*a^2)*sin(c+d*x))*sec(c+d*x)+2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-1*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-6*b^2+6*a^2)^-1*(-3*b^2+4*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^3//2*cos(c+d*x)^5, x) == :(2//13*b^-5*d^-1*(a+b*sin(c+d*x))^13//2+-8//11*a*b^-5*d^-1*(a+b*sin(c+d*x))^11//2+1//9*b^-5*d^-1*(a+b*sin(c+d*x))^9//2*(-4*b^2+12*a^2)+2//5*b^-5*d^-1*(a+b*sin(c+d*x))^5//2*(a^2+-1*b^2)^2+-8//7*a*b^-5*d^-1*(a+b*sin(c+d*x))^7//2*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^3//2*cos(c+d*x)^3, x) == :(-2//9*b^-3*d^-1*(a+b*sin(c+d*x))^9//2+1//5*b^-3*d^-1*(a+b*sin(c+d*x))^5//2*(-2*a^2+2*b^2)+4//7*a*b^-3*d^-1*(a+b*sin(c+d*x))^7//2)
@test integrate((a+b*sin(c+d*x))^3//2*cos(c+d*x), x) == :(2//5*b^-1*d^-1*(a+b*sin(c+d*x))^5//2)
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x), x) == :(d^-1*(a+b)^3//2*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^3//2*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-2*b*d^-1*(a+b*sin(c+d*x))^(1/2))
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x)^3, x) == :((1/2)*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^2*(b+a*sin(c+d*x))+-1//4*d^-1*(a+-1b)^(1/2)*(b+2a)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^(1/2)*(-1b+2a)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x)^5, x) == :(-1//16*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^2*(b+-6*a*sin(c+d*x))+1//4*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^4*(b+a*sin(c+d*x))+1//32*d^-1*(a+b)^-1//2*(-3*b^2+12*a^2+6*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//32*d^-1*(a+-1b)^-1//2*(-12*a^2+3*b^2+6*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^3//2*cos(c+d*x)^4, x) == :(-2//11*b*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^5+-4//1155*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(-15*b^4+4*a^4+-21*a^2*b^2+-3*a*b*(a^2+31*b^2)*sin(c+d*x))*cos(c+d*x)+2//231*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3*(a^2+3*b^2+28*a*b*sin(c+d*x))+1//1155*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(32*a^6+120*b^6+-200*a^4*b^2+48*a^2*b^4)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-32//1155*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^4+-27*b^4+-6*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^3//2*cos(c+d*x)^2, x) == :(-2//7*b*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3+2//105*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(3*a^2+5*b^2+24*a*b*sin(c+d*x))*cos(c+d*x)+-1//105*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-20*b^4+12*a^4+8*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+4//105*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(3*a^2+29*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x)^2, x) == :(d^-1*(a+b*sin(c+d*x))^(1/2)*(b+a*sin(c+d*x))*sec(c+d*x)+d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-1*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x)^4, x) == :(-1//6*d^-1*(a+b*sin(c+d*x))^(1/2)*(b+-4*a*sin(c+d*x))*sec(c+d*x)+1//3*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^3*(b+a*sin(c+d*x))+-2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+1//6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-1*b^2+4*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^3//2*sec(c+d*x)^6, x) == :(-1//30*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^3*(b+-8*a*sin(c+d*x))+1//5*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^5*(b+a*sin(c+d*x))+-1//60*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(b*(5*b^4+8*a^4+-13*a^2*b^2)+-1*a*(29*b^4+32*a^4+-61*a^2*b^2)*sin(c+d*x))*sec(c+d*x)+1//60*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-5*b^2+32*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-1*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-60*b^2+60*a^2)^-1*(-29*b^2+32*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*cos(c+d*x)^5, x) == :(2//15*b^-5*d^-1*(a+b*sin(c+d*x))^15//2+-8//13*a*b^-5*d^-1*(a+b*sin(c+d*x))^13//2+1//11*b^-5*d^-1*(a+b*sin(c+d*x))^11//2*(-4*b^2+12*a^2)+2//7*b^-5*d^-1*(a+b*sin(c+d*x))^7//2*(a^2+-1*b^2)^2+-8//9*a*b^-5*d^-1*(a+b*sin(c+d*x))^9//2*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^5//2*cos(c+d*x)^3, x) == :(-2//11*b^-3*d^-1*(a+b*sin(c+d*x))^11//2+1//7*b^-3*d^-1*(a+b*sin(c+d*x))^7//2*(-2*a^2+2*b^2)+4//9*a*b^-3*d^-1*(a+b*sin(c+d*x))^9//2)
