# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.10 (c+d x)^m (a+b sin)^n.txt"

lst:=[

# Integrands of the form (e+f x)^m (a+b Sin[c+d x])^p

# Integrands of the form (c+d x)^m (b Sin[e+f x])^n

# Integrands of the form (c+d x)^m Sin[e+f x]^n

# n>0
[(c+d*x)^4*sin(a+b*x),x,5,-24*d^4*cos(a+b*x)/b^5+12*d^2*(c+d*x)^2*cos(a+b*x)/b^3-(c+d*x)^4*cos(a+b*x)/b-24*d^3*(c+d*x)*sin(a+b*x)/b^4+4*d*(c+d*x)^3*sin(a+b*x)/b^2],
[(c+d*x)^3*sin(a+b*x),x,4,6*d^2*(c+d*x)*cos(a+b*x)/b^3-(c+d*x)^3*cos(a+b*x)/b-6*d^3*sin(a+b*x)/b^4+3*d*(c+d*x)^2*sin(a+b*x)/b^2],
[(c+d*x)^2*sin(a+b*x),x,3,2*d^2*cos(a+b*x)/b^3-(c+d*x)^2*cos(a+b*x)/b+2*d*(c+d*x)*sin(a+b*x)/b^2],
[(c+d*x)*sin(a+b*x),x,2,-(c+d*x)*cos(a+b*x)/b+d*sin(a+b*x)/b^2],
[sin(a+b*x)/(c+d*x),x,3,cos(a-b*c/d)*Si(b*c/d+b*x)/d+Ci(b*c/d+b*x)*sin(a-b*c/d)/d],
[sin(a+b*x)/(c+d*x)^2,x,4,b*Ci(b*c/d+b*x)*cos(a-b*c/d)/d^2-b*Si(b*c/d+b*x)*sin(a-b*c/d)/d^2-sin(a+b*x)/(d*(c+d*x))],
[sin(a+b*x)/(c+d*x)^3,x,5,-1/2*b*cos(a+b*x)/(d^2*(c+d*x))-1/2*b^2*cos(a-b*c/d)*Si(b*c/d+b*x)/d^3-1/2*b^2*Ci(b*c/d+b*x)*sin(a-b*c/d)/d^3-1/2*sin(a+b*x)/(d*(c+d*x)^2)],
[(c+d*x)^4*sin(a+b*x)^2,x,6,3/4*d^4*x/b^4-1/2*d*(c+d*x)^3/b^2+1/10*(c+d*x)^5/d-3/4*d^4*cos(a+b*x)*sin(a+b*x)/b^5+3/2*d^2*(c+d*x)^2*cos(a+b*x)*sin(a+b*x)/b^3-1/2*(c+d*x)^4*cos(a+b*x)*sin(a+b*x)/b-3/2*d^3*(c+d*x)*sin(a+b*x)^2/b^4+d*(c+d*x)^3*sin(a+b*x)^2/b^2],
[(c+d*x)^3*sin(a+b*x)^2,x,4,-3/4*c*d^2*x/b^2-3/8*d^3*x^2/b^2+1/8*(c+d*x)^4/d+3/4*d^2*(c+d*x)*cos(a+b*x)*sin(a+b*x)/b^3-1/2*(c+d*x)^3*cos(a+b*x)*sin(a+b*x)/b-3/8*d^3*sin(a+b*x)^2/b^4+3/4*d*(c+d*x)^2*sin(a+b*x)^2/b^2],
[(c+d*x)^2*sin(a+b*x)^2,x,4,-1/4*d^2*x/b^2+1/6*(c+d*x)^3/d+1/4*d^2*cos(a+b*x)*sin(a+b*x)/b^3-1/2*(c+d*x)^2*cos(a+b*x)*sin(a+b*x)/b+1/2*d*(c+d*x)*sin(a+b*x)^2/b^2],
[(c+d*x)*sin(a+b*x)^2,x,2,1/2*c*x+1/4*d*x^2-1/2*(c+d*x)*cos(a+b*x)*sin(a+b*x)/b+1/4*d*sin(a+b*x)^2/b^2],
[sin(a+b*x)^2/(c+d*x),x,5,-1/2*Ci(2*b*c/d+2*b*x)*cos(2*a-2*b*c/d)/d+1/2*log(c+d*x)/d+1/2*Si(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d],
[sin(a+b*x)^2/(c+d*x)^2,x,5,b*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d^2+b*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^2-sin(a+b*x)^2/(d*(c+d*x))],
[sin(a+b*x)^2/(c+d*x)^3,x,7,b^2*Ci(2*b*c/d+2*b*x)*cos(2*a-2*b*c/d)/d^3-b^2*Si(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^3-b*cos(a+b*x)*sin(a+b*x)/(d^2*(c+d*x))-1/2*sin(a+b*x)^2/(d*(c+d*x)^2)],
[sin(a+b*x)^2/(c+d*x)^4,x,7,-1/3*b^2/(d^3*(c+d*x))-2/3*b^3*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d^4-2/3*b^3*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^4-1/3*b*cos(a+b*x)*sin(a+b*x)/(d^2*(c+d*x)^2)-1/3*sin(a+b*x)^2/(d*(c+d*x)^3)+2/3*b^2*sin(a+b*x)^2/(d^3*(c+d*x))],
[(c+d*x)^4*sin(a+b*x)^3,x,12,-488/27*d^4*cos(a+b*x)/b^5+80/9*d^2*(c+d*x)^2*cos(a+b*x)/b^3-2/3*(c+d*x)^4*cos(a+b*x)/b+8/81*d^4*cos(a+b*x)^3/b^5-160/9*d^3*(c+d*x)*sin(a+b*x)/b^4+8/3*d*(c+d*x)^3*sin(a+b*x)/b^2+4/9*d^2*(c+d*x)^2*cos(a+b*x)*sin(a+b*x)^2/b^3-1/3*(c+d*x)^4*cos(a+b*x)*sin(a+b*x)^2/b-8/27*d^3*(c+d*x)*sin(a+b*x)^3/b^4+4/9*d*(c+d*x)^3*sin(a+b*x)^3/b^2],
[(c+d*x)^3*sin(a+b*x)^3,x,8,40/9*d^2*(c+d*x)*cos(a+b*x)/b^3-2/3*(c+d*x)^3*cos(a+b*x)/b-40/9*d^3*sin(a+b*x)/b^4+2*d*(c+d*x)^2*sin(a+b*x)/b^2+2/9*d^2*(c+d*x)*cos(a+b*x)*sin(a+b*x)^2/b^3-1/3*(c+d*x)^3*cos(a+b*x)*sin(a+b*x)^2/b-2/27*d^3*sin(a+b*x)^3/b^4+1/3*d*(c+d*x)^2*sin(a+b*x)^3/b^2],
[(c+d*x)^2*sin(a+b*x)^3,x,6,14/9*d^2*cos(a+b*x)/b^3-2/3*(c+d*x)^2*cos(a+b*x)/b-2/27*d^2*cos(a+b*x)^3/b^3+4/3*d*(c+d*x)*sin(a+b*x)/b^2-1/3*(c+d*x)^2*cos(a+b*x)*sin(a+b*x)^2/b+2/9*d*(c+d*x)*sin(a+b*x)^3/b^2],
[(c+d*x)*sin(a+b*x)^3,x,3,-2/3*(c+d*x)*cos(a+b*x)/b+2/3*d*sin(a+b*x)/b^2-1/3*(c+d*x)*cos(a+b*x)*sin(a+b*x)^2/b+1/9*d*sin(a+b*x)^3/b^2],
[sin(a+b*x)^3/(c+d*x),x,8,3/4*cos(a-b*c/d)*Si(b*c/d+b*x)/d-1/4*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d-1/4*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d+3/4*Ci(b*c/d+b*x)*sin(a-b*c/d)/d],
[sin(a+b*x)^3/(c+d*x)^2,x,8,-3/4*b*Ci(3*b*c/d+3*b*x)*cos(3*a-3*b*c/d)/d^2+3/4*b*Ci(b*c/d+b*x)*cos(a-b*c/d)/d^2+3/4*b*Si(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d^2-3/4*b*Si(b*c/d+b*x)*sin(a-b*c/d)/d^2-sin(a+b*x)^3/(d*(c+d*x))],
[sin(a+b*x)^3/(c+d*x)^3,x,12,-3/8*b^2*cos(a-b*c/d)*Si(b*c/d+b*x)/d^3+9/8*b^2*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d^3+9/8*b^2*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d^3-3/8*b^2*Ci(b*c/d+b*x)*sin(a-b*c/d)/d^3-3/2*b*cos(a+b*x)*sin(a+b*x)^2/(d^2*(c+d*x))-1/2*sin(a+b*x)^3/(d*(c+d*x)^2)],

# n<0
[(c+d*x)^3*csc(a+b*x),x,9,-2*(c+d*x)^3*arctanh(exp(I*(a+b*x)))/b+3*I*d*(c+d*x)^2*polylog(2,-exp(I*(a+b*x)))/b^2-3*I*d*(c+d*x)^2*polylog(2,exp(I*(a+b*x)))/b^2-6*d^2*(c+d*x)*polylog(3,-exp(I*(a+b*x)))/b^3+6*d^2*(c+d*x)*polylog(3,exp(I*(a+b*x)))/b^3-6*I*d^3*polylog(4,-exp(I*(a+b*x)))/b^4+6*I*d^3*polylog(4,exp(I*(a+b*x)))/b^4],
[(c+d*x)^2*csc(a+b*x),x,7,-2*(c+d*x)^2*arctanh(exp(I*(a+b*x)))/b+2*I*d*(c+d*x)*polylog(2,-exp(I*(a+b*x)))/b^2-2*I*d*(c+d*x)*polylog(2,exp(I*(a+b*x)))/b^2-2*d^2*polylog(3,-exp(I*(a+b*x)))/b^3+2*d^2*polylog(3,exp(I*(a+b*x)))/b^3],
[(c+d*x)*csc(a+b*x),x,5,-2*(c+d*x)*arctanh(exp(I*(a+b*x)))/b+I*d*polylog(2,-exp(I*(a+b*x)))/b^2-I*d*polylog(2,exp(I*(a+b*x)))/b^2],
[csc(a+b*x)/(c+d*x),x,0,Unintegrable(csc(a+b*x)/(c+d*x),x)],
[csc(a+b*x)/(c+d*x)^2,x,0,Unintegrable(csc(a+b*x)/(c+d*x)^2,x)],
[(c+d*x)^3*csc(a+b*x)^2,x,6,-I*(c+d*x)^3/b-(c+d*x)^3*cot(a+b*x)/b+3*d*(c+d*x)^2*log(1-exp(2*I*(a+b*x)))/b^2-3*I*d^2*(c+d*x)*polylog(2,exp(2*I*(a+b*x)))/b^3+3/2*d^3*polylog(3,exp(2*I*(a+b*x)))/b^4],
[(c+d*x)^2*csc(a+b*x)^2,x,5,-I*(c+d*x)^2/b-(c+d*x)^2*cot(a+b*x)/b+2*d*(c+d*x)*log(1-exp(2*I*(a+b*x)))/b^2-I*d^2*polylog(2,exp(2*I*(a+b*x)))/b^3],
[(c+d*x)*csc(a+b*x)^2,x,2,-(c+d*x)*cot(a+b*x)/b+d*log(sin(a+b*x))/b^2],
[csc(a+b*x)^2/(c+d*x),x,0,Unintegrable(csc(a+b*x)^2/(c+d*x),x)],
[csc(a+b*x)^2/(c+d*x)^2,x,0,Unintegrable(csc(a+b*x)^2/(c+d*x)^2,x)],
[(c+d*x)^3*csc(a+b*x)^3,x,15,-6*d^2*(c+d*x)*arctanh(exp(I*(a+b*x)))/b^3-(c+d*x)^3*arctanh(exp(I*(a+b*x)))/b-3/2*d*(c+d*x)^2*csc(a+b*x)/b^2-1/2*(c+d*x)^3*cot(a+b*x)*csc(a+b*x)/b+3*I*d^3*polylog(2,-exp(I*(a+b*x)))/b^4+3/2*I*d*(c+d*x)^2*polylog(2,-exp(I*(a+b*x)))/b^2-3*I*d^3*polylog(2,exp(I*(a+b*x)))/b^4-3/2*I*d*(c+d*x)^2*polylog(2,exp(I*(a+b*x)))/b^2-3*d^2*(c+d*x)*polylog(3,-exp(I*(a+b*x)))/b^3+3*d^2*(c+d*x)*polylog(3,exp(I*(a+b*x)))/b^3-3*I*d^3*polylog(4,-exp(I*(a+b*x)))/b^4+3*I*d^3*polylog(4,exp(I*(a+b*x)))/b^4],
[(c+d*x)^2*csc(a+b*x)^3,x,9,-(c+d*x)^2*arctanh(exp(I*(a+b*x)))/b-d^2*arctanh(cos(a+b*x))/b^3-d*(c+d*x)*csc(a+b*x)/b^2-1/2*(c+d*x)^2*cot(a+b*x)*csc(a+b*x)/b+I*d*(c+d*x)*polylog(2,-exp(I*(a+b*x)))/b^2-I*d*(c+d*x)*polylog(2,exp(I*(a+b*x)))/b^2-d^2*polylog(3,-exp(I*(a+b*x)))/b^3+d^2*polylog(3,exp(I*(a+b*x)))/b^3],
[(c+d*x)*csc(a+b*x)^3,x,6,-(c+d*x)*arctanh(exp(I*(a+b*x)))/b-1/2*d*csc(a+b*x)/b^2-1/2*(c+d*x)*cot(a+b*x)*csc(a+b*x)/b+1/2*I*d*polylog(2,-exp(I*(a+b*x)))/b^2-1/2*I*d*polylog(2,exp(I*(a+b*x)))/b^2],
[csc(a+b*x)^3/(c+d*x),x,0,Unintegrable(csc(a+b*x)^3/(c+d*x),x)],
[csc(a+b*x)^3/(c+d*x)^2,x,0,Unintegrable(csc(a+b*x)^3/(c+d*x)^2,x)],