@test integrate((a+b*sin(c+d*x))^5//2*cos(c+d*x), x) == :(2//7*b^-1*d^-1*(a+b*sin(c+d*x))^7//2)
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x), x) == :(d^-1*(a+b)^5//2*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^5//2*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-2//3*b*d^-1*(a+b*sin(c+d*x))^3//2+-4*a*b*d^-1*(a+b*sin(c+d*x))^(1/2))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^3, x) == :((1/2)*a*b*d^-1*(a+b*sin(c+d*x))^(1/2)+(1/2)*d^-1*(a+b*sin(c+d*x))^3//2*sec(c+d*x)^2*(b+a*sin(c+d*x))+-1//4*d^-1*(a+-1b)^3//2*(2a+3b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^3//2*(-3b+2a)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^5, x) == :(-3//32*d^-1*(a+-1b)^(1/2)*(-1*b^2+4*a^2+2*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b*sin(c+d*x))^3//2*sec(c+d*x)^4*(b+a*sin(c+d*x))+3//16*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^2*(a*b+(-1*b^2+2*a^2)*sin(c+d*x))+3//32*d^-1*(a+b)^(1/2)*(-1*b^2+4*a^2+-2*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^5//2*cos(c+d*x)^4, x) == :(-2//13*b*d^-1*(a+b*sin(c+d*x))^3//2*cos(c+d*x)^5+-32//143*a*b*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^5+-4//15015*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(4*a*(-93*b^4+5*a^4+-40*a^2*b^2)+-3*b*(5*a^4+77*b^4+430*a^2*b^2)*sin(c+d*x))*cos(c+d*x)+2//3003*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3*(a*(5*a^2+59*b^2)+7*b*(11*b^2+53*a^2)*sin(c+d*x))+-1//15015*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-1848*b^6+160*a^6+-13296*a^2*b^4+-1400*a^4*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+32//15015*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(5*a^6+93*b^6+-53*a^2*b^4+-45*a^4*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*cos(c+d*x)^2, x) == :(-2//9*b*d^-1*(a+b*sin(c+d*x))^3//2*cos(c+d*x)^3+-8//21*a*b*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3+2//315*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(a*(5*a^2+27*b^2)+3*b*(7*b^2+25*a^2)*sin(c+d*x))*cos(c+d*x)+1//315*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(20*a^4+84*b^4+408*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-4//315*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-27*b^4+5*a^4+22*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^2, x) == :(d^-1*(a+b*sin(c+d*x))^3//2*(b+a*sin(c+d*x))*sec(c+d*x)+a*b*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)+-1*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+3*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^4, x) == :(1//3*d^-1*(a+b*sin(c+d*x))^3//2*sec(c+d*x)^3*(b+a*sin(c+d*x))+1//6*d^-1*(a+b*sin(c+d*x))^(1/2)*(a*b+(-3*b^2+4*a^2)*sin(c+d*x))*sec(c+d*x)+-1//6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-3*b^2+4*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^6, x) == :(1//5*d^-1*(a+b*sin(c+d*x))^3//2*sec(c+d*x)^5*(b+a*sin(c+d*x))+1//30*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^3*((-3*b^2+8*a^2)*sin(c+d*x)+5*a*b)+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-60*b^2+60*a^2)^-1*(-1*(9*b^4+32*a^4+-41*a^2*b^2)*sin(c+d*x)+8*a*b*(a^2+-1*b^2))*sec(c+d*x)+-1//60*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-9*b^2+32*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+1//60*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-17*b^2+32*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^5//2*sec(c+d*x)^8, x) == :(1//7*d^-1*(a+b*sin(c+d*x))^3//2*sec(c+d*x)^7*(b+a*sin(c+d*x))+3//70*d^-1*(a+b*sin(c+d*x))^(1/2)*sec(c+d*x)^5*((-1*b^2+4*a^2)*sin(c+d*x)+3*a*b)+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-140*b^2+140*a^2)^-1*sec(c+d*x)^3*(-1*(7*b^4+32*a^4+-39*a^2*b^2)*sin(c+d*x)+4*a*b*(a^2+-1*b^2))+-1//280*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(-1*(-21*b^6+128*a^6+-272*a^4*b^2+165*a^2*b^4)*sin(c+d*x)+a*b*(27*b^4+32*a^4+-59*a^2*b^2))*sec(c+d*x)+-1*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-280*b^2+280*a^2)^-1*(21*b^4+128*a^4+-144*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+2//35*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-3*b^2+8*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-1//2*cos(c+d*x)^5, x) == :(2//9*b^-5*d^-1*(a+b*sin(c+d*x))^9//2+2*b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^2+-8//7*a*b^-5*d^-1*(a+b*sin(c+d*x))^7//2+1//5*b^-5*d^-1*(a+b*sin(c+d*x))^5//2*(-4*b^2+12*a^2)+-8//3*a*b^-5*d^-1*(a+b*sin(c+d*x))^3//2*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-1//2*cos(c+d*x)^3, x) == :(-2//5*b^-3*d^-1*(a+b*sin(c+d*x))^5//2+b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(-2*a^2+2*b^2)+4//3*a*b^-3*d^-1*(a+b*sin(c+d*x))^3//2)
@test integrate((a+b*sin(c+d*x))^-1//2*cos(c+d*x), x) == :(2*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2))
@test integrate((a+b*sin(c+d*x))^-1//2*sec(c+d*x), x) == :(d^-1*(a+b)^-1//2*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^-1//2*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2)))
@test integrate((a+b*sin(c+d*x))^-1//2*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+-1b)^-3//2*(-3b+2a)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^-3//2*(2a+3b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1//2*sec(c+d*x)^5, x) == :(1//32*d^-1*(a+b)^-5//2*(12*a^2+21*b^2+30*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//32*d^-1*(a+-1b)^-5//2*(-21*b^2+-12*a^2+30*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-1//16*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(b*(a^2+-7*b^2)+-6*a*(a^2+-2*b^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-1//2*cos(c+d*x)^4, x) == :(2//7*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3+-4//35*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(-5*b^2+4*a^2+-3*a*b*sin(c+d*x))*cos(c+d*x)+1//35*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(32*a^4+40*b^4+-72*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-32//35*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-1//2*cos(c+d*x)^2, x) == :(2//3*b^-1*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)+-1//3*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-4*b^2+4*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+4//3*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-1//2*sec(c+d*x)^2, x) == :(d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-1*(b+-1*a*sin(c+d*x))*sec(c+d*x)+-1*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-1*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-1//2*sec(c+d*x)^4, x) == :(-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(-3*b^2+3*a^2)^-1*sec(c+d*x)^3*(b+-1*a*sin(c+d*x))+-1//6*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(b*(a^2+-5*b^2)+-4*a*(a^2+-2*b^2)*sin(c+d*x))*sec(c+d*x)+d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-6*b^2+6*a^2)^-1*(-5*b^2+4*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(a^2+-2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x)^5, x) == :(2//7*b^-5*d^-1*(a+b*sin(c+d*x))^7//2+-2*b^-5*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^2+-8//5*a*b^-5*d^-1*(a+b*sin(c+d*x))^5//2+1//3*b^-5*d^-1*(a+b*sin(c+d*x))^3//2*(-4*b^2+12*a^2)+-8*a*b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x)^3, x) == :(-2//3*b^-3*d^-1*(a+b*sin(c+d*x))^3//2+b^-3*d^-1*(a+b*sin(c+d*x))^-1//2*(-2*b^2+2*a^2)+4*a*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x), x) == :(-2*b^-1*d^-1*(a+b*sin(c+d*x))^-1//2)
@test integrate((a+b*sin(c+d*x))^-3//2*sec(c+d*x), x) == :(d^-1*(a+b)^-3//2*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^-3//2*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+2*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-1)
@test integrate((a+b*sin(c+d*x))^-3//2*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+-1b)^-5//2*(-5b+2a)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^-5//2*(2a+5b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^-1//2*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+-1//2*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*(a^2+5*b^2))