# Integrands of the form (c+d x)^(m/2) Sin[e+f x]^n

# n>0
[(c+d*x)^(5/2)*sin(a+b*x),x,8,-(c+d*x)^(5/2)*cos(a+b*x)/b+5/2*d*(c+d*x)^(3/2)*sin(a+b*x)/b^2-15/4*d^(5/2)*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/2*Pi)/b^(7/2)+15/4*d^(5/2)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(1/2*Pi)/b^(7/2)+15/4*d^2*cos(a+b*x)*sqrt(c+d*x)/b^3],
[(c+d*x)^(3/2)*sin(a+b*x),x,7,-(c+d*x)^(3/2)*cos(a+b*x)/b-3/2*d^(3/2)*cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/2*Pi)/b^(5/2)-3/2*d^(3/2)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(1/2*Pi)/b^(5/2)+3/2*d*sin(a+b*x)*sqrt(c+d*x)/b^2],
[(c+d*x)^(1/2)*sin(a+b*x),x,6,cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(d)*sqrt(1/2*Pi)/b^(3/2)-FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(d)*sqrt(1/2*Pi)/b^(3/2)-cos(a+b*x)*sqrt(c+d*x)/b],
[sin(a+b*x)/(c+d*x)^(1/2),x,5,cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(2*Pi)/(sqrt(b)*sqrt(d))+FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(2*Pi)/(sqrt(b)*sqrt(d))],
[sin(a+b*x)/(c+d*x)^(3/2),x,6,2*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(b)*sqrt(2*Pi)/d^(3/2)-2*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(b)*sqrt(2*Pi)/d^(3/2)-2*sin(a+b*x)/(d*sqrt(c+d*x))],
[sin(a+b*x)/(c+d*x)^(5/2),x,7,-2/3*sin(a+b*x)/(d*(c+d*x)^(3/2))-4/3*b^(3/2)*cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(2*Pi)/d^(5/2)-4/3*b^(3/2)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(2*Pi)/d^(5/2)-4/3*b*cos(a+b*x)/(d^2*sqrt(c+d*x))],
[sin(a+b*x)/(c+d*x)^(7/2),x,8,-4/15*b*cos(a+b*x)/(d^2*(c+d*x)^(3/2))-2/5*sin(a+b*x)/(d*(c+d*x)^(5/2))-8/15*b^(5/2)*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(2*Pi)/d^(7/2)+8/15*b^(5/2)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(2*Pi)/d^(7/2)+8/15*b^2*sin(a+b*x)/(d^3*sqrt(c+d*x))],
[(c+d*x)^(5/2)*sin(a+b*x)^2,x,10,-5/16*d*(c+d*x)^(3/2)/b^2+1/7*(c+d*x)^(7/2)/d-1/2*(c+d*x)^(5/2)*cos(a+b*x)*sin(a+b*x)/b+5/8*d*(c+d*x)^(3/2)*sin(a+b*x)^2/b^2-15/128*d^(5/2)*cos(2*a-2*b*c/d)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/b^(7/2)-15/128*d^(5/2)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/b^(7/2)+15/64*d^2*sin(2*a+2*b*x)*sqrt(c+d*x)/b^3],
[(c+d*x)^(3/2)*sin(a+b*x)^2,x,9,1/5*(c+d*x)^(5/2)/d-1/2*(c+d*x)^(3/2)*cos(a+b*x)*sin(a+b*x)/b+3/32*d^(3/2)*cos(2*a-2*b*c/d)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/b^(5/2)-3/32*d^(3/2)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/b^(5/2)-3/16*d*sqrt(c+d*x)/b^2+3/8*d*sin(a+b*x)^2*sqrt(c+d*x)/b^2],
[(c+d*x)^(1/2)*sin(a+b*x)^2,x,8,1/3*(c+d*x)^(3/2)/d+1/8*cos(2*a-2*b*c/d)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(d)*sqrt(Pi)/b^(3/2)+1/8*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(d)*sqrt(Pi)/b^(3/2)-1/4*sin(2*a+2*b*x)*sqrt(c+d*x)/b],
[sin(a+b*x)^2/(c+d*x)^(1/2),x,7,-1/2*cos(2*a-2*b*c/d)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/(sqrt(b)*sqrt(d))+1/2*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/(sqrt(b)*sqrt(d))+sqrt(c+d*x)/d],
[sin(a+b*x)^2/(c+d*x)^(3/2),x,7,2*cos(2*a-2*b*c/d)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(b)*sqrt(Pi)/d^(3/2)+2*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(b)*sqrt(Pi)/d^(3/2)-2*sin(a+b*x)^2/(d*sqrt(c+d*x))],
[sin(a+b*x)^2/(c+d*x)^(5/2),x,9,-2/3*sin(a+b*x)^2/(d*(c+d*x)^(3/2))+8/3*b^(3/2)*cos(2*a-2*b*c/d)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/d^(5/2)-8/3*b^(3/2)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/d^(5/2)-8/3*b*cos(a+b*x)*sin(a+b*x)/(d^2*sqrt(c+d*x))],
[sin(a+b*x)^2/(c+d*x)^(7/2),x,9,-8/15*b*cos(a+b*x)*sin(a+b*x)/(d^2*(c+d*x)^(3/2))-2/5*sin(a+b*x)^2/(d*(c+d*x)^(5/2))-32/15*b^(5/2)*cos(2*a-2*b*c/d)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/d^(7/2)-32/15*b^(5/2)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/d^(7/2)-16/15*b^2/(d^3*sqrt(c+d*x))+32/15*b^2*sin(a+b*x)^2/(d^3*sqrt(c+d*x))],
[sin(a+b*x)^2/(c+d*x)^(9/2),x,11,-16/105*b^2/(d^3*(c+d*x)^(3/2))-8/35*b*cos(a+b*x)*sin(a+b*x)/(d^2*(c+d*x)^(5/2))-2/7*sin(a+b*x)^2/(d*(c+d*x)^(7/2))+32/105*b^2*sin(a+b*x)^2/(d^3*(c+d*x)^(3/2))-128/105*b^(7/2)*cos(2*a-2*b*c/d)*FresnelC(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sqrt(Pi)/d^(9/2)+128/105*b^(7/2)*FresnelS(2*sqrt(b)*sqrt(c+d*x)/(sqrt(d)*sqrt(Pi)))*sin(2*a-2*b*c/d)*sqrt(Pi)/d^(9/2)+128/105*b^3*cos(a+b*x)*sin(a+b*x)/(d^4*sqrt(c+d*x))],
[(c+d*x)^(5/2)*sin(a+b*x)^3,x,23,-2/3*(c+d*x)^(5/2)*cos(a+b*x)/b+5/3*d*(c+d*x)^(3/2)*sin(a+b*x)/b^2-1/3*(c+d*x)^(5/2)*cos(a+b*x)*sin(a+b*x)^2/b+5/18*d*(c+d*x)^(3/2)*sin(a+b*x)^3/b^2+5/144*d^(5/2)*cos(3*a-3*b*c/d)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/6*Pi)/b^(7/2)-5/144*d^(5/2)*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(1/6*Pi)/b^(7/2)-45/16*d^(5/2)*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/2*Pi)/b^(7/2)+45/16*d^(5/2)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(1/2*Pi)/b^(7/2)+45/16*d^2*cos(a+b*x)*sqrt(c+d*x)/b^3-5/144*d^2*cos(3*a+3*b*x)*sqrt(c+d*x)/b^3],
[(c+d*x)^(3/2)*sin(a+b*x)^3,x,20,-2/3*(c+d*x)^(3/2)*cos(a+b*x)/b-1/3*(c+d*x)^(3/2)*cos(a+b*x)*sin(a+b*x)^2/b+1/24*d^(3/2)*cos(3*a-3*b*c/d)*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/6*Pi)/b^(5/2)+1/24*d^(3/2)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(1/6*Pi)/b^(5/2)-9/8*d^(3/2)*cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/2*Pi)/b^(5/2)-9/8*d^(3/2)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(1/2*Pi)/b^(5/2)+d*sin(a+b*x)*sqrt(c+d*x)/b^2+1/6*d*sin(a+b*x)^3*sqrt(c+d*x)/b^2],
[(c+d*x)^(1/2)*sin(a+b*x)^3,x,14,-1/12*cos(3*a-3*b*c/d)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(d)*sqrt(1/6*Pi)/b^(3/2)+1/12*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(d)*sqrt(1/6*Pi)/b^(3/2)+3/4*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(d)*sqrt(1/2*Pi)/b^(3/2)-3/4*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(d)*sqrt(1/2*Pi)/b^(3/2)-3/4*cos(a+b*x)*sqrt(c+d*x)/b+1/12*cos(3*a+3*b*x)*sqrt(c+d*x)/b],
[sin(a+b*x)^3/(c+d*x)^(1/2),x,12,-1/2*cos(3*a-3*b*c/d)*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/6*Pi)/(sqrt(b)*sqrt(d))-1/2*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(1/6*Pi)/(sqrt(b)*sqrt(d))+3/2*cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(1/2*Pi)/(sqrt(b)*sqrt(d))+3/2*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(1/2*Pi)/(sqrt(b)*sqrt(d))],
[sin(a+b*x)^3/(c+d*x)^(3/2),x,12,3*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(b)*sqrt(1/2*Pi)/d^(3/2)-3*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(b)*sqrt(1/2*Pi)/d^(3/2)-cos(3*a-3*b*c/d)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(b)*sqrt(3/2*Pi)/d^(3/2)+FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(b)*sqrt(3/2*Pi)/d^(3/2)-2*sin(a+b*x)^3/(d*sqrt(c+d*x))],
[sin(a+b*x)^3/(c+d*x)^(5/2),x,18,-2/3*sin(a+b*x)^3/(d*(c+d*x)^(3/2))-b^(3/2)*cos(a-b*c/d)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(2*Pi)/d^(5/2)-b^(3/2)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(2*Pi)/d^(5/2)+b^(3/2)*cos(3*a-3*b*c/d)*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(6*Pi)/d^(5/2)+b^(3/2)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(6*Pi)/d^(5/2)-4*b*cos(a+b*x)*sin(a+b*x)^2/(d^2*sqrt(c+d*x))],
[sin(a+b*x)^3/(c+d*x)^(7/2),x,19,-4/5*b*cos(a+b*x)*sin(a+b*x)^2/(d^2*(c+d*x)^(3/2))-2/5*sin(a+b*x)^3/(d*(c+d*x)^(5/2))-2/5*b^(5/2)*cos(a-b*c/d)*FresnelC(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(2*Pi)/d^(7/2)+2/5*b^(5/2)*FresnelS(sqrt(b)*sqrt(2/Pi)*sqrt(c+d*x)/sqrt(d))*sin(a-b*c/d)*sqrt(2*Pi)/d^(7/2)+6/5*b^(5/2)*cos(3*a-3*b*c/d)*FresnelC(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sqrt(6*Pi)/d^(7/2)-6/5*b^(5/2)*FresnelS(sqrt(b)*sqrt(6/Pi)*sqrt(c+d*x)/sqrt(d))*sin(3*a-3*b*c/d)*sqrt(6*Pi)/d^(7/2)-16/5*b^2*sin(a+b*x)/(d^3*sqrt(c+d*x))+24/5*b^2*sin(a+b*x)^3/(d^3*sqrt(c+d*x))],
[(d*x)^(3/2)*sin(f*x),x,4,-(d*x)^(3/2)*cos(f*x)/f-3/2*d^(3/2)*FresnelS(sqrt(f)*sqrt(2/Pi)*sqrt(d*x)/sqrt(d))*sqrt(1/2*Pi)/f^(5/2)+3/2*d*sin(f*x)*sqrt(d*x)/f^2],
[(d*x)^(1/2)*sin(f*x),x,3,FresnelC(sqrt(f)*sqrt(2/Pi)*sqrt(d*x)/sqrt(d))*sqrt(d)*sqrt(1/2*Pi)/f^(3/2)-cos(f*x)*sqrt(d*x)/f],
[sin(f*x)/(d*x)^(1/2),x,2,FresnelS(sqrt(f)*sqrt(2/Pi)*sqrt(d*x)/sqrt(d))*sqrt(2*Pi)/(sqrt(d)*sqrt(f))],
[sin(f*x)/(d*x)^(3/2),x,3,2*FresnelC(sqrt(f)*sqrt(2/Pi)*sqrt(d*x)/sqrt(d))*sqrt(f)*sqrt(2*Pi)/d^(3/2)-2*sin(f*x)/(d*sqrt(d*x))],
[sin(f*x)/(d*x)^(5/2),x,4,-2/3*sin(f*x)/(d*(d*x)^(3/2))-4/3*f^(3/2)*FresnelS(sqrt(f)*sqrt(2/Pi)*sqrt(d*x)/sqrt(d))*sqrt(2*Pi)/d^(5/2)-4/3*f*cos(f*x)/(d^2*sqrt(d*x))],

# n<0
[(c+d*x)^(1/2)*csc(a+b*x),x,0,Unintegrable(csc(a+b*x)*sqrt(c+d*x),x)],
[csc(a+b*x)/(c+d*x)^(1/2),x,0,Unintegrable(csc(a+b*x)/sqrt(c+d*x),x)],

# Integrands of the form (c+d x)^m (b Sin[e+f x])^(n/2)

# n>0

# n<0
[x/sin(e+f*x)^(3/2)+x*sqrt(sin(e+f*x)),x,2,-2*x*cos(e+f*x)/(f*sqrt(sin(e+f*x)))+4*sqrt(sin(e+f*x))/f^2],
[x^2/sin(e+f*x)^(3/2)+x^2*sqrt(sin(e+f*x)),x,3,-16*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/f^3-2*x^2*cos(e+f*x)/(f*sqrt(sin(e+f*x)))+8*x*sqrt(sin(e+f*x))/f^2],
[x/sin(e+f*x)^(5/2)-1/3*x/sqrt(sin(e+f*x)),x,2,-2/3*x*cos(e+f*x)/(f*sin(e+f*x)^(3/2))+(-4/3)/(f^2*sqrt(sin(e+f*x)))],
[x/sin(e+f*x)^(7/2)+3/5*x*sqrt(sin(e+f*x)),x,3,-2/5*x*cos(e+f*x)/(f*sin(e+f*x)^(5/2))+(-4/15)/(f^2*sin(e+f*x)^(3/2))-6/5*x*cos(e+f*x)/(f*sqrt(sin(e+f*x)))+12/5*sqrt(sin(e+f*x))/f^2],