@test integrate((a+b*sin(c+d*x))^-3//2*sec(c+d*x)^5, x) == :(1//32*d^-1*(a+b)^-7//2*(12*a^2+45*b^2+42*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//32*d^-1*(a+-1b)^-7//2*(-45*b^2+-12*a^2+42*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^-1//2*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-3//16*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-3*(-15*b^4+2*a^4+-7*a^2*b^2)+1//16*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(b*(a^2+9*b^2)+2*a*(-8*b^2+3*a^2)*sin(c+d*x)))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x)^6, x) == :(-2*b^-1*d^-1*(a+b*sin(c+d*x))^-1//2*cos(c+d*x)^5+-8//63*b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*(a*(-33*b^2+32*a^2)+-3*b*(-7*b^2+8*a^2)*sin(c+d*x))*cos(c+d*x)+20//63*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3*(8a+-7*b*sin(c+d*x))+-1//63*b^-6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(336*b^4+512*a^4+-912*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+16//63*a*b^-6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(32*a^4+33*b^4+-65*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x)^4, x) == :(-2*b^-1*d^-1*(a+b*sin(c+d*x))^-1//2*cos(c+d*x)^3+4//5*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)*(4a+-3*b*sin(c+d*x))*cos(c+d*x)+1//5*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-24*b^2+32*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-32//5*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-3//2*cos(c+d*x)^2, x) == :(-2*b^-1*d^-1*(a+b*sin(c+d*x))^-1//2*cos(c+d*x)+-4*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+4*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-3//2*sec(c+d*x)^2, x) == :(-1*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(-1*(a^2+3*b^2)*sin(c+d*x)+4*a*b)*sec(c+d*x)+2*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-1*sec(c+d*x)+a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-1*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-1*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*(a^2+3*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-3//2*sec(c+d*x)^4, x) == :(2*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-1*sec(c+d*x)^3+-1//3*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-2*sec(c+d*x)^3*(-1*(a^2+7*b^2)*sin(c+d*x)+8*a*b)+-1//6*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-3*(-1*(-21*b^4+4*a^4+-15*a^2*b^2)*sin(c+d*x)+a*b*(a^2+-33*b^2))*sec(c+d*x)+-1//6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-3*(-21*b^4+4*a^4+-15*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*(a^2+-3*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^5, x) == :(2//5*b^-5*d^-1*(a+b*sin(c+d*x))^5//2+b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*(-4*b^2+12*a^2)+-8//3*a*b^-5*d^-1*(a+b*sin(c+d*x))^3//2+-2//3*b^-5*d^-1*(a+b*sin(c+d*x))^-3//2*(a^2+-1*b^2)^2+8*a*b^-5*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^3, x) == :(-2*b^-3*d^-1*(a+b*sin(c+d*x))^(1/2)+-4*a*b^-3*d^-1*(a+b*sin(c+d*x))^-1//2+1//3*b^-3*d^-1*(a+b*sin(c+d*x))^-3//2*(-2*b^2+2*a^2))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x), x) == :(-2//3*b^-1*d^-1*(a+b*sin(c+d*x))^-3//2)
@test integrate((a+b*sin(c+d*x))^-5//2*sec(c+d*x), x) == :(d^-1*(a+b)^-5//2*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+-1b)^-5//2*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+2*b*d^-1*(a+b*sin(c+d*x))^-3//2*(-3*b^2+3*a^2)^-1+4*a*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2)
@test integrate((a+b*sin(c+d*x))^-5//2*sec(c+d*x)^3, x) == :(-1//4*d^-1*(a+-1b)^-7//2*(-7b+2a)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//4*d^-1*(a+b)^-7//2*(2a+7b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^-3//2*(-2*b^2+2*a^2)^-1*sec(c+d*x)^2*(b+-1*a*sin(c+d*x))+-1//6*b*d^-1*(a+b*sin(c+d*x))^-3//2*(a^2+-1*b^2)^-2*(3*a^2+7*b^2)+-1//2*a*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-3*(a^2+19*b^2))
@test integrate((a+b*sin(c+d*x))^-5//2*sec(c+d*x)^5, x) == :(-1//32*d^-1*(a+-1b)^-9//2*(12*a^2+77*b^2+-54*a*b)*arctanh((a+-1b)^-1//2*(a+b*sin(c+d*x))^(1/2))+1//32*d^-1*(a+b)^-9//2*(12*a^2+77*b^2+54*a*b)*arctanh((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2))+-1*d^-1*(a+b*sin(c+d*x))^-3//2*(-4*b^2+4*a^2)^-1*sec(c+d*x)^4*(b+-1*a*sin(c+d*x))+-1//48*b*d^-1*(a+b*sin(c+d*x))^-3//2*(a^2+-1*b^2)^-3*(-77*b^4+18*a^4+-81*a^2*b^2)+1//16*d^-1*(a+b*sin(c+d*x))^-3//2*(a^2+-1*b^2)^-2*sec(c+d*x)^2*(b*(3*a^2+11*b^2)+2*a*(-10*b^2+3*a^2)*sin(c+d*x))+-1//8*a*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-4*(-127*b^4+3*a^4+-16*a^2*b^2))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^8, x) == :(-2//3*b^-1*d^-1*(a+b*sin