# Integrands of the form (c+d x)^m (b Sin[e+f x])^n with m symbolic
[(c+d*x)^m*(b*sin(e+f*x))^n,x,0,Unintegrable((c+d*x)^m*(b*sin(e+f*x))^n,x)],
[(c+d*x)^m*sin(a+b*x)^3,x,8,-3/8*exp(I*(a-b*c/d))*(c+d*x)^m*GAMMA(1+m,-I*b*(c+d*x)/d)/(b*(-I*b*(c+d*x)/d)^m)-3/8*(c+d*x)^m*GAMMA(1+m,I*b*(c+d*x)/d)/(b*exp(I*(a-b*c/d))*(I*b*(c+d*x)/d)^m)+1/8*3^(-1-m)*exp(3*I*(a-b*c/d))*(c+d*x)^m*GAMMA(1+m,-3*I*b*(c+d*x)/d)/(b*(-I*b*(c+d*x)/d)^m)+1/8*3^(-1-m)*(c+d*x)^m*GAMMA(1+m,3*I*b*(c+d*x)/d)/(b*exp(3*I*(a-b*c/d))*(I*b*(c+d*x)/d)^m)],
[(c+d*x)^m*sin(a+b*x)^2,x,5,1/2*(c+d*x)^(1+m)/(d*(1+m))+I*2^(-3-m)*exp(2*I*(a-b*c/d))*(c+d*x)^m*GAMMA(1+m,-2*I*b*(c+d*x)/d)/(b*(-I*b*(c+d*x)/d)^m)-I*2^(-3-m)*(c+d*x)^m*GAMMA(1+m,2*I*b*(c+d*x)/d)/(b*exp(2*I*(a-b*c/d))*(I*b*(c+d*x)/d)^m)],
[(c+d*x)^m*sin(a+b*x),x,3,-1/2*exp(I*(a-b*c/d))*(c+d*x)^m*GAMMA(1+m,-I*b*(c+d*x)/d)/(b*(-I*b*(c+d*x)/d)^m)-1/2*(c+d*x)^m*GAMMA(1+m,I*b*(c+d*x)/d)/(b*exp(I*(a-b*c/d))*(I*b*(c+d*x)/d)^m)],
[(c+d*x)^m*csc(a+b*x),x,0,Unintegrable((c+d*x)^m*csc(a+b*x),x)],
[(c+d*x)^m*csc(a+b*x)^2,x,0,Unintegrable((c+d*x)^m*csc(a+b*x)^2,x)],
[x^(3+m)*sin(a+b*x),x,3,1/2*I*exp(I*a)*x^m*GAMMA(4+m,-I*b*x)/(b^4*(-I*b*x)^m)-1/2*I*x^m*GAMMA(4+m,I*b*x)/(b^4*exp(I*a)*(I*b*x)^m)],
[x^(2+m)*sin(a+b*x),x,3,1/2*exp(I*a)*x^m*GAMMA(3+m,-I*b*x)/(b^3*(-I*b*x)^m)+1/2*x^m*GAMMA(3+m,I*b*x)/(b^3*exp(I*a)*(I*b*x)^m)],
[x^(1+m)*sin(a+b*x),x,3,-1/2*I*exp(I*a)*x^m*GAMMA(2+m,-I*b*x)/(b^2*(-I*b*x)^m)+1/2*I*x^m*GAMMA(2+m,I*b*x)/(b^2*exp(I*a)*(I*b*x)^m)],
[x^m*sin(a+b*x),x,3,-1/2*exp(I*a)*x^m*GAMMA(1+m,-I*b*x)/(b*(-I*b*x)^m)-1/2*x^m*GAMMA(1+m,I*b*x)/(b*exp(I*a)*(I*b*x)^m)],
[x^(-1+m)*sin(a+b*x),x,3,1/2*I*exp(I*a)*x^m*GAMMA(m,-I*b*x)/(-I*b*x)^m-1/2*I*x^m*GAMMA(m,I*b*x)/(exp(I*a)*(I*b*x)^m)],
[x^(-2+m)*sin(a+b*x),x,3,1/2*b*exp(I*a)*x^m*GAMMA(-1+m,-I*b*x)/(-I*b*x)^m+1/2*b*x^m*GAMMA(-1+m,I*b*x)/(exp(I*a)*(I*b*x)^m)],
[x^(-3+m)*sin(a+b*x),x,3,-1/2*I*b^2*exp(I*a)*x^m*GAMMA(-2+m,-I*b*x)/(-I*b*x)^m+1/2*I*b^2*x^m*GAMMA(-2+m,I*b*x)/(exp(I*a)*(I*b*x)^m)],
[x^(3+m)*sin(a+b*x)^2,x,5,1/2*x^(4+m)/(4+m)+2^(-6-m)*exp(2*I*a)*x^m*GAMMA(4+m,-2*I*b*x)/(b^4*(-I*b*x)^m)+2^(-6-m)*x^m*GAMMA(4+m,2*I*b*x)/(b^4*exp(2*I*a)*(I*b*x)^m)],
[x^(2+m)*sin(a+b*x)^2,x,5,1/2*x^(3+m)/(3+m)-I*2^(-5-m)*exp(2*I*a)*x^m*GAMMA(3+m,-2*I*b*x)/(b^3*(-I*b*x)^m)+I*2^(-5-m)*x^m*GAMMA(3+m,2*I*b*x)/(b^3*exp(2*I*a)*(I*b*x)^m)],
[x^(1+m)*sin(a+b*x)^2,x,5,1/2*x^(2+m)/(2+m)-2^(-4-m)*exp(2*I*a)*x^m*GAMMA(2+m,-2*I*b*x)/(b^2*(-I*b*x)^m)-2^(-4-m)*x^m*GAMMA(2+m,2*I*b*x)/(b^2*exp(2*I*a)*(I*b*x)^m)],
[x^m*sin(a+b*x)^2,x,5,1/2*x^(1+m)/(1+m)+I*2^(-3-m)*exp(2*I*a)*x^m*GAMMA(1+m,-2*I*b*x)/(b*(-I*b*x)^m)-I*2^(-3-m)*x^m*GAMMA(1+m,2*I*b*x)/(b*exp(2*I*a)*(I*b*x)^m)],
[x^(-1+m)*sin(a+b*x)^2,x,5,1/2*x^m/m+2^(-2-m)*exp(2*I*a)*x^m*GAMMA(m,-2*I*b*x)/(-I*b*x)^m+2^(-2-m)*x^m*GAMMA(m,2*I*b*x)/(exp(2*I*a)*(I*b*x)^m)],
[x^(-2+m)*sin(a+b*x)^2,x,5,-1/2*x^(-1+m)/(1-m)-I*2^(-1-m)*b*exp(2*I*a)*x^m*GAMMA(-1+m,-2*I*b*x)/(-I*b*x)^m+I*2^(-1-m)*b*x^m*GAMMA(-1+m,2*I*b*x)/(exp(2*I*a)*(I*b*x)^m)],
[x^(-3+m)*sin(a+b*x)^2,x,5,-1/2*x^(-2+m)/(2-m)-b^2*exp(2*I*a)*x^m*GAMMA(-2+m,-2*I*b*x)/(2^m*(-I*b*x)^m)-b^2*x^m*GAMMA(-2+m,2*I*b*x)/(2^m*exp(2*I*a)*(I*b*x)^m)],

# Integrands of the form (c+d x)^m (b Csc[e+f x])^n

# Integrands of the form (c+d x)^m (b Csc[e+f x])^(n/2)

# n>0

# n<0
[x/csc(e+f*x)^(3/2)-1/3*x*sqrt(csc(e+f*x)),x,4,4/9/(f^2*csc(e+f*x)^(3/2))-2/3*x*cos(e+f*x)/(f*sqrt(csc(e+f*x)))],
[x^2/csc(e+f*x)^(3/2)-1/3*x^2*sqrt(csc(e+f*x)),x,7,8/9*x/(f^2*csc(e+f*x)^(3/2))+16/27*cos(e+f*x)/(f^3*sqrt(csc(e+f*x)))-2/3*x^2*cos(e+f*x)/(f*sqrt(csc(e+f*x)))-16/27*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(csc(e+f*x))*sqrt(sin(e+f*x))/f^3],
[x/csc(e+f*x)^(5/2)-3/5*x/sqrt(csc(e+f*x)),x,4,4/25/(f^2*csc(e+f*x)^(5/2))-2/5*x*cos(e+f*x)/(f*csc(e+f*x)^(3/2))],
[x/csc(e+f*x)^(7/2)-5/21*x*sqrt(csc(e+f*x)),x,5,4/49/(f^2*csc(e+f*x)^(7/2))-2/7*x*cos(e+f*x)/(f*csc(e+f*x)^(5/2))+20/63/(f^2*csc(e+f*x)^(3/2))-10/21*x*cos(e+f*x)/(f*sqrt(csc(e+f*x)))],

# Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n with a^2-b^2=0

# Integrands of the form (c+d x)^m (a+a Sin[e+f x])^n

# n>0
[(c+d*x)^3*(a+a*sin(e+f*x)),x,6,1/4*a*(c+d*x)^4/d+6*a*d^2*(c+d*x)*cos(e+f*x)/f^3-a*(c+d*x)^3*cos(e+f*x)/f-6*a*d^3*sin(e+f*x)/f^4+3*a*d*(c+d*x)^2*sin(e+f*x)/f^2],
[(c+d*x)^2*(a+a*sin(e+f*x)),x,5,1/3*a*(c+d*x)^3/d+2*a*d^2*cos(e+f*x)/f^3-a*(c+d*x)^2*cos(e+f*x)/f+2*a*d*(c+d*x)*sin(e+f*x)/f^2],
[(c+d*x)*(a+a*sin(e+f*x)),x,4,1/2*a*(c+d*x)^2/d-a*(c+d*x)*cos(e+f*x)/f+a*d*sin(e+f*x)/f^2],
[(a+a*sin(e+f*x))/(c+d*x),x,5,a*log(c+d*x)/d+a*cos(e-c*f/d)*Si(c*f/d+f*x)/d+a*Ci(c*f/d+f*x)*sin(e-c*f/d)/d],
[(a+a*sin(e+f*x))/(c+d*x)^2,x,6,-a/(d*(c+d*x))+a*f*Ci(c*f/d+f*x)*cos(e-c*f/d)/d^2-a*f*Si(c*f/d+f*x)*sin(e-c*f/d)/d^2-a*sin(e+f*x)/(d*(c+d*x))],
[(a+a*sin(e+f*x))/(c+d*x)^3,x,7,-1/2*a/(d*(c+d*x)^2)-1/2*a*f*cos(e+f*x)/(d^2*(c+d*x))-1/2*a*f^2*cos(e-c*f/d)*Si(c*f/d+f*x)/d^3-1/2*a*f^2*Ci(c*f/d+f*x)*sin(e-c*f/d)/d^3-1/2*a*sin(e+f*x)/(d*(c+d*x)^2)],
[(c+d*x)^3*(a+a*sin(e+f*x))^2,x,10,-3/4*a^2*c*d^2*x/f^2-3/8*a^2*d^3*x^2/f^2+3/8*a^2*(c+d*x)^4/d+12*a^2*d^2*(c+d*x)*cos(e+f*x)/f^3-2*a^2*(c+d*x)^3*cos(e+f*x)/f-12*a^2*d^3*sin(e+f*x)/f^4+6*a^2*d*(c+d*x)^2*sin(e+f*x)/f^2+3/4*a^2*d^2*(c+d*x)*cos(e+f*x)*sin(e+f*x)/f^3-1/2*a^2*(c+d*x)^3*cos(e+f*x)*sin(e+f*x)/f-3/8*a^2*d^3*sin(e+f*x)^2/f^4+3/4*a^2*d*(c+d*x)^2*sin(e+f*x)^2/f^2],
[(c+d*x)^2*(a+a*sin(e+f*x))^2,x,9,-1/4*a^2*d^2*x/f^2+1/2*a^2*(c+d*x)^3/d+4*a^2*d^2*cos(e+f*x)/f^3-2*a^2*(c+d*x)^2*cos(e+f*x)/f+4*a^2*d*(c+d*x)*sin(e+f*x)/f^2+1/4*a^2*d^2*cos(e+f*x)*sin(e+f*x)/f^3-1/2*a^2*(c+d*x)^2*cos(e+f*x)*sin(e+f*x)/f+1/2*a^2*d*(c+d*x)*sin(e+f*x)^2/f^2],
[(c+d*x)*(a+a*sin(e+f*x))^2,x,6,1/2*a^2*c*x+1/4*a^2*d*x^2+1/2*a^2*(c+d*x)^2/d-2*a^2*(c+d*x)*cos(e+f*x)/f+2*a^2*d*sin(e+f*x)/f^2-1/2*a^2*(c+d*x)*cos(e+f*x)*sin(e+f*x)/f+1/4*a^2*d*sin(e+f*x)^2/f^2],
[(a+a*sin(e+f*x))^2/(c+d*x),x,9,-1/2*a^2*Ci(2*c*f/d+2*f*x)*cos(2*e-2*c*f/d)/d+3/2*a^2*log(c+d*x)/d+2*a^2*cos(e-c*f/d)*Si(c*f/d+f*x)/d+1/2*a^2*Si(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d+2*a^2*Ci(c*f/d+f*x)*sin(e-c*f/d)/d],
[(a+a*sin(e+f*x))^2/(c+d*x)^2,x,9,2*a^2*f*Ci(c*f/d+f*x)*cos(e-c*f/d)/d^2+a^2*f*cos(2*e-2*c*f/d)*Si(2*c*f/d+2*f*x)/d^2+a^2*f*Ci(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d^2-2*a^2*f*Si(c*f/d+f*x)*sin(e-c*f/d)/d^2-4*a^2*sin(1/2*e+1/4*Pi+1/2*f*x)^4/(d*(c+d*x))],
[(a+a*sin(e+f*x))^2/(c+d*x)^3,x,15,a^2*f^2*Ci(2*c*f/d+2*f*x)*cos(2*e-2*c*f/d)/d^3-a^2*f^2*cos(e-c*f/d)*Si(c*f/d+f*x)/d^3-a^2*f^2*Si(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d^3-a^2*f^2*Ci(c*f/d+f*x)*sin(e-c*f/d)/d^3-4*a^2*f*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)^3/(d^2*(c+d*x))-2*a^2*sin(1/2*e+1/4*Pi+1/2*f*x)^4/(d*(c+d*x)^2)],

# n<0
[(c+d*x)^3/(a+a*sin(e+f*x)),x,7,-I*(c+d*x)^3/(a*f)-(c+d*x)^3*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f)+6*d*(c+d*x)^2*log(1-I*exp(I*(e+f*x)))/(a*f^2)-12*I*d^2*(c+d*x)*polylog(2,I*exp(I*(e+f*x)))/(a*f^3)+12*d^3*polylog(3,I*exp(I*(e+f*x)))/(a*f^4)],
[(c+d*x)^2/(a+a*sin(e+f*x)),x,6,-I*(c+d*x)^2/(a*f)-(c+d*x)^2*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f)+4*d*(c+d*x)*log(1-I*exp(I*(e+f*x)))/(a*f^2)-4*I*d^2*polylog(2,I*exp(I*(e+f*x)))/(a*f^3)],
[(c+d*x)/(a+a*sin(e+f*x)),x,3,-(c+d*x)*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f)+2*d*log(sin(1/2*e+1/4*Pi+1/2*f*x))/(a*f^2)],
[1/((c+d*x)*(a+a*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)*(a+a*sin(e+f*x))),x)],
[1/((c+d*x)^2*(a+a*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)^2*(a+a*sin(e+f*x))),x)],
[(c+d*x)^3/(a+a*sin(e+f*x))^2,x,10,-1/3*I*(c+d*x)^3/(a^2*f)-2*d^2*(c+d*x)*cot(1/2*e+1/4*Pi+1/2*f*x)/(a^2*f^3)-1/3*(c+d*x)^3*cot(1/2*e+1/4*Pi+1/2*f*x)/(a^2*f)-1/2*d*(c+d*x)^2*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f^2)-1/6*(c+d*x)^3*cot(1/2*e+1/4*Pi+1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f)+2*d*(c+d*x)^2*log(1-I*exp(I*(e+f*x)))/(a^2*f^2)+4*d^3*log(sin(1/2*e+1/4*Pi+1/2*f*x))/(a^2*f^4)-4*I*d^2*(c+d*x)*polylog(2,I*exp(I*(e+f*x)))/(a^2*f^3)+4*d^3*polylog(3,I*exp(I*(e+f*x)))/(a^2*f^4)],
[(c+d*x)^2/(a+a*sin(e+f*x))^2,x,9,-1/3*I*(c+d*x)^2/(a^2*f)-2/3*d^2*cot(1/2*e+1/4*Pi+1/2*f*x)/(a^2*f^3)-1/3*(c+d*x)^2*cot(1/2*e+1/4*Pi+1/2*f*x)/(a^2*f)-1/3*d*(c+d*x)*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f^2)-1/6*(c+d*x)^2*cot(1/2*e+1/4*Pi+1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f)+4/3*d*(c+d*x)*log(1-I*exp(I*(e+f*x)))/(a^2*f^2)-4/3*I*d^2*polylog(2,I*exp(I*(e+f*x)))/(a^2*f^3)],
[(c+d*x)/(a+a*sin(e+f*x))^2,x,4,-1/3*(c+d*x)*cot(1/2*e+1/4*Pi+1/2*f*x)/(a^2*f)-1/6*d*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f^2)-1/6*(c+d*x)*cot(1/2*e+1/4*Pi+1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)^2/(a^2*f)+2/3*d*log(sin(1/2*e+1/4*Pi+1/2*f*x))/(a^2*f^2)],
[1/((c+d*x)*(a+a*sin(e+f*x))^2),x,0,Unintegrable(1/((c+d*x)*(a+a*sin(e+f*x))^2),x)],
[1/((c+d*x)^2*(a+a*sin(e+f*x))^2),x,0,Unintegrable(1/((c+d*x)^2*(a+a*sin(e+f*x))^2),x)],
[(c+d*x)^3/(a-a*sin(e+f*x)),x,7,-I*(c+d*x)^3/(a*f)+6*d*(c+d*x)^2*log(1+I*exp(I*(e+f*x)))/(a*f^2)-12*I*d^2*(c+d*x)*polylog(2,-I*exp(I*(e+f*x)))/(a*f^3)+12*d^3*polylog(3,-I*exp(I*(e+f*x)))/(a*f^4)+(c+d*x)^3*tan(1/2*e+1/4*Pi+1/2*f*x)/(a*f)],
[(c+d*x)^2/(a-a*sin(e+f*x)),x,6,-I*(c+d*x)^2/(a*f)+4*d*(c+d*x)*log(1+I*exp(I*(e+f*x)))/(a*f^2)-4*I*d^2*polylog(2,-I*exp(I*(e+f*x)))/(a*f^3)+(c+d*x)^2*tan(1/2*e+1/4*Pi+1/2*f*x)/(a*f)],
[(c+d*x)/(a-a*sin(e+f*x)),x,3,2*d*log(cos(1/2*e+1/4*Pi+1/2*f*x))/(a*f^2)+(c+d*x)*tan(1/2*e+1/4*Pi+1/2*f*x)/(a*f)],
[1/((c+d*x)*(a-a*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)*(a-a*sin(e+f*x))),x)],
[1/((c+d*x)^2*(a-a*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)^2*(a-a*sin(e+f*x))),x)],

# Integrands of the form (c+d x)^m (a+a Sin[e+f x])^(n/2)

# n>0
[x^3*sqrt(a+a*sin(c+d*x)),x,5,-96*sqrt(a+a*sin(c+d*x))/d^4+12*x^2*sqrt(a+a*sin(c+d*x))/d^2+48*x*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/d^3-2*x^3*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/d],
[x^2*sqrt(a+a*sin(c+d*x)),x,4,8*x*sqrt(a+a*sin(c+d*x))/d^2+16*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/d^3-2*x^2*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/d],
[x*sqrt(a+a*sin(c+d*x)),x,3,4*sqrt(a+a*sin(c+d*x))/d^2-2*x*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/d],
[sqrt(a+a*sin(c+d*x))/x,x,4,cos(1/4*(2*c+Pi))*csc(1/2*c+1/4*Pi+1/2*d*x)*Si(1/2*d*x)*sqrt(a+a*sin(c+d*x))+Ci(1/2*d*x)*csc(1/2*c+1/4*Pi+1/2*d*x)*sin(1/4*(2*c+Pi))*sqrt(a+a*sin(c+d*x))],
[sqrt(a+a*sin(c+d*x))/x^2,x,5,-sqrt(a+a*sin(c+d*x))/x-1/2*d*Ci(1/2*d*x)*csc(1/2*c+1/4*Pi+1/2*d*x)*sin(1/4*(2*c-Pi))*sqrt(a+a*sin(c+d*x))-1/2*d*csc(1/2*c+1/4*Pi+1/2*d*x)*Si(1/2*d*x)*sin(1/4*(2*c+Pi))*sqrt(a+a*sin(c+d*x))],
[sqrt(a+a*sin(c+d*x))/x^3,x,6,-1/2*sqrt(a+a*sin(c+d*x))/x^2-1/4*d*cot(1/2*c+1/4*Pi+1/2*d*x)*sqrt(a+a*sin(c+d*x))/x-1/8*d^2*cos(1/4*(2*c+Pi))*csc(1/2*c+1/4*Pi+1/2*d*x)*Si(1/2*d*x)*sqrt(a+a*sin(c+d*x))-1/8*d^2*Ci(1/2*d*x)*csc(1/2*c+1/4*Pi+1/2*d*x)*sin(1/4*(2*c+Pi))*sqrt(a+a*sin(c+d*x))],
[x^3*(a+a*sin(e+f*x))^(3/2),x,9,-1280/9*a*sqrt(a+a*sin(e+f*x))/f^4+16*a*x^2*sqrt(a+a*sin(e+f*x))/f^2+640/9*a*x*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f^3-8/3*a*x^3*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f+32/9*a*x*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f^3-4/3*a*x^3*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f-64/27*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/f^4+8/3*a*x^2*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/f^2],
[x^2*(a+a*sin(e+f*x))^(3/2),x,7,32/3*a*x*sqrt(a+a*sin(e+f*x))/f^2+224/9*a*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f^3-8/3*a*x^2*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f-32/27*a*cos(1/2*e+1/4*Pi+1/2*f*x)^2*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f^3-4/3*a*x^2*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f+16/9*a*x*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/f^2],
[x*(a+a*sin(e+f*x))^(3/2),x,4,16/3*a*sqrt(a+a*sin(e+f*x))/f^2-8/3*a*x*cot(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f-4/3*a*x*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/f+8/9*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/f^2],
[(a+a*sin(e+f*x))^(3/2)/x,x,9,1/2*a*Ci(3/2*f*x)*cos(3/4*(2*e-Pi))*csc(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))+3/2*a*cos(1/4*(2*e+Pi))*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(1/2*f*x)*sqrt(a+a*sin(e+f*x))-1/2*a*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(3/2*f*x)*sin(3/4*(2*e-Pi))*sqrt(a+a*sin(e+f*x))+3/2*a*Ci(1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/4*(2*e+Pi))*sqrt(a+a*sin(e+f*x))],
[(a+a*sin(e+f*x))^(3/2)/x^2,x,9,3/4*a*f*cos(1/4*(6*e+Pi))*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(3/2*f*x)*sqrt(a+a*sin(e+f*x))-3/4*a*f*Ci(1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/4*(2*e-Pi))*sqrt(a+a*sin(e+f*x))-3/4*a*f*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(1/2*f*x)*sin(1/4*(2*e+Pi))*sqrt(a+a*sin(e+f*x))+3/4*a*f*Ci(3/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/4*(6*e+Pi))*sqrt(a+a*sin(e+f*x))-2*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/x],
[(a+a*sin(e+f*x))^(3/2)/x^3,x,13,-9/16*a*f^2*Ci(3/2*f*x)*cos(3/4*(2*e-Pi))*csc(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))-3/16*a*f^2*cos(1/4*(2*e+Pi))*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(1/2*f*x)*sqrt(a+a*sin(e+f*x))+9/16*a*f^2*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(3/2*f*x)*sin(3/4*(2*e-Pi))*sqrt(a+a*sin(e+f*x))-3/16*a*f^2*Ci(1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/4*(2*e+Pi))*sqrt(a+a*sin(e+f*x))-3/2*a*f*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2*f*x)*sqrt(a+a*sin(e+f*x))/x-a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*sqrt(a+a*sin(e+f*x))/x^2],

# n<0
[x^3/sqrt(a+a*sin(c+d*x)),x,10,-4*x^3*arctanh(exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d*sqrt(a+a*sin(c+d*x)))+12*I*x^2*polylog(2,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))-12*I*x^2*polylog(2,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))-48*x*polylog(3,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^3*sqrt(a+a*sin(c+d*x)))+48*x*polylog(3,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^3*sqrt(a+a*sin(c+d*x)))-96*I*polylog(4,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^4*sqrt(a+a*sin(c+d*x)))+96*I*polylog(4,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^4*sqrt(a+a*sin(c+d*x)))],
[x^2/sqrt(a+a*sin(c+d*x)),x,8,-4*x^2*arctanh(exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d*sqrt(a+a*sin(c+d*x)))+8*I*x*polylog(2,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))-8*I*x*polylog(2,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))-16*polylog(3,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^3*sqrt(a+a*sin(c+d*x)))+16*polylog(3,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^3*sqrt(a+a*sin(c+d*x)))],
[x/sqrt(a+a*sin(c+d*x)),x,6,-4*x*arctanh(exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d*sqrt(a+a*sin(c+d*x)))+4*I*polylog(2,-exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))-4*I*polylog(2,exp(1/4*I*(2*c+Pi+2*d*x)))*sin(1/2*c+1/4*Pi+1/2*d*x)/(d^2*sqrt(a+a*sin(c+d*x)))],
[1/(x*sqrt(a+a*sin(c+d*x))),x,0,Unintegrable(1/(x*sqrt(a+a*sin(c+d*x))),x)],
[1/(x^2*sqrt(a+a*sin(c+d*x))),x,0,Unintegrable(1/(x^2*sqrt(a+a*sin(c+d*x))),x)],
[x^3/(a+a*sin(e+f*x))^(3/2),x,16,-3*x^2/(a*f^2*sqrt(a+a*sin(e+f*x)))-1/2*x^3*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))-24*x*arctanh(exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))-x^3*arctanh(exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))+24*I*polylog(2,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^4*sqrt(a+a*sin(e+f*x)))+3*I*x^2*polylog(2,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))-24*I*polylog(2,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^4*sqrt(a+a*sin(e+f*x)))-3*I*x^2*polylog(2,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))-12*x*polylog(3,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))+12*x*polylog(3,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))-24*I*polylog(4,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^4*sqrt(a+a*sin(e+f*x)))+24*I*polylog(4,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^4*sqrt(a+a*sin(e+f*x)))],
[x^2/(a+a*sin(e+f*x))^(3/2),x,10,-2*x/(a*f^2*sqrt(a+a*sin(e+f*x)))-1/2*x^2*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))-x^2*arctanh(exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))-4*arctanh(cos(1/2*e+1/4*Pi+1/2*f*x))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))+2*I*x*polylog(2,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))-2*I*x*polylog(2,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))-4*polylog(3,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))+4*polylog(3,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^3*sqrt(a+a*sin(e+f*x)))],
[x/(a+a*sin(e+f*x))^(3/2),x,7,(-1)/(a*f^2*sqrt(a+a*sin(e+f*x)))-1/2*x*cot(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))-x*arctanh(exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f*sqrt(a+a*sin(e+f*x)))+I*polylog(2,-exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))-I*polylog(2,exp(1/4*I*(2*e+Pi+2*f*x)))*sin(1/2*e+1/4*Pi+1/2*f*x)/(a*f^2*sqrt(a+a*sin(e+f*x)))],
[1/(x*(a+a*sin(e+f*x))^(3/2)),x,0,Unintegrable(1/(x*(a+a*sin(e+f*x))^(3/2)),x)],
[1/(x^2*(a+a*sin(e+f*x))^(3/2)),x,0,Unintegrable(1/(x^2*(a+a*sin(e+f*x))^(3/2)),x)],

# Integrands of the form (c+d x)^m (a+a Sin[e+f x])^(n/3)

#  Used to hang Rubi 
[(a+a*sin(c+d*x))^(1/3)/x,x,0,Unintegrable((a+a*sin(c+d*x))^(1/3)/x,x)],

# Integrands of the form (c+d x)^m (a+a Sin[e+f x])^n with m symbolic
[(c+d*x)^m*(a+a*sin(e+f*x))^n,x,0,Unintegrable((c+d*x)^m*(a+a*sin(e+f*x))^n,x)],
[(c+d*x)^m*(a+a*sin(e+f*x))^3,x,12,5/2*a^3*(c+d*x)^(1+m)/(d*(1+m))-15/8*a^3*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-15/8*a^3*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+3*I*2^(-3-m)*a^3*exp(2*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-2*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-3*I*2^(-3-m)*a^3*(c+d*x)^m*GAMMA(1+m,2*I*f*(c+d*x)/d)/(exp(2*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+1/8*3^(-1-m)*a^3*exp(3*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-3*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)+1/8*3^(-1-m)*a^3*(c+d*x)^m*GAMMA(1+m,3*I*f*(c+d*x)/d)/(exp(3*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m*(a+a*sin(e+f*x))^2,x,9,3/2*a^2*(c+d*x)^(1+m)/(d*(1+m))-a^2*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-a^2*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+I*2^(-3-m)*a^2*exp(2*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-2*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-I*2^(-3-m)*a^2*(c+d*x)^m*GAMMA(1+m,2*I*f*(c+d*x)/d)/(exp(2*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m*(a+a*sin(e+f*x)),x,5,a*(c+d*x)^(1+m)/(d*(1+m))-1/2*a*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-1/2*a*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m/(a+a*sin(e+f*x)),x,0,Unintegrable((c+d*x)^m/(a+a*sin(e+f*x)),x)],
[(c+d*x)^m/(a+a*sin(e+f*x))^2,x,0,Unintegrable((c+d*x)^m/(a+a*sin(e+f*x))^2,x)],

# Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n

# Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n

# n>0
[(c+d*x)^3*(a+b*sin(e+f*x)),x,6,1/4*a*(c+d*x)^4/d+6*b*d^2*(c+d*x)*cos(e+f*x)/f^3-b*(c+d*x)^3*cos(e+f*x)/f-6*b*d^3*sin(e+f*x)/f^4+3*b*d*(c+d*x)^2*sin(e+f*x)/f^2],
[(c+d*x)^2*(a+b*sin(e+f*x)),x,5,1/3*a*(c+d*x)^3/d+2*b*d^2*cos(e+f*x)/f^3-b*(c+d*x)^2*cos(e+f*x)/f+2*b*d*(c+d*x)*sin(e+f*x)/f^2],
[(c+d*x)*(a+b*sin(e+f*x)),x,4,1/2*a*(c+d*x)^2/d-b*(c+d*x)*cos(e+f*x)/f+b*d*sin(e+f*x)/f^2],
[(a+b*sin(e+f*x))/(c+d*x),x,5,a*log(c+d*x)/d+b*cos(e-c*f/d)*Si(c*f/d+f*x)/d+b*Ci(c*f/d+f*x)*sin(e-c*f/d)/d],
[(a+b*sin(e+f*x))/(c+d*x)^2,x,6,-a/(d*(c+d*x))+b*f*Ci(c*f/d+f*x)*cos(e-c*f/d)/d^2-b*f*Si(c*f/d+f*x)*sin(e-c*f/d)/d^2-b*sin(e+f*x)/(d*(c+d*x))],
[(a+b*sin(e+f*x))/(c+d*x)^3,x,7,-1/2*a/(d*(c+d*x)^2)-1/2*b*f*cos(e+f*x)/(d^2*(c+d*x))-1/2*b*f^2*cos(e-c*f/d)*Si(c*f/d+f*x)/d^3-1/2*b*f^2*Ci(c*f/d+f*x)*sin(e-c*f/d)/d^3-1/2*b*sin(e+f*x)/(d*(c+d*x)^2)],
[(c+d*x)^3*(a+b*sin(e+f*x))^2,x,10,-3/4*b^2*c*d^2*x/f^2-3/8*b^2*d^3*x^2/f^2+1/4*a^2*(c+d*x)^4/d+1/8*b^2*(c+d*x)^4/d+12*a*b*d^2*(c+d*x)*cos(e+f*x)/f^3-2*a*b*(c+d*x)^3*cos(e+f*x)/f-12*a*b*d^3*sin(e+f*x)/f^4+6*a*b*d*(c+d*x)^2*sin(e+f*x)/f^2+3/4*b^2*d^2*(c+d*x)*cos(e+f*x)*sin(e+f*x)/f^3-1/2*b^2*(c+d*x)^3*cos(e+f*x)*sin(e+f*x)/f-3/8*b^2*d^3*sin(e+f*x)^2/f^4+3/4*b^2*d*(c+d*x)^2*sin(e+f*x)^2/f^2],
[(c+d*x)^2*(a+b*sin(e+f*x))^2,x,9,-1/4*b^2*d^2*x/f^2+1/3*a^2*(c+d*x)^3/d+1/6*b^2*(c+d*x)^3/d+4*a*b*d^2*cos(e+f*x)/f^3-2*a*b*(c+d*x)^2*cos(e+f*x)/f+4*a*b*d*(c+d*x)*sin(e+f*x)/f^2+1/4*b^2*d^2*cos(e+f*x)*sin(e+f*x)/f^3-1/2*b^2*(c+d*x)^2*cos(e+f*x)*sin(e+f*x)/f+1/2*b^2*d*(c+d*x)*sin(e+f*x)^2/f^2],
[(c+d*x)*(a+b*sin(e+f*x))^2,x,6,1/2*b^2*c*x+1/4*b^2*d*x^2+1/2*a^2*(c+d*x)^2/d-2*a*b*(c+d*x)*cos(e+f*x)/f+2*a*b*d*sin(e+f*x)/f^2-1/2*b^2*(c+d*x)*cos(e+f*x)*sin(e+f*x)/f+1/4*b^2*d*sin(e+f*x)^2/f^2],
[(a+b*sin(e+f*x))^2/(c+d*x),x,10,-1/2*b^2*Ci(2*c*f/d+2*f*x)*cos(2*e-2*c*f/d)/d+a^2*log(c+d*x)/d+1/2*b^2*log(c+d*x)/d+2*a*b*cos(e-c*f/d)*Si(c*f/d+f*x)/d+1/2*b^2*Si(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d+2*a*b*Ci(c*f/d+f*x)*sin(e-c*f/d)/d],
[(a+b*sin(e+f*x))^2/(c+d*x)^2,x,11,-a^2/(d*(c+d*x))+2*a*b*f*Ci(c*f/d+f*x)*cos(e-c*f/d)/d^2+b^2*f*cos(2*e-2*c*f/d)*Si(2*c*f/d+2*f*x)/d^2+b^2*f*Ci(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d^2-2*a*b*f*Si(c*f/d+f*x)*sin(e-c*f/d)/d^2-2*a*b*sin(e+f*x)/(d*(c+d*x))-b^2*sin(e+f*x)^2/(d*(c+d*x))],
[(a+b*sin(e+f*x))^2/(c+d*x)^3,x,14,-1/2*a^2/(d*(c+d*x)^2)+b^2*f^2*Ci(2*c*f/d+2*f*x)*cos(2*e-2*c*f/d)/d^3-a*b*f*cos(e+f*x)/(d^2*(c+d*x))-a*b*f^2*cos(e-c*f/d)*Si(c*f/d+f*x)/d^3-b^2*f^2*Si(2*c*f/d+2*f*x)*sin(2*e-2*c*f/d)/d^3-a*b*f^2*Ci(c*f/d+f*x)*sin(e-c*f/d)/d^3-a*b*sin(e+f*x)/(d*(c+d*x)^2)-b^2*f*cos(e+f*x)*sin(e+f*x)/(d^2*(c+d*x))-1/2*b^2*sin(e+f*x)^2/(d*(c+d*x)^2)],