(c+d*x))^-3//2*cos(c+d*x)^7+-28//33*b^-3*d^-1*(a+b*sin(c+d*x))^-1//2*cos(c+d*x)^5*(12a+b*sin(c+d*x))+-16//99*b^-7*d^-1*(a+b*sin(c+d*x))^(1/2)*(15*b^4+128*a^4+-144*a^2*b^2+-3*a*b*(-31*b^2+32*a^2)*sin(c+d*x))*cos(c+d*x)+40//99*b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*cos(c+d*x)^3*(-3*b^2+32*a^2+-28*a*b*sin(c+d*x))+1//99*b^-8*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-480*b^6+4096*a^6+-8704*a^4*b^2+5088*a^2*b^4)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-128//99*a*b^-8*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-9*b^2+8*a^2)*(-3*b^2+4*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^6, x) == :(-2//3*b^-1*d^-1*(a+b*sin(c+d*x))^-3//2*cos(c+d*x)^5+-20//21*b^-3*d^-1*(a+b*sin(c+d*x))^-1//2*cos(c+d*x)^3*(8a+b*sin(c+d*x))+8//21*b^-5*d^-1*(a+b*sin(c+d*x))^(1/2)*(-5*b^2+32*a^2+-24*a*b*sin(c+d*x))*cos(c+d*x)+-1//21*b^-6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(80*b^4+512*a^4+-592*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+16//21*a*b^-6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(-29*b^2+32*a^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^4, x) == :(-2//3*b^-1*d^-1*(a+b*sin(c+d*x))^-3//2*cos(c+d*x)^3+-4//3*b^-3*d^-1*(a+b*sin(c+d*x))^-1//2*(4a+b*sin(c+d*x))*cos(c+d*x)+-32//3*a*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+1//3*b^-4*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(-8*b^2+32*a^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*cos(c+d*x)^2, x) == :(-2//3*b^-1*d^-1*(a+b*sin(c+d*x))^-3//2*cos(c+d*x)+-4//3*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+4//3*a*b^-1*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-1*cos(c+d*x)+4//3*a*b^-2*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-1*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*sec(c+d*x)^2, x) == :(2*b*d^-1*(a+b*sin(c+d*x))^-3//2*(-3*b^2+3*a^2)^-1*sec(c+d*x)+-1//3*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-3*(b*(5*b^2+27*a^2)+-1*a*(3*a^2+29*b^2)*sin(c+d*x))*sec(c+d*x)+1//3*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*(3*a^2+5*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+16//3*a*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*sec(c+d*x)+-1//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-3*(3*a^2+29*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-5//2*sec(c+d*x)^4, x) == :(2*b*d^-1*(a+b*sin(c+d*x))^-3//2*(-3*b^2+3*a^2)^-1*sec(c+d*x)^3+-1//3*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-3*sec(c+d*x)^3*(b*(3*b^2+29*a^2)+-1*a*(a^2+31*b^2)*sin(c+d*x))+-1//6*d^-1*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-4*(b*(a^4+-15*b^4+-114*a^2*b^2)+-4*a*(a^4+-27*b^4+-6*a^2*b^2)*sin(c+d*x))*sec(c+d*x)+8*a*b*d^-1*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-2*sec(c+d*x)^3+1//6*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^(1/2)*(a+b*sin(c+d*x))^-1//2*(a^2+-1*b^2)^-3*(-15*b^4+4*a^4+-21*a^2*b^2)*Elliptic.F((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1)+-2//3*a*d^-1*((a+b)^-1*(a+b*sin(c+d*x)))^-1//2*(a+b*sin(c+d*x))^(1/2)*(a^2+-1*b^2)^-4*(a^4+-27*b^4+-6*a^2*b^2)*Elliptic.E((1/2)*c+-1//4*pi+(1/2)*d*x,2*b*(a+b)^-1))
@test integrate((e*cos(c+d*x))^7//2*(a+b*sin(c+d*x)), x) == :(-2//9*b*d^-1*e^-1*(e*cos(c+d*x))^9//2+2//7*a*e*d^-1*(e*cos(c+d*x))^5//2*sin(c+d*x)+10//21*a*d^-1*e^3*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+10//21*a*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+b*sin(c+d*x)), x) == :(-2//7*b*d^-1*e^-1*(e*cos(c+d*x))^7//2+2//5*a*e*d^-1*(e*cos(c+d*x))^3//2*sin(c+d*x)+6//5*a*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+b*sin(c+d*x)), x) == :(-2//5*b*d^-1*e^-1*(e*cos(c+d*x))^5//2+2//3*a*e*d^-1*(e*cos(c+d*x))^(1/2)*sin(c+d*x)+2//3*a*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x)), x) == :(-2//3*b*d^-1*e^-1*(e*cos(c+d*x))^3//2+2*a*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x)), x) == :(-2*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+2*a*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x)), x) == :(2*b*d^-1*e^-1*(e*cos(c+d*x))^-1//2+2*a*d^-1*e^-1*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-2*a*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x)), x) == :(2//3*b*d^-1*e^-1*(e*cos(c+d*x))^-3//2+2//3*a*d^-1*e^-1*(e*cos(c+d*x))^-3//2*sin(c+d*x)+2//3*a*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x)), x) == :(2//5*b*d^-1*e^-1*(e*cos(c+d*x))^-5//2+2//5*a*d^-1*e^-1*(e*cos(c+d*x))^-5//2*sin(c+d*x)+6//5*a*d^-1*e^-3*(e*cos(c+d*x))^-1//2*sin(c+d*x)+-6//5*a*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^2, x) == :(-26//99*a*b*d^-1*e^-1*(e*cos(c+d*x))^9//2+-2//11*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))+1//77*e*d^-1*(e*cos(c+d*x))^5//2*(4*b^2+22*a^2)*sin(c+d*x)+1//231*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(20*b^2+110*a^2)*sin(c+d*x)+1//231*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(20*b^2+110*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^2, x) == :(-22//63*a*b*d^-1*e^-1*(e*cos(c+d*x))^7//2+-2//9*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))+1//45*e*d^-1*(e*cos(c+d*x))^3//2*(4*b^2+18*a^2)*sin(c+d*x)+1//15*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(4*b^2+18*a^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^2, x) == :(-18//35*a*b*d^-1*e^-1*(e*cos(c+d*x))^5//2+-2//7*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))+1//21*e*d^-1*(e*cos(c+d*x))^(1/2)*(4*b^2+14*a^2)*sin(c+d*x)+1//21*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(4*b^2+14*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^2, x) == :(-14//15*a*b*d^-1*e^-1*(e*cos(c+d*x))^3//2+-2//5*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))+1//5*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(4*b^2+10*a^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^2, x) == :(-10//3*a*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)+-2//3*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))+1//3*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(4*b^2+6*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^2, x) == :(d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))*(2b+2*a*sin(c+d*x))+2*a*b*d^-1*e^-3*(e*cos(c+d*x))^3//2+-1*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(2*a^2+4*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^2, x) == :(1//3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))*(2b+2*a*sin(c+d*x))+2//3*a*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)+1//3*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(-4*b^2+2*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x))^2, x) == :(1//5*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))*(2b+2*a*sin(c+d*x))+1//5*d^-1*e^-3*(e*cos(c+d*x))^-1//2*(-4*b^2+6*a^2)*sin(c+d*x)+2//5*a*b*d^-1*e^-3*(e*cos(c+d*x))^-1//2+-1//5*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(-4*b^2+6*a^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^3, x) == :(-2//13*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))^2+-2//1287*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(44*b^2+177*a^2)+-34//143*a*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))+2//77*a*e*d^-1*(e*cos(c+d*x))^5//2*(6*b^2+11*a^2)*sin(c+d*x)+10//231*a*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(6*b^2+11*a^2)*sin(c+d*x)+10//231*a*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(6*b^2+11*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^3, x) == :(-2//11*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^2+-2//231*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(12*b^2+43*a^2)+-10//33*a*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))+2//15*a*e*d^-1*(e*cos(c+d*x))^3//2*(2*b^2+3*a^2)*sin(c+d*x)+2//5*a*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(2*b^2+3*a^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^3, x) == :(-2//9*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^2+-2//315*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(28*b^2+89*a^2)+-26//63*a*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))+2//21*a*e*d^-1*(e*cos(c+d*x))^(1/2)*(6*b^2+7*a^2)*sin(c+d*x)+2//21*a*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(6*b^2+7*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^3, x) == :(-2//7*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^2+-2//105*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(20*b^2+57*a^2)+-22//35*a*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))+2//5*a*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(5*a^2+6*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^3, x) == :(-2//5*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^2+-2//5*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(4*b^2+11*a^2)+2*a*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(a^2+2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2)+-6//5*a*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x)))