# n<0
[(c+d*x)^3/(a+b*sin(e+f*x)),x,12,-I*(c+d*x)^3*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))+I*(c+d*x)^3*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))-3*d*(c+d*x)^2*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))+3*d*(c+d*x)^2*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))-6*I*d^2*(c+d*x)*polylog(3,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^3*sqrt(a^2-b^2))+6*I*d^2*(c+d*x)*polylog(3,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^3*sqrt(a^2-b^2))+6*d^3*polylog(4,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^4*sqrt(a^2-b^2))-6*d^3*polylog(4,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^4*sqrt(a^2-b^2))],
[(c+d*x)^2/(a+b*sin(e+f*x)),x,10,-I*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))+I*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))-2*d*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))+2*d*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))-2*I*d^2*polylog(3,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^3*sqrt(a^2-b^2))+2*I*d^2*polylog(3,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^3*sqrt(a^2-b^2))],
[(c+d*x)/(a+b*sin(e+f*x)),x,8,-I*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))+I*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f*sqrt(a^2-b^2))-d*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))+d*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/(f^2*sqrt(a^2-b^2))],
[1/((c+d*x)*(a+b*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)*(a+b*sin(e+f*x))),x)],
[1/((c+d*x)^2*(a+b*sin(e+f*x))),x,0,Unintegrable(1/((c+d*x)^2*(a+b*sin(e+f*x))),x)],
[(c+d*x)^3/(a+b*sin(e+f*x))^2,x,22,I*(c+d*x)^3/((a^2-b^2)*f)-3*d*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*f^2)-I*a*(c+d*x)^3*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)-3*d*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*f^2)+I*a*(c+d*x)^3*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)+6*I*d^2*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*f^3)-3*a*d*(c+d*x)^2*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)+6*I*d^2*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*f^3)+3*a*d*(c+d*x)^2*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)-6*d^3*polylog(3,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*f^4)-6*I*a*d^2*(c+d*x)*polylog(3,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^3)-6*d^3*polylog(3,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*f^4)+6*I*a*d^2*(c+d*x)*polylog(3,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^3)+6*a*d^3*polylog(4,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^4)-6*a*d^3*polylog(4,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^4)+b*(c+d*x)^3*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x)))],
[(c+d*x)^2/(a+b*sin(e+f*x))^2,x,18,I*(c+d*x)^2/((a^2-b^2)*f)-2*d*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*f^2)-I*a*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)-2*d*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*f^2)+I*a*(c+d*x)^2*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)+2*I*d^2*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*f^3)-2*a*d*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)+2*I*d^2*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*f^3)+2*a*d*(c+d*x)*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)-2*I*a*d^2*polylog(3,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^3)+2*I*a*d^2*polylog(3,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^3)+b*(c+d*x)^2*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x)))],
[(c+d*x)/(a+b*sin(e+f*x))^2,x,11,-d*log(a+b*sin(e+f*x))/((a^2-b^2)*f^2)-I*a*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)+I*a*(c+d*x)*log(1-I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f)-a*d*polylog(2,I*b*exp(I*(e+f*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)+a*d*polylog(2,I*b*exp(I*(e+f*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*f^2)+b*(c+d*x)*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x)))],
[1/((c+d*x)*(a+b*sin(e+f*x))^2),x,0,Unintegrable(1/((c+d*x)*(a+b*sin(e+f*x))^2),x)],
[1/((c+d*x)^2*(a+b*sin(e+f*x))^2),x,0,Unintegrable(1/((c+d*x)^2*(a+b*sin(e+f*x))^2),x)],

# Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n with m symbolic
[(c+d*x)^m*(a+b*sin(e+f*x))^n,x,0,Unintegrable((c+d*x)^m*(a+b*sin(e+f*x))^n,x)],
[(c+d*x)^m*(a+b*sin(e+f*x))^3,x,18,a^3*(c+d*x)^(1+m)/(d*(1+m))+3/2*a*b^2*(c+d*x)^(1+m)/(d*(1+m))-3/2*a^2*b*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-3/8*b^3*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-3/2*a^2*b*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)-3/8*b^3*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+3*I*2^(-3-m)*a*b^2*exp(2*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-2*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-3*I*2^(-3-m)*a*b^2*(c+d*x)^m*GAMMA(1+m,2*I*f*(c+d*x)/d)/(exp(2*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+1/8*3^(-1-m)*b^3*exp(3*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-3*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)+1/8*3^(-1-m)*b^3*(c+d*x)^m*GAMMA(1+m,3*I*f*(c+d*x)/d)/(exp(3*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m*(a+b*sin(e+f*x))^2,x,10,a^2*(c+d*x)^(1+m)/(d*(1+m))+1/2*b^2*(c+d*x)^(1+m)/(d*(1+m))-a*b*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-a*b*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)+I*2^(-3-m)*b^2*exp(2*I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-2*I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-I*2^(-3-m)*b^2*(c+d*x)^m*GAMMA(1+m,2*I*f*(c+d*x)/d)/(exp(2*I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m*(a+b*sin(e+f*x)),x,5,a*(c+d*x)^(1+m)/(d*(1+m))-1/2*b*exp(I*(e-c*f/d))*(c+d*x)^m*GAMMA(1+m,-I*f*(c+d*x)/d)/(f*(-I*f*(c+d*x)/d)^m)-1/2*b*(c+d*x)^m*GAMMA(1+m,I*f*(c+d*x)/d)/(exp(I*(e-c*f/d))*f*(I*f*(c+d*x)/d)^m)],
[(c+d*x)^m/(a+b*sin(e+f*x)),x,0,Unintegrable((c+d*x)^m/(a+b*sin(e+f*x)),x)],
[(c+d*x)^m/(a+b*sin(e+f*x))^2,x,0,Unintegrable((c+d*x)^m/(a+b*sin(e+f*x))^2,x)],

# Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p

# Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p with a^2-b^2=0

# Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+a Sin[c+d x])

# n>0
[(e+f*x)^3*sin(c+d*x)/(a+a*sin(c+d*x)),x,9,I*(e+f*x)^3/(a*d)+1/4*(e+f*x)^4/(a*f)+(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)+12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)],
[(e+f*x)^2*sin(c+d*x)/(a+a*sin(c+d*x)),x,8,I*(e+f*x)^2/(a*d)+1/3*(e+f*x)^3/(a*f)+(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)+4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)],
[(e+f*x)*sin(c+d*x)/(a+a*sin(c+d*x)),x,5,e*x/a+1/2*f*x^2/a+(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)],
[sin(c+d*x)/(a+a*sin(c+d*x)),x,2,x/a+cos(c+d*x)/(d*(a+a*sin(c+d*x)))],
[sin(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sin(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,14,-I*(e+f*x)^3/(a*d)-1/4*(e+f*x)^4/(a*f)+6*f^2*(e+f*x)*cos(c+d*x)/(a*d^3)-(e+f*x)^3*cos(c+d*x)/(a*d)-(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)-12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)+12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)-6*f^3*sin(c+d*x)/(a*d^4)+3*f*(e+f*x)^2*sin(c+d*x)/(a*d^2)],
[(e+f*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,12,-I*(e+f*x)^2/(a*d)-1/3*(e+f*x)^3/(a*f)+2*f^2*cos(c+d*x)/(a*d^3)-(e+f*x)^2*cos(c+d*x)/(a*d)-(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)-4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)+2*f*(e+f*x)*sin(c+d*x)/(a*d^2)],
[(e+f*x)*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,8,-e*x/a-1/2*f*x^2/a-(e+f*x)*cos(c+d*x)/(a*d)-(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)+f*sin(c+d*x)/(a*d^2)],
[sin(c+d*x)^2/(a+a*sin(c+d*x)),x,4,-x/a-cos(c+d*x)/(a*d)-cos(c+d*x)/(a*d*(1+sin(c+d*x)))],
[sin(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sin(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,19,-3/4*e*f^2*x/(a*d^2)-3/8*f^3*x^2/(a*d^2)+I*(e+f*x)^3/(a*d)+3/8*(e+f*x)^4/(a*f)-6*f^2*(e+f*x)*cos(c+d*x)/(a*d^3)+(e+f*x)^3*cos(c+d*x)/(a*d)+(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)+12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)+6*f^3*sin(c+d*x)/(a*d^4)-3*f*(e+f*x)^2*sin(c+d*x)/(a*d^2)+3/4*f^2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(a*d^3)-1/2*(e+f*x)^3*cos(c+d*x)*sin(c+d*x)/(a*d)-3/8*f^3*sin(c+d*x)^2/(a*d^4)+3/4*f*(e+f*x)^2*sin(c+d*x)^2/(a*d^2)],
[(e+f*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,17,-1/4*f^2*x/(a*d^2)+I*(e+f*x)^2/(a*d)+1/2*(e+f*x)^3/(a*f)-2*f^2*cos(c+d*x)/(a*d^3)+(e+f*x)^2*cos(c+d*x)/(a*d)+(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)+4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-2*f*(e+f*x)*sin(c+d*x)/(a*d^2)+1/4*f^2*cos(c+d*x)*sin(c+d*x)/(a*d^3)-1/2*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(a*d)+1/2*f*(e+f*x)*sin(c+d*x)^2/(a*d^2)],
[(e+f*x)*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,11,3/2*e*x/a+3/4*f*x^2/a+(e+f*x)*cos(c+d*x)/(a*d)+(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)-f*sin(c+d*x)/(a*d^2)-1/2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(a*d)+1/4*f*sin(c+d*x)^2/(a*d^2)],
[sin(c+d*x)^3/(a+a*sin(c+d*x)),x,2,3/2*x/a+2*cos(c+d*x)/(a*d)-3/2*cos(c+d*x)*sin(c+d*x)/(a*d)+cos(c+d*x)*sin(c+d*x)^2/(d*(a+a*sin(c+d*x)))],
[sin(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sin(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sin(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x)],

# n<0
[(e+f*x)^3*csc(c+d*x)/(a+a*sin(c+d*x)),x,17,I*(e+f*x)^3/(a*d)-2*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)+3*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)+12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-3*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)-6*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)-12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)+6*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)-6*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)+6*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)],
[(e+f*x)^2*csc(c+d*x)/(a+a*sin(c+d*x)),x,14,I*(e+f*x)^2/(a*d)-2*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)+2*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)+4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-2*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)-2*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+2*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)],
[(e+f*x)*csc(c+d*x)/(a+a*sin(c+d*x)),x,9,-2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)+I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)],
[csc(c+d*x)/(a+a*sin(c+d*x)),x,3,-arctanh(cos(c+d*x))/(a*d)+cos(c+d*x)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x)],
[csc(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,24,-2*I*(e+f*x)^3/(a*d)+2*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)-(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-(e+f*x)^3*cot(c+d*x)/(a*d)+6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)+3*f*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a*d^2)-3*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)+3*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)-3*I*f^2*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)+6*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)-6*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)+3/2*f^3*polylog(3,exp(2*I*(c+d*x)))/(a*d^4)+6*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)-6*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)],
[(e+f*x)^2*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,20,-2*I*(e+f*x)^2/(a*d)+2*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)-(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-(e+f*x)^2*cot(c+d*x)/(a*d)+4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)+2*f*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a*d^2)-2*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)+2*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)-I*f^2*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)+2*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)-2*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)],
[(e+f*x)*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,12,2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)-(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)-(e+f*x)*cot(c+d*x)/(a*d)+2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)+f*log(sin(c+d*x))/(a*d^2)-I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)+I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)],
[csc(c+d*x)^2/(a+a*sin(c+d*x)),x,5,arctanh(cos(c+d*x))/(a*d)-2*cot(c+d*x)/(a*d)+cot(c+d*x)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x)],
[csc(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,40,2*I*(e+f*x)^3/(a*d)-6*f^2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d^3)-3*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+(e+f*x)^3*cot(c+d*x)/(a*d)-3/2*f*(e+f*x)^2*csc(c+d*x)/(a*d^2)-1/2*(e+f*x)^3*cot(c+d*x)*csc(c+d*x)/(a*d)-6*f*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d^2)-3*f*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a*d^2)+3*I*f^3*polylog(2,-exp(I*(c+d*x)))/(a*d^4)+9/2*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)+12*I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-3*I*f^3*polylog(2,exp(I*(c+d*x)))/(a*d^4)-9/2*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)+3*I*f^2*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)-9*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)-12*f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)+9*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)-3/2*f^3*polylog(3,exp(2*I*(c+d*x)))/(a*d^4)-9*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)+9*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)],
[(e+f*x)^2*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,30,2*I*(e+f*x)^2/(a*d)-3*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)-f^2*arctanh(cos(c+d*x))/(a*d^3)+(e+f*x)^2*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+(e+f*x)^2*cot(c+d*x)/(a*d)-f*(e+f*x)*csc(c+d*x)/(a*d^2)-1/2*(e+f*x)^2*cot(c+d*x)*csc(c+d*x)/(a*d)-4*f*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d^2)-2*f*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a*d^2)+3*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)+4*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-3*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)+I*f^2*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)-3*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+3*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)],
[(e+f*x)*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,19,-3*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)*cot(1/2*c+1/4*Pi+1/2*d*x)/(a*d)+(e+f*x)*cot(c+d*x)/(a*d)-1/2*f*csc(c+d*x)/(a*d^2)-1/2*(e+f*x)*cot(c+d*x)*csc(c+d*x)/(a*d)-2*f*log(sin(1/2*c+1/4*Pi+1/2*d*x))/(a*d^2)-f*log(sin(c+d*x))/(a*d^2)+3/2*I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-3/2*I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)],
[csc(c+d*x)^3/(a+a*sin(c+d*x)),x,6,-3/2*arctanh(cos(c+d*x))/(a*d)+2*cot(c+d*x)/(a*d)-3/2*cot(c+d*x)*csc(c+d*x)/(a*d)+cot(c+d*x)*csc(c+d*x)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x)],
[csc(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(csc(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x)],

# Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+a Sin[c+d x]) with m symbolic
[(e+f*x)^m*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sin(c+d*x)^2/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m*sin(c+d*x)/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sin(c+d*x)/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m*csc(c+d*x)/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*csc(c+d*x)/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*csc(c+d*x)^2/(a+a*sin(c+d*x)),x)],

# Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p

# Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^1

# n>0
[(e+f*x)^3*sin(c+d*x)/(a+b*sin(c+d*x)),x,14,1/4*(e+f*x)^4/(b*f)+I*a*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-I*a*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+3*a*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-3*a*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+6*I*a*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))-6*I*a*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))-6*a*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))+6*a*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))],
[(e+f*x)^2*sin(c+d*x)/(a+b*sin(c+d*x)),x,12,1/3*(e+f*x)^3/(b*f)+I*a*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-I*a*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+2*a*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-2*a*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+2*I*a*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))-2*I*a*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))],
[(e+f*x)*sin(c+d*x)/(a+b*sin(c+d*x)),x,10,e*x/b+1/2*f*x^2/b+I*a*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-I*a*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+a*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-a*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))],
[sin(c+d*x)/(a+b*sin(c+d*x)),x,4,x/b-2*a*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*d*sqrt(a^2-b^2))],
[(e+f*x)^3*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,19,-1/4*a*(e+f*x)^4/(b^2*f)+6*f^2*(e+f*x)*cos(c+d*x)/(b*d^3)-(e+f*x)^3*cos(c+d*x)/(b*d)-6*f^3*sin(c+d*x)/(b*d^4)+3*f*(e+f*x)^2*sin(c+d*x)/(b*d^2)-I*a^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))+I*a^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))-3*a^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))+3*a^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))-6*I*a^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^3*sqrt(a^2-b^2))+6*I*a^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^3*sqrt(a^2-b^2))+6*a^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^4*sqrt(a^2-b^2))-6*a^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^4*sqrt(a^2-b^2))],
[(e+f*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,16,-1/3*a*(e+f*x)^3/(b^2*f)+2*f^2*cos(c+d*x)/(b*d^3)-(e+f*x)^2*cos(c+d*x)/(b*d)+2*f*(e+f*x)*sin(c+d*x)/(b*d^2)-I*a^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))+I*a^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))-2*a^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))+2*a^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))-2*I*a^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^3*sqrt(a^2-b^2))+2*I*a^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^3*sqrt(a^2-b^2))],
[(e+f*x)*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,13,-a*e*x/b^2-1/2*a*f*x^2/b^2-(e+f*x)*cos(c+d*x)/(b*d)+f*sin(c+d*x)/(b*d^2)-I*a^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))+I*a^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d*sqrt(a^2-b^2))-a^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))+a^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^2*d^2*sqrt(a^2-b^2))],
[sin(c+d*x)^2/(a+b*sin(c+d*x)),x,6,-a*x/b^2-cos(c+d*x)/(b*d)+2*a^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*d*sqrt(a^2-b^2))],
[(e+f*x)^3*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,24,-3/4*e*f^2*x/(b*d^2)-3/8*f^3*x^2/(b*d^2)+1/4*a^2*(e+f*x)^4/(b^3*f)+1/8*(e+f*x)^4/(b*f)-6*a*f^2*(e+f*x)*cos(c+d*x)/(b^2*d^3)+a*(e+f*x)^3*cos(c+d*x)/(b^2*d)+6*a*f^3*sin(c+d*x)/(b^2*d^4)-3*a*f*(e+f*x)^2*sin(c+d*x)/(b^2*d^2)+3/4*f^2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(b*d^3)-1/2*(e+f*x)^3*cos(c+d*x)*sin(c+d*x)/(b*d)-3/8*f^3*sin(c+d*x)^2/(b*d^4)+3/4*f*(e+f*x)^2*sin(c+d*x)^2/(b*d^2)+I*a^3*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))-I*a^3*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))+3*a^3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))-3*a^3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))+6*I*a^3*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^3*sqrt(a^2-b^2))-6*I*a^3*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^3*sqrt(a^2-b^2))-6*a^3*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^4*sqrt(a^2-b^2))+6*a^3*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^4*sqrt(a^2-b^2))],
[(e+f*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,21,-1/4*f^2*x/(b*d^2)+1/3*a^2*(e+f*x)^3/(b^3*f)+1/6*(e+f*x)^3/(b*f)-2*a*f^2*cos(c+d*x)/(b^2*d^3)+a*(e+f*x)^2*cos(c+d*x)/(b^2*d)-2*a*f*(e+f*x)*sin(c+d*x)/(b^2*d^2)+1/4*f^2*cos(c+d*x)*sin(c+d*x)/(b*d^3)-1/2*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(b*d)+1/2*f*(e+f*x)*sin(c+d*x)^2/(b*d^2)+I*a^3*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))-I*a^3*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))+2*a^3*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))-2*a^3*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))+2*I*a^3*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^3*sqrt(a^2-b^2))-2*I*a^3*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^3*sqrt(a^2-b^2))],
[(e+f*x)*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,16,a^2*e*x/b^3+1/2*e*x/b+1/2*a^2*f*x^2/b^3+1/4*f*x^2/b+a*(e+f*x)*cos(c+d*x)/(b^2*d)-a*f*sin(c+d*x)/(b^2*d^2)-1/2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(b*d)+1/4*f*sin(c+d*x)^2/(b*d^2)+I*a^3*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))-I*a^3*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d*sqrt(a^2-b^2))+a^3*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))-a^3*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2*sqrt(a^2-b^2))],
[sin(c+d*x)^3/(a+b*sin(c+d*x)),x,6,1/2*(2*a^2+b^2)*x/b^3+a*cos(c+d*x)/(b^2*d)-1/2*cos(c+d*x)*sin(c+d*x)/(b*d)-2*a^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^3*d*sqrt(a^2-b^2))],

# n<0
[(e+f*x)^3*csc(c+d*x)/(a+b*sin(c+d*x)),x,22,-2*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)+3*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-3*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)-6*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+6*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)-6*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)+6*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)+I*b*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))-I*b*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))+3*b*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))-3*b*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))+6*I*b*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^3*sqrt(a^2-b^2))-6*I*b*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^3*sqrt(a^2-b^2))-6*b*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^4*sqrt(a^2-b^2))+6*b*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^4*sqrt(a^2-b^2))],
[(e+f*x)^2*csc(c+d*x)/(a+b*sin(c+d*x)),x,18,-2*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)+2*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-2*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)-2*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+2*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)+I*b*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))-I*b*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))+2*b*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))-2*b*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))+2*I*b*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^3*sqrt(a^2-b^2))-2*I*b*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^3*sqrt(a^2-b^2))],
[(e+f*x)*csc(c+d*x)/(a+b*sin(c+d*x)),x,14,-2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)+I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)+I*b*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))-I*b*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d*sqrt(a^2-b^2))+b*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))-b*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*d^2*sqrt(a^2-b^2))],
[csc(c+d*x)/(a+b*sin(c+d*x)),x,5,-arctanh(cos(c+d*x))/(a*d)-2*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*d*sqrt(a^2-b^2))],
[(e+f*x)^3*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,29,-I*(e+f*x)^3/(a*d)+2*b*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a^2*d)-(e+f*x)^3*cot(c+d*x)/(a*d)+3*f*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a*d^2)-3*I*b*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+3*I*b*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-3*I*f^2*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)+6*b*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a^2*d^3)-6*b*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a^2*d^3)+3/2*f^3*polylog(3,exp(2*I*(c+d*x)))/(a*d^4)+6*I*b*f^3*polylog(4,-exp(I*(c+d*x)))/(a^2*d^4)-6*I*b*f^3*polylog(4,exp(I*(c+d*x)))/(a^2*d^4)-I*b^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))+I*b^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))-3*b^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))+3*b^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))-6*I*b^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^3*sqrt(a^2-b^2))+6*I*b^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^3*sqrt(a^2-b^2))+6*b^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^4*sqrt(a^2-b^2))-6*b^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^4*sqrt(a^2-b^2))],
[(e+f*x)^2*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,24,-I*(e+f*x)^2/(a*d)+2*b*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a^2*d)-(e+f*x)^2*cot(c+d*x)/(a*d)+2*f*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a*d^2)-2*I*b*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+2*I*b*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-I*f^2*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)+2*b*f^2*polylog(3,-exp(I*(c+d*x)))/(a^2*d^3)-2*b*f^2*polylog(3,exp(I*(c+d*x)))/(a^2*d^3)-I*b^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))+I*b^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))-2*b^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))+2*b^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))-2*I*b^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^3*sqrt(a^2-b^2))+2*I*b^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^3*sqrt(a^2-b^2))],
[(e+f*x)*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,17,2*b*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a^2*d)-(e+f*x)*cot(c+d*x)/(a*d)+f*log(sin(c+d*x))/(a*d^2)-I*b*f*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+I*b*f*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-I*b^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))+I*b^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d*sqrt(a^2-b^2))-b^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))+b^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*d^2*sqrt(a^2-b^2))],
[csc(c+d*x)^2/(a+b*sin(c+d*x)),x,7,b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)+2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*d*sqrt(a^2-b^2))],

# m symbolic
[(e+f*x)^m*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sin(c+d*x)^2/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*sin(c+d*x)/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sin(c+d*x)/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*csc(c+d*x)/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*csc(c+d*x)/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*csc(c+d*x)^2/(a+b*sin(c+d*x)),x)],

# Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^2
[(e+f*x)*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,21,a*f*log(a+b*sin(c+d*x))/(b*(a^2-b^2)*d^2)+I*a^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-I*a^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+a^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-a^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-a*(e+f*x)*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))-I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))],
[(e+f*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,30,-I*a*(e+f*x)^2/(b*(a^2-b^2)*d)+2*a*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)+I*a^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+2*a*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)-I*a^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-2*I*a*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)+2*a^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-2*I*a*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)-2*a^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+2*I*a^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-2*I*a^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-a*(e+f*x)^2*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))-I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))+2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))],
[(e+f*x)^3*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,36,-I*a*(e+f*x)^3/(b*(a^2-b^2)*d)+3*a*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)+I*a^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+3*a*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)-I*a^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-6*I*a*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)+3*a^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-6*I*a*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)-3*a^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+6*a*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)+6*I*a^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+6*a*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)-6*I*a^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-6*a^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)+6*a^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)-a*(e+f*x)^3*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))-I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))+I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d*sqrt(a^2-b^2))-3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))+6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))+6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))-6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))],

# Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^3
[(e+f*x)*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,48,3/2*a^2*f*log(a+b*sin(c+d*x))/(b*(a^2-b^2)^2*d^2)-f*log(a+b*sin(c+d*x))/(b*(a^2-b^2)*d^2)+3/2*I*a^3*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)-3/2*I*a*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-3/2*I*a^3*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)+3/2*I*a*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+3/2*a^3*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)-3/2*a*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-3/2*a^3*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)+3/2*a*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-1/2*a*(e+f*x)*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-1/2*a*f/(b*(a^2-b^2)*d^2*(a+b*sin(c+d*x)))-3/2*a^2*(e+f*x)*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))+(e+f*x)*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[(e+f*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,73,-3/2*I*a^2*(e+f*x)^2/(b*(a^2-b^2)^2*d)+I*(e+f*x)^2/(b*(a^2-b^2)*d)+2*a*f^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(3/2)*d^3)+3*a^2*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^2)-2*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)+3/2*I*a^3*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)-3/2*I*a*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+3*a^2*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^2)-2*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)-3/2*I*a^3*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)+3/2*I*a*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-3*I*a^2*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^3)+2*I*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)+3*a^3*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)-3*a*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-3*I*a^2*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^3)+2*I*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)-3*a^3*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)+3*a*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+3*I*a^3*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^3)-3*I*a*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-3*I*a^3*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^3)+3*I*a*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-1/2*a*(e+f*x)^2*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-a*f*(e+f*x)/(b*(a^2-b^2)*d^2*(a+b*sin(c+d*x)))-3/2*a^2*(e+f*x)^2*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))+(e+f*x)^2*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[(e+f*x)^3*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,92,-3/2*I*a^2*(e+f*x)^3/(b*(a^2-b^2)^2*d)+I*(e+f*x)^3/(b*(a^2-b^2)*d)-3*I*a*f^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+9/2*a^2*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^2)-3*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)+3/2*I*a^3*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)-3/2*I*a*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)+3*I*a*f^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+9/2*a^2*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^2)-3*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^2)-3/2*I*a^3*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d)+3/2*I*a*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d)-3*a*f^3*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)-9*I*a^2*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^3)+6*I*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)+9/2*a^3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)-9/2*a*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+3*a*f^3*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)-9*I*a^2*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^3)+6*I*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)-9/2*a^3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^2)+9/2*a*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+9*a^2*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^4)-6*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)+9*I*a^3*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^3)-9*I*a*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+9*a^2*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^2*d^4)-6*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)-9*I*a^3*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^3)+9*I*a*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-9*a^3*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^4)+9*a*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)+9*a^3*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(5/2)*d^4)-9*a*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)-1/2*a*(e+f*x)^3*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-3/2*a*f*(e+f*x)^2/(b*(a^2-b^2)*d^2*(a+b*sin(c+d*x)))-3/2*a^2*(e+f*x)^3*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))+(e+f*x)^3*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))],

# Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p

# Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p with a^2-b^2=0

# Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+a Sin[c+d x])