@test integrate((e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^3, x) == :(d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^2*(2b+2*a*sin(c+d*x))+2//3*b*d^-1*e^-3*(e*cos(c+d*x))^3//2*(3*a^2+4*b^2)+2*a*b*d^-1*e^-3*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))+-2*a*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(a^2+6*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^3, x) == :(1//3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^2*(2b+2*a*sin(c+d*x))+2//3*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a^2+4*b^2)+2//3*a*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))+2//3*a*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(a^2+-6*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x))^3, x) == :(-1//5*d^-1*e^-3*(e*cos(c+d*x))^-1//2*(2a+2*b*sin(c+d*x))*(a*b+-1*(-4*b^2+3*a^2)*sin(c+d*x))+1//5*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^2*(2b+2*a*sin(c+d*x))+2//5*b*d^-1*e^-5*(e*cos(c+d*x))^3//2*(-4*b^2+3*a^2)+-6//5*a*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(a^2+-2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^-9//2*(a+b*sin(c+d*x))^3, x) == :(1//7*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x))^2*(2b+2*a*sin(c+d*x))+1//21*d^-1*e^-3*(e*cos(c+d*x))^-3//2*(2a+2*b*sin(c+d*x))*(a*b+(-4*b^2+5*a^2)*sin(c+d*x))+2//21*b*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(-4*b^2+5*a^2)+2//21*a*d^-1*e^-4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(-6*b^2+5*a^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^4, x) == :(-2//15*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))^3+1//231*d^-1*e^3*(e*cos(c+d*x))^(1/2)*(8*b^4+110*a^4+120*a^2*b^2)*sin(c+d*x)+1//385*e*d^-1*(e*cos(c+d*x))^5//2*(8*b^4+110*a^4+120*a^2*b^2)*sin(c+d*x)+-34//6435*a*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(38*b^2+53*a^2)+-14//65*a*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))^2+-2//715*b*d^-1*e^-1*(e*cos(c+d*x))^9//2*(a+b*sin(c+d*x))*(26*b^2+93*a^2)+1//231*d^-1*e^4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(8*b^4+110*a^4+120*a^2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^4, x) == :(-2//13*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^3+1//195*e*d^-1*(e*cos(c+d*x))^3//2*(8*b^4+78*a^4+104*a^2*b^2)*sin(c+d*x)+-38//143*a*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))^2+-10//3003*a*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(94*b^2+115*a^2)+-2//429*b*d^-1*e^-1*(e*cos(c+d*x))^7//2*(a+b*sin(c+d*x))*(22*b^2+73*a^2)+1//65*d^-1*e^2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(8*b^4+78*a^4+104*a^2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^4, x) == :(-2//11*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^3+1//231*e*d^-1*(e*cos(c+d*x))^(1/2)*(24*b^4+154*a^4+264*a^2*b^2)*sin(c+d*x)+-34//99*a*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))^2+-26//3465*a*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(74*b^2+79*a^2)+-2//693*b*d^-1*e^-1*(e*cos(c+d*x))^5//2*(a+b*sin(c+d*x))*(54*b^2+167*a^2)+1//231*d^-1*e^2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(24*b^4+154*a^4+264*a^2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2))
@test integrate((e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^4, x) == :(-2//9*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^3+1//15*d^-1*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(8*b^4+30*a^4+72*a^2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2)+-22//315*a*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(17*a^2+18*b^2)+-10//21*a*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^2+-2//105*b*d^-1*e^-1*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))*(14*b^2+41*a^2))
@test integrate((e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^4, x) == :(-2//7*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^3+1//7*d^-1*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(8*b^4+14*a^4+56*a^2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2)+-26//35*a*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^2+-6//35*a*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(31*a^2+34*b^2)+-2//35*b*d^-1*e^-1*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))*(10*b^2+29*a^2))
@test integrate((e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^4, x) == :(d^-1*e^-1*(e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^3*(2b+2*a*sin(c+d*x))+2*a*b*d^-1*e^-3*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))^2+-1//5*d^-1*e^-2*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(10*a^4+24*b^4+120*a^2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2)+2//5*b*d^-1*e^-3*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))*(5*a^2+6*b^2)+2//15*a*b*d^-1*e^-3*(e*cos(c+d*x))^3//2*(15*a^2+62*b^2))
@test integrate((e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^4, x) == :(1//3*d^-1*e^-1*(e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^3*(2b+2*a*sin(c+d*x))+1//3*d^-1*e^-2*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(-8*b^4+2*a^4+-24*a^2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2)+2//3*a*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^2+2//3*a*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a^2+14*b^2)+2//3*b*d^-1*e^-3*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))*(a^2+2*b^2))
@test integrate((e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x))^4, x) == :(-6//5*d^-1*e^-3*(e*cos(c+d*x))^-1//2*(a+b*sin(c+d*x))^2*(a*b+-1*(a^2+-2*b^2)*sin(c+d*x))+1//5*d^-1*e^-1*(e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^3*(2b+2*a*sin(c+d*x))+-1//5*d^-1*e^-4*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(-24*b^4+6*a^4+-24*a^2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2)+2//5*a*b*d^-1*e^-5*(e*cos(c+d*x))^3//2*(-10*b^2+3*a^2)+6//5*b*d^-1*e^-5*(e*cos(c+d*x))^3//2*(a+b*sin(c+d*x))*(a^2+-2*b^2))
@test integrate((e*cos(c+d*x))^-9//2*(a+b*sin(c+d*x))^4, x) == :(-2//21*d^-1*e^-3*(e*cos(c+d*x))^-3//2*(a+b*sin(c+d*x))^2*(a*b+-1*(-6*b^2+5*a^2)*sin(c+d*x))+1//7*d^-1*e^-1*(e*cos(c+d*x))^-7//2*(a+b*sin(c+d*x))^3*(2b+2*a*sin(c+d*x))+1//21*d^-1*e^-4*(e*cos(c+d*x))^-1//2*cos(c+d*x)^(1/2)*(10*a^4+24*b^4+-24*a^2*b^2)*Elliptic.F((1/2)*c+(1/2)*d*x,2)+2//21*b*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))*(-6*b^2+5*a^2)+10//21*a*b*d^-1*e^-5*(e*cos(c+d*x))^(1/2)*(a^2+-2*b^2))
@test integrate((e*cos(c+d*x))^-11//2*(a+b*sin(c+d*x))^4, x) == :(-1//45*d^-1*e^-5*(e*cos(c+d*x))^-1//2*(2a+2*b*sin(c+d*x))*(b*(-6*b^2+7*a^2)+-1*a*(-22*b^2+21*a^2)*sin(c+d*x))+1//9*d^-1*e^-1*(e*cos(c+d*x))^-9//2*(a+b*sin(c+d*x))^3*(2b+2*a*sin(c+d*x))+2//45*d^-1*e^-3*(e*cos(c+d*x))^-5//2*(a+b*sin(c+d*x))^2*(a*b+(-6*b^2+7*a^2)*sin(c+d*x))+-1//15*d^-1*e^-6*(e*cos(c+d*x))^(1/2)*cos(c+d*x)^-1//2*(8*b^4+14*a^4+-24*a^2*b^2)*Elliptic.E((1/2)*c+(1/2)*d*x,2)+2//45*a*b*d^-1*e^-7*(e*cos(c+d*x))^3//2*(-22*b^2+21*a^2))
@test integrate((a+b*sin(c+d*x))^m*cos(c+d*x)^7, x) == :(-1*b^-7*d^-1*(7+m)^-1*(a+b*sin(c+d*x))^(7+m)+-1*b^-7*d^-1*(1+m)^-1*(a+b*sin(c+d*x))^(1+m)*(a^2+-1*b^2)^3+-1*b^-7*d^-1*(3+m)^-1*(a+b*sin(c+d*x))^(3+m)*(3*b^4+15*a^4+-18*a^2*b^2)+-1*b^-7*d^-1*(5+m)^-1*(a+b*sin(c+d*x))^(5+m)*(-3*b^2+15*a^2)+6*a*b^-7*d^-1*(6+m)^-1*(a+b*sin(c+d*x))^(6+m)+4*a*b^-7*d^-1*(4+m)^-1*(a+b*sin(c+d*x))^(4+m)*(-3*b^2+5*a^2)+6*a*b^-7*d^-1*(2+m)^-1*(a+b*sin(c+d*x))^(2+m)*(a^2+-1*b^2)^2)
@test integrate((a+b*sin(c+d*x))^m*cos(c+d*x)^5, x) == :(b^-5*d^-1*(5+m)^-1*(a+b*sin(c+d*x))^(5+m)+b^-5*d^-1*(1+m)^-1*(a+b*sin(c+d*x))^(1+m)*(a^2+-1*b^2)^2+b^-5*d^-1*(3+m)^-1*(a+b*sin(c+d*x))^(3+m)*(-2*b^2+6*a^2)+-4*a*b^-5*d^-1*(4+m)^-1*(a+b*sin(c+d*x))^(4+m)+-4*a*b^-5*d^-1*(2+m)^-1*(a+b*sin(c+d*x))^(2+m)*(a^2+-1*b^2))
@test integrate((a+b*sin(c+d*x))^m*cos(c+d*x)^3, x) == :(-1*b^-3*d^-1*(3+m)^-1*(a+b*sin(c+d*x))^(3+m)+-1*b^-3*d^-1*(1+m)^-1*(a+b*sin(c+d*x))^(1+m)*(a^2+-1*b^2)+2*a*b^-3*d^-1*(2+m)^-1*(a+b*sin(c+d*x))^(2+m))
@test integrate((a+b*sin(c+d*x))^m*cos(c+d*x), x) == :(b^-1*d^-1*(1+m)^-1*(a+b*sin(c+d*x))^(1+m))

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.jl