# n>0
[(e+f*x)^3*cos(c+d*x)/(a+a*sin(c+d*x)),x,6,-1/4*I*(e+f*x)^4/(a*f)+2*(e+f*x)^3*log(1-I*exp(I*(c+d*x)))/(a*d)-6*I*f*(e+f*x)^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)+12*f^2*(e+f*x)*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)+12*I*f^3*polylog(4,I*exp(I*(c+d*x)))/(a*d^4)],
[(e+f*x)^2*cos(c+d*x)/(a+a*sin(c+d*x)),x,5,-1/3*I*(e+f*x)^3/(a*f)+2*(e+f*x)^2*log(1-I*exp(I*(c+d*x)))/(a*d)-4*I*f*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)+4*f^2*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)],
[(e+f*x)*cos(c+d*x)/(a+a*sin(c+d*x)),x,4,-1/2*I*(e+f*x)^2/(a*f)+2*(e+f*x)*log(1-I*exp(I*(c+d*x)))/(a*d)-2*I*f*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)],
[cos(c+d*x)/(a+a*sin(c+d*x)),x,2,log(1+sin(c+d*x))/(a*d)],
[cos(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(cos(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x)],
[cos(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(cos(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*cos(c+d*x)^2/(a+a*sin(c+d*x)),x,6,1/4*(e+f*x)^4/(a*f)-6*f^2*(e+f*x)*cos(c+d*x)/(a*d^3)+(e+f*x)^3*cos(c+d*x)/(a*d)+6*f^3*sin(c+d*x)/(a*d^4)-3*f*(e+f*x)^2*sin(c+d*x)/(a*d^2)],
[(e+f*x)^2*cos(c+d*x)^2/(a+a*sin(c+d*x)),x,5,1/3*(e+f*x)^3/(a*f)-2*f^2*cos(c+d*x)/(a*d^3)+(e+f*x)^2*cos(c+d*x)/(a*d)-2*f*(e+f*x)*sin(c+d*x)/(a*d^2)],
[(e+f*x)*cos(c+d*x)^2/(a+a*sin(c+d*x)),x,4,e*x/a+1/2*f*x^2/a+(e+f*x)*cos(c+d*x)/(a*d)-f*sin(c+d*x)/(a*d^2)],
[cos(c+d*x)^2/(a+a*sin(c+d*x)),x,2,x/a+cos(c+d*x)/(a*d)],
[cos(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x,5,log(e+f*x)/(a*f)-cos(c-d*e/f)*Si(d*e/f+d*x)/(a*f)-Ci(d*e/f+d*x)*sin(c-d*e/f)/(a*f)],
[cos(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x,6,(-1)/(a*f*(e+f*x))-d*Ci(d*e/f+d*x)*cos(c-d*e/f)/(a*f^2)+d*Si(d*e/f+d*x)*sin(c-d*e/f)/(a*f^2)+sin(c+d*x)/(a*f*(e+f*x))],
[(e+f*x)^3*cos(c+d*x)^3/(a+a*sin(c+d*x)),x,10,-3/8*f^3*x/(a*d^3)+1/4*(e+f*x)^3/(a*d)-6*f^3*cos(c+d*x)/(a*d^4)+3*f*(e+f*x)^2*cos(c+d*x)/(a*d^2)-6*f^2*(e+f*x)*sin(c+d*x)/(a*d^3)+(e+f*x)^3*sin(c+d*x)/(a*d)+3/8*f^3*cos(c+d*x)*sin(c+d*x)/(a*d^4)-3/4*f*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(a*d^2)+3/4*f^2*(e+f*x)*sin(c+d*x)^2/(a*d^3)-1/2*(e+f*x)^3*sin(c+d*x)^2/(a*d)],
[(e+f*x)^2*cos(c+d*x)^3/(a+a*sin(c+d*x)),x,7,1/2*e*f*x/(a*d)+1/4*f^2*x^2/(a*d)+2*f*(e+f*x)*cos(c+d*x)/(a*d^2)-2*f^2*sin(c+d*x)/(a*d^3)+(e+f*x)^2*sin(c+d*x)/(a*d)-1/2*f*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(a*d^2)+1/4*f^2*sin(c+d*x)^2/(a*d^3)-1/2*(e+f*x)^2*sin(c+d*x)^2/(a*d)],
[(e+f*x)*cos(c+d*x)^3/(a+a*sin(c+d*x)),x,6,1/4*f*x/(a*d)+f*cos(c+d*x)/(a*d^2)+(e+f*x)*sin(c+d*x)/(a*d)-1/4*f*cos(c+d*x)*sin(c+d*x)/(a*d^2)-1/2*(e+f*x)*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^3/(a+a*sin(c+d*x)),x,2,sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x,9,Ci(d*e/f+d*x)*cos(c-d*e/f)/(a*f)-1/2*cos(2*c-2*d*e/f)*Si(2*d*e/f+2*d*x)/(a*f)-1/2*Ci(2*d*e/f+2*d*x)*sin(2*c-2*d*e/f)/(a*f)-Si(d*e/f+d*x)*sin(c-d*e/f)/(a*f)],
[cos(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x,11,-d*Ci(2*d*e/f+2*d*x)*cos(2*c-2*d*e/f)/(a*f^2)-cos(c+d*x)/(a*f*(e+f*x))-d*cos(c-d*e/f)*Si(d*e/f+d*x)/(a*f^2)+d*Si(2*d*e/f+2*d*x)*sin(2*c-2*d*e/f)/(a*f^2)-d*Ci(d*e/f+d*x)*sin(c-d*e/f)/(a*f^2)+1/2*sin(2*c+2*d*x)/(a*f*(e+f*x))],

# n<0
[(e+f*x)^3*sec(c+d*x)/(a+a*sin(c+d*x)),x,22,-3/2*I*f*(e+f*x)^2/(a*d^2)-6*I*f^2*(e+f*x)*arctan(exp(I*(c+d*x)))/(a*d^3)-I*(e+f*x)^3*arctan(exp(I*(c+d*x)))/(a*d)+3*f^2*(e+f*x)*log(1+exp(2*I*(c+d*x)))/(a*d^3)+3*I*f^3*polylog(2,-I*exp(I*(c+d*x)))/(a*d^4)+3/2*I*f*(e+f*x)^2*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-3*I*f^3*polylog(2,I*exp(I*(c+d*x)))/(a*d^4)-3/2*I*f*(e+f*x)^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-3/2*I*f^3*polylog(2,-exp(2*I*(c+d*x)))/(a*d^4)-3*f^2*(e+f*x)*polylog(3,-I*exp(I*(c+d*x)))/(a*d^3)+3*f^2*(e+f*x)*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)-3*I*f^3*polylog(4,-I*exp(I*(c+d*x)))/(a*d^4)+3*I*f^3*polylog(4,I*exp(I*(c+d*x)))/(a*d^4)-3/2*f*(e+f*x)^2*sec(c+d*x)/(a*d^2)-1/2*(e+f*x)^3*sec(c+d*x)^2/(a*d)+3/2*f*(e+f*x)^2*tan(c+d*x)/(a*d^2)+1/2*(e+f*x)^3*sec(c+d*x)*tan(c+d*x)/(a*d)],
[(e+f*x)^2*sec(c+d*x)/(a+a*sin(c+d*x)),x,13,-I*(e+f*x)^2*arctan(exp(I*(c+d*x)))/(a*d)+f^2*arctanh(sin(c+d*x))/(a*d^3)+f^2*log(cos(c+d*x))/(a*d^3)+I*f*(e+f*x)*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-I*f*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-f^2*polylog(3,-I*exp(I*(c+d*x)))/(a*d^3)+f^2*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)-f*(e+f*x)*sec(c+d*x)/(a*d^2)-1/2*(e+f*x)^2*sec(c+d*x)^2/(a*d)+f*(e+f*x)*tan(c+d*x)/(a*d^2)+1/2*(e+f*x)^2*sec(c+d*x)*tan(c+d*x)/(a*d)],
[(e+f*x)*sec(c+d*x)/(a+a*sin(c+d*x)),x,10,-I*(e+f*x)*arctan(exp(I*(c+d*x)))/(a*d)+1/2*I*f*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-1/2*I*f*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-1/2*f*sec(c+d*x)/(a*d^2)-1/2*(e+f*x)*sec(c+d*x)^2/(a*d)+1/2*f*tan(c+d*x)/(a*d^2)+1/2*(e+f*x)*sec(c+d*x)*tan(c+d*x)/(a*d)],
[sec(c+d*x)/(a+a*sin(c+d*x)),x,4,1/2*arctanh(sin(c+d*x))/(a*d)+(-1/2)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sec(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,20,-2/3*I*(e+f*x)^3/(a*d)-I*f*(e+f*x)^2*arctan(exp(I*(c+d*x)))/(a*d^2)+f^3*arctanh(sin(c+d*x))/(a*d^4)+2*f*(e+f*x)^2*log(1+exp(2*I*(c+d*x)))/(a*d^2)+f^3*log(cos(c+d*x))/(a*d^4)+I*f^2*(e+f*x)*polylog(2,-I*exp(I*(c+d*x)))/(a*d^3)-I*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-2*I*f^2*(e+f*x)*polylog(2,-exp(2*I*(c+d*x)))/(a*d^3)-f^3*polylog(3,-I*exp(I*(c+d*x)))/(a*d^4)+f^3*polylog(3,I*exp(I*(c+d*x)))/(a*d^4)+f^3*polylog(3,-exp(2*I*(c+d*x)))/(a*d^4)-f^2*(e+f*x)*sec(c+d*x)/(a*d^3)-1/2*f*(e+f*x)^2*sec(c+d*x)^2/(a*d^2)-1/3*(e+f*x)^3*sec(c+d*x)^3/(a*d)+f^2*(e+f*x)*tan(c+d*x)/(a*d^3)+2/3*(e+f*x)^3*tan(c+d*x)/(a*d)+1/2*f*(e+f*x)^2*sec(c+d*x)*tan(c+d*x)/(a*d^2)+1/3*(e+f*x)^3*sec(c+d*x)^2*tan(c+d*x)/(a*d)],
[(e+f*x)^2*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,16,-2/3*I*(e+f*x)^2/(a*d)-2/3*I*f*(e+f*x)*arctan(exp(I*(c+d*x)))/(a*d^2)+4/3*f*(e+f*x)*log(1+exp(2*I*(c+d*x)))/(a*d^2)+1/3*I*f^2*polylog(2,-I*exp(I*(c+d*x)))/(a*d^3)-1/3*I*f^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^3)-2/3*I*f^2*polylog(2,-exp(2*I*(c+d*x)))/(a*d^3)-1/3*f^2*sec(c+d*x)/(a*d^3)-1/3*f*(e+f*x)*sec(c+d*x)^2/(a*d^2)-1/3*(e+f*x)^2*sec(c+d*x)^3/(a*d)+1/3*f^2*tan(c+d*x)/(a*d^3)+2/3*(e+f*x)^2*tan(c+d*x)/(a*d)+1/3*f*(e+f*x)*sec(c+d*x)*tan(c+d*x)/(a*d^2)+1/3*(e+f*x)^2*sec(c+d*x)^2*tan(c+d*x)/(a*d)],
[(e+f*x)*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,7,1/6*f*arctanh(sin(c+d*x))/(a*d^2)+2/3*f*log(cos(c+d*x))/(a*d^2)-1/6*f*sec(c+d*x)^2/(a*d^2)-1/3*(e+f*x)*sec(c+d*x)^3/(a*d)+2/3*(e+f*x)*tan(c+d*x)/(a*d)+1/6*f*sec(c+d*x)*tan(c+d*x)/(a*d^2)+1/3*(e+f*x)*sec(c+d*x)^2*tan(c+d*x)/(a*d)],
[sec(c+d*x)^2/(a+a*sin(c+d*x)),x,3,-1/3*sec(c+d*x)/(d*(a+a*sin(c+d*x)))+2/3*tan(c+d*x)/(a*d)],
[sec(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)^2/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sec(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)^2/((e+f*x)^2*(a+a*sin(c+d*x))),x)],
[(e+f*x)^3*sec(c+d*x)^3/(a+a*sin(c+d*x)),x,32,-1/2*I*f*(e+f*x)^2/(a*d^2)-5*I*f^2*(e+f*x)*arctan(exp(I*(c+d*x)))/(a*d^3)-3/4*I*(e+f*x)^3*arctan(exp(I*(c+d*x)))/(a*d)+f^2*(e+f*x)*log(1+exp(2*I*(c+d*x)))/(a*d^3)+5/2*I*f^3*polylog(2,-I*exp(I*(c+d*x)))/(a*d^4)+9/8*I*f*(e+f*x)^2*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-5/2*I*f^3*polylog(2,I*exp(I*(c+d*x)))/(a*d^4)-9/8*I*f*(e+f*x)^2*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-1/2*I*f^3*polylog(2,-exp(2*I*(c+d*x)))/(a*d^4)-9/4*f^2*(e+f*x)*polylog(3,-I*exp(I*(c+d*x)))/(a*d^3)+9/4*f^2*(e+f*x)*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)-9/4*I*f^3*polylog(4,-I*exp(I*(c+d*x)))/(a*d^4)+9/4*I*f^3*polylog(4,I*exp(I*(c+d*x)))/(a*d^4)-1/4*f^3*sec(c+d*x)/(a*d^4)-9/8*f*(e+f*x)^2*sec(c+d*x)/(a*d^2)-1/4*f^2*(e+f*x)*sec(c+d*x)^2/(a*d^3)-1/4*f*(e+f*x)^2*sec(c+d*x)^3/(a*d^2)-1/4*(e+f*x)^3*sec(c+d*x)^4/(a*d)+1/4*f^3*tan(c+d*x)/(a*d^4)+1/2*f*(e+f*x)^2*tan(c+d*x)/(a*d^2)+1/4*f^2*(e+f*x)*sec(c+d*x)*tan(c+d*x)/(a*d^3)+3/8*(e+f*x)^3*sec(c+d*x)*tan(c+d*x)/(a*d)+1/4*f*(e+f*x)^2*sec(c+d*x)^2*tan(c+d*x)/(a*d^2)+1/4*(e+f*x)^3*sec(c+d*x)^3*tan(c+d*x)/(a*d)],
[(e+f*x)^2*sec(c+d*x)^3/(a+a*sin(c+d*x)),x,17,-3/4*I*(e+f*x)^2*arctan(exp(I*(c+d*x)))/(a*d)+5/6*f^2*arctanh(sin(c+d*x))/(a*d^3)+1/3*f^2*log(cos(c+d*x))/(a*d^3)+3/4*I*f*(e+f*x)*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-3/4*I*f*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-3/4*f^2*polylog(3,-I*exp(I*(c+d*x)))/(a*d^3)+3/4*f^2*polylog(3,I*exp(I*(c+d*x)))/(a*d^3)-3/4*f*(e+f*x)*sec(c+d*x)/(a*d^2)-1/12*f^2*sec(c+d*x)^2/(a*d^3)-1/6*f*(e+f*x)*sec(c+d*x)^3/(a*d^2)-1/4*(e+f*x)^2*sec(c+d*x)^4/(a*d)+1/3*f*(e+f*x)*tan(c+d*x)/(a*d^2)+1/12*f^2*sec(c+d*x)*tan(c+d*x)/(a*d^3)+3/8*(e+f*x)^2*sec(c+d*x)*tan(c+d*x)/(a*d)+1/6*f*(e+f*x)*sec(c+d*x)^2*tan(c+d*x)/(a*d^2)+1/4*(e+f*x)^2*sec(c+d*x)^3*tan(c+d*x)/(a*d)],
[(e+f*x)*sec(c+d*x)^3/(a+a*sin(c+d*x)),x,11,-3/4*I*(e+f*x)*arctan(exp(I*(c+d*x)))/(a*d)+3/8*I*f*polylog(2,-I*exp(I*(c+d*x)))/(a*d^2)-3/8*I*f*polylog(2,I*exp(I*(c+d*x)))/(a*d^2)-3/8*f*sec(c+d*x)/(a*d^2)-1/12*f*sec(c+d*x)^3/(a*d^2)-1/4*(e+f*x)*sec(c+d*x)^4/(a*d)+1/4*f*tan(c+d*x)/(a*d^2)+3/8*(e+f*x)*sec(c+d*x)*tan(c+d*x)/(a*d)+1/4*(e+f*x)*sec(c+d*x)^3*tan(c+d*x)/(a*d)+1/12*f*tan(c+d*x)^3/(a*d^2)],
[sec(c+d*x)^3/(a+a*sin(c+d*x)),x,4,3/8*arctanh(sin(c+d*x))/(a*d)+1/8/(d*(a-a*sin(c+d*x)))-1/8*a/(d*(a+a*sin(c+d*x))^2)+(-1/4)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)^3/((e+f*x)*(a+a*sin(c+d*x))),x)],
[sec(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x,0,Unintegrable(sec(c+d*x)^3/((e+f*x)^2*(a+a*sin(c+d*x))),x)],

# Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+a Sin[c+d x]) with m symbolic
[(e+f*x)^m*cos(c+d*x)^4/(a+a*sin(c+d*x)),x,14,1/2*(e+f*x)^(1+m)/(a*f*(1+m))+1/8*exp(I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+1/8*(e+f*x)^m*GAMMA(1+m,I*d*(e+f*x)/f)/(a*d*exp(I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)-I*2^(-3-m)*exp(2*I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-2*I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+I*2^(-3-m)*(e+f*x)^m*GAMMA(1+m,2*I*d*(e+f*x)/f)/(a*d*exp(2*I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)+1/8*3^(-1-m)*exp(3*I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-3*I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+1/8*3^(-1-m)*(e+f*x)^m*GAMMA(1+m,3*I*d*(e+f*x)/f)/(a*d*exp(3*I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)],
[(e+f*x)^m*cos(c+d*x)^3/(a+a*sin(c+d*x)),x,9,-1/2*I*exp(I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+1/2*I*(e+f*x)^m*GAMMA(1+m,I*d*(e+f*x)/f)/(a*d*exp(I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)+2^(-3-m)*exp(2*I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-2*I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+2^(-3-m)*(e+f*x)^m*GAMMA(1+m,2*I*d*(e+f*x)/f)/(a*d*exp(2*I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)],
[(e+f*x)^m*cos(c+d*x)^2/(a+a*sin(c+d*x)),x,5,(e+f*x)^(1+m)/(a*f*(1+m))+1/2*exp(I*(c-d*e/f))*(e+f*x)^m*GAMMA(1+m,-I*d*(e+f*x)/f)/(a*d*(-I*d*(e+f*x)/f)^m)+1/2*(e+f*x)^m*GAMMA(1+m,I*d*(e+f*x)/f)/(a*d*exp(I*(c-d*e/f))*(I*d*(e+f*x)/f)^m)],
[(e+f*x)^m*cos(c+d*x)/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*cos(c+d*x)/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m*sec(c+d*x)/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sec(c+d*x)/(a+a*sin(c+d*x)),x)],
[(e+f*x)^m*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sec(c+d*x)^2/(a+a*sin(c+d*x)),x)],

# Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p

# Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^1

# n>0
[(e+f*x)^3*cos(c+d*x)/(a+b*sin(c+d*x)),x,11,-1/4*I*(e+f*x)^4/(b*f)+(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d)+(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d)-3*I*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2)-3*I*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2)+6*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3)+6*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3)+6*I*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^4)+6*I*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^4)],
[(e+f*x)^2*cos(c+d*x)/(a+b*sin(c+d*x)),x,9,-1/3*I*(e+f*x)^3/(b*f)+(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d)+(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d)-2*I*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2)-2*I*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2)+2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3)+2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3)],
[(e+f*x)*cos(c+d*x)/(a+b*sin(c+d*x)),x,7,-1/2*I*(e+f*x)^2/(b*f)+(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d)+(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d)-I*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2)-I*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2)],
[cos(c+d*x)/(a+b*sin(c+d*x)),x,2,log(a+b*sin(c+d*x))/(b*d)],
[(e+f*x)^3*cos(c+d*x)^2/(a+b*sin(c+d*x)),x,18,1/4*a*(e+f*x)^4/(b^2*f)-6*f^2*(e+f*x)*cos(c+d*x)/(b*d^3)+(e+f*x)^3*cos(c+d*x)/(b*d)+6*f^3*sin(c+d*x)/(b*d^4)-3*f*(e+f*x)^2*sin(c+d*x)/(b*d^2)+I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)-I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)+3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)-3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)+6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^3)-6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^3)-6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^4)+6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^4)],
[(e+f*x)^2*cos(c+d*x)^2/(a+b*sin(c+d*x)),x,15,1/3*a*(e+f*x)^3/(b^2*f)-2*f^2*cos(c+d*x)/(b*d^3)+(e+f*x)^2*cos(c+d*x)/(b*d)-2*f*(e+f*x)*sin(c+d*x)/(b*d^2)+I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)-I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)+2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)-2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)+2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^3)-2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^3)],
[(e+f*x)*cos(c+d*x)^2/(a+b*sin(c+d*x)),x,12,a*e*x/b^2+1/2*a*f*x^2/b^2+(e+f*x)*cos(c+d*x)/(b*d)-f*sin(c+d*x)/(b*d^2)+I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)-I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d)+f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)-f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(b^2*d^2)],
[cos(c+d*x)^2/(a+b*sin(c+d*x)),x,5,a*x/b^2+cos(c+d*x)/(b*d)-2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^2*d)],
[(e+f*x)^3*cos(c+d*x)^3/(a+b*sin(c+d*x)),x,21,-3/8*f^3*x/(b*d^3)+1/4*(e+f*x)^3/(b*d)+1/4*I*(a^2-b^2)*(e+f*x)^4/(b^3*f)-6*a*f^3*cos(c+d*x)/(b^2*d^4)+3*a*f*(e+f*x)^2*cos(c+d*x)/(b^2*d^2)-(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d)-(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d)+3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2)+3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2)-6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^3)-6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^3)-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^4)-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^4)-6*a*f^2*(e+f*x)*sin(c+d*x)/(b^2*d^3)+a*(e+f*x)^3*sin(c+d*x)/(b^2*d)+3/8*f^3*cos(c+d*x)*sin(c+d*x)/(b*d^4)-3/4*f*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(b*d^2)+3/4*f^2*(e+f*x)*sin(c+d*x)^2/(b*d^3)-1/2*(e+f*x)^3*sin(c+d*x)^2/(b*d)],
[(e+f*x)^2*cos(c+d*x)^3/(a+b*sin(c+d*x)),x,16,1/2*e*f*x/(b*d)+1/4*f^2*x^2/(b*d)+1/3*I*(a^2-b^2)*(e+f*x)^3/(b^3*f)+2*a*f*(e+f*x)*cos(c+d*x)/(b^2*d^2)-(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d)-(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d)+2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2)+2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2)-2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^3)-2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^3)-2*a*f^2*sin(c+d*x)/(b^2*d^3)+a*(e+f*x)^2*sin(c+d*x)/(b^2*d)-1/2*f*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(b*d^2)+1/4*f^2*sin(c+d*x)^2/(b*d^3)-1/2*(e+f*x)^2*sin(c+d*x)^2/(b*d)],
[(e+f*x)*cos(c+d*x)^3/(a+b*sin(c+d*x)),x,13,1/4*f*x/(b*d)+1/2*I*(a^2-b^2)*(e+f*x)^2/(b^3*f)+a*f*cos(c+d*x)/(b^2*d^2)-(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d)-(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d)+I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b^3*d^2)+I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b^3*d^2)+a*(e+f*x)*sin(c+d*x)/(b^2*d)-1/4*f*cos(c+d*x)*sin(c+d*x)/(b*d^2)-1/2*(e+f*x)*sin(c+d*x)^2/(b*d)],
[cos(c+d*x)^3/(a+b*sin(c+d*x)),x,3,-(a^2-b^2)*log(a+b*sin(c+d*x))/(b^3*d)+a*sin(c+d*x)/(b^2*d)-1/2*sin(c+d*x)^2/(b*d)],

# n<0
[(e+f*x)^3*sec(c+d*x)/(a+b*sin(c+d*x)),x,29,-2*I*a*(e+f*x)^3*arctan(exp(I*(c+d*x)))/((a^2-b^2)*d)+b*(e+f*x)^3*log(1+exp(2*I*(c+d*x)))/((a^2-b^2)*d)-b*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d)-b*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d)+3*I*a*f*(e+f*x)^2*polylog(2,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-3*I*a*f*(e+f*x)^2*polylog(2,I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-3/2*I*b*f*(e+f*x)^2*polylog(2,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^2)+3*I*b*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^2)+3*I*b*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^2)-6*a*f^2*(e+f*x)*polylog(3,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)+6*a*f^2*(e+f*x)*polylog(3,I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)+3/2*b*f^2*(e+f*x)*polylog(3,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^3)-6*b*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^3)-6*b*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^3)-6*I*a*f^3*polylog(4,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^4)+6*I*a*f^3*polylog(4,I*exp(I*(c+d*x)))/((a^2-b^2)*d^4)+3/4*I*b*f^3*polylog(4,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^4)-6*I*b*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^4)-6*I*b*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^4)],
[(e+f*x)^2*sec(c+d*x)/(a+b*sin(c+d*x)),x,24,-2*I*a*(e+f*x)^2*arctan(exp(I*(c+d*x)))/((a^2-b^2)*d)+b*(e+f*x)^2*log(1+exp(2*I*(c+d*x)))/((a^2-b^2)*d)-b*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d)-b*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d)+2*I*a*f*(e+f*x)*polylog(2,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-2*I*a*f*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-I*b*f*(e+f*x)*polylog(2,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^2)+2*I*b*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^2)+2*I*b*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^2)-2*a*f^2*polylog(3,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)+2*a*f^2*polylog(3,I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)+1/2*b*f^2*polylog(3,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^3)-2*b*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^3)-2*b*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^3)],
[(e+f*x)*sec(c+d*x)/(a+b*sin(c+d*x)),x,19,-2*I*a*(e+f*x)*arctan(exp(I*(c+d*x)))/((a^2-b^2)*d)+b*(e+f*x)*log(1+exp(2*I*(c+d*x)))/((a^2-b^2)*d)-b*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d)-b*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d)+I*a*f*polylog(2,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-I*a*f*polylog(2,I*exp(I*(c+d*x)))/((a^2-b^2)*d^2)-1/2*I*b*f*polylog(2,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^2)+I*b*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)*d^2)+I*b*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)*d^2)],
[sec(c+d*x)/(a+b*sin(c+d*x)),x,6,-1/2*log(1-sin(c+d*x))/((a+b)*d)+1/2*log(1+sin(c+d*x))/((a-b)*d)-b*log(a+b*sin(c+d*x))/((a^2-b^2)*d)],
[(e+f*x)^3*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,29,-I*a*(e+f*x)^3/((a^2-b^2)*d)-6*I*b*f*(e+f*x)^2*arctan(exp(I*(c+d*x)))/((a^2-b^2)*d^2)+3*a*f*(e+f*x)^2*log(1+exp(2*I*(c+d*x)))/((a^2-b^2)*d^2)+I*b^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)-I*b^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)+6*I*b*f^2*(e+f*x)*polylog(2,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)-6*I*b*f^2*(e+f*x)*polylog(2,I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)-3*I*a*f^2*(e+f*x)*polylog(2,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^3)+3*b^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)-3*b^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)-6*b*f^3*polylog(3,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^4)+6*b*f^3*polylog(3,I*exp(I*(c+d*x)))/((a^2-b^2)*d^4)+3/2*a*f^3*polylog(3,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^4)+6*I*b^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^3)-6*I*b^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^3)-6*b^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^4)+6*b^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^4)-b*(e+f*x)^3*sec(c+d*x)/((a^2-b^2)*d)+a*(e+f*x)^3*tan(c+d*x)/((a^2-b^2)*d)],
[(e+f*x)^2*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,24,-I*a*(e+f*x)^2/((a^2-b^2)*d)-4*I*b*f*(e+f*x)*arctan(exp(I*(c+d*x)))/((a^2-b^2)*d^2)+2*a*f*(e+f*x)*log(1+exp(2*I*(c+d*x)))/((a^2-b^2)*d^2)+I*b^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)-I*b^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)+2*I*b*f^2*polylog(2,-I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)-2*I*b*f^2*polylog(2,I*exp(I*(c+d*x)))/((a^2-b^2)*d^3)-I*a*f^2*polylog(2,-exp(2*I*(c+d*x)))/((a^2-b^2)*d^3)+2*b^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)-2*b^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)+2*I*b^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^3)-2*I*b^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^3)-b*(e+f*x)^2*sec(c+d*x)/((a^2-b^2)*d)+a*(e+f*x)^2*tan(c+d*x)/((a^2-b^2)*d)],
[(e+f*x)*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,15,b*f*arctanh(sin(c+d*x))/((a^2-b^2)*d^2)+a*f*log(cos(c+d*x))/((a^2-b^2)*d^2)+I*b^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)-I*b^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d)+b^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)-b^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/((a^2-b^2)^(3/2)*d^2)-b*(e+f*x)*sec(c+d*x)/((a^2-b^2)*d)+a*(e+f*x)*tan(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^2/(a+b*sin(c+d*x)),x,5,-2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*d)-sec(c+d*x)*(b-a*sin(c+d*x))/((a^2-b^2)*d)],

# m symbolic
[(e+f*x)^m*cos(c+d*x)^2/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*cos(c+d*x)^2/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*cos(c+d*x)/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*cos(c+d*x)/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*sec(c+d*x)/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sec(c+d*x)/(a+b*sin(c+d*x)),x)],
[(e+f*x)^m*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,0,Unintegrable((e+f*x)^m*sec(c+d*x)^2/(a+b*sin(c+d*x)),x)],

# Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^2
[(e+f*x)*cos(c+d*x)/(a+b*sin(c+d*x))^2,x,4,(-e-f*x)/(b*d*(a+b*sin(c+d*x)))+2*f*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*d^2*sqrt(a^2-b^2))],
[(e+f*x)^2*cos(c+d*x)/(a+b*sin(c+d*x))^2,x,9,-(e+f*x)^2/(b*d*(a+b*sin(c+d*x)))-2*I*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+2*I*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-2*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))+2*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))],
[(e+f*x)^3*cos(c+d*x)/(a+b*sin(c+d*x))^2,x,11,-(e+f*x)^3/(b*d*(a+b*sin(c+d*x)))-3*I*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))+3*I*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^2*sqrt(a^2-b^2))-6*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))+6*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^3*sqrt(a^2-b^2))-6*I*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))+6*I*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*d^4*sqrt(a^2-b^2))],

# Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^3
[(e+f*x)*cos(c+d*x)/(a+b*sin(c+d*x))^3,x,6,a*f*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(3/2)*d^2)+1/2*(-e-f*x)/(b*d*(a+b*sin(c+d*x))^2)+1/2*f*cos(c+d*x)/((a^2-b^2)*d^2*(a+b*sin(c+d*x)))],
[(e+f*x)^2*cos(c+d*x)/(a+b*sin(c+d*x))^3,x,12,-f^2*log(a+b*sin(c+d*x))/(b*(a^2-b^2)*d^3)-I*a*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+I*a*f*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-a*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+a*f^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-1/2*(e+f*x)^2/(b*d*(a+b*sin(c+d*x))^2)+f*(e+f*x)*cos(c+d*x)/((a^2-b^2)*d^2*(a+b*sin(c+d*x)))],
[(e+f*x)^3*cos(c+d*x)/(a+b*sin(c+d*x))^3,x,19,3/2*I*f*(e+f*x)^2/(b*(a^2-b^2)*d^2)-3*f^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)-3/2*I*a*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)-3*f^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^3)+3/2*I*a*f*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^2)+3*I*f^3*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)-3*a*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)+3*I*f^3*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)*d^4)+3*a*f^2*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^3)-3*I*a*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)+3*I*a*f^3*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(b*(a^2-b^2)^(3/2)*d^4)-1/2*(e+f*x)^3/(b*d*(a+b*sin(c+d*x))^2)+3/2*f*(e+f*x)^2*cos(c+d*x)/((a^2-b^2)*d^2*(a+b*sin(c+d*x)))],

# Integrands of the form (e+f x)^m Cos[c+d x]^n Sin[c+d x]^p (a+b Sin[c+d x])^q

# Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^p / (a+b Sin[c+d x])^1

# Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^1 / (a+b Sin[c+d x])
[(e+f*x)^3*cos(c+d*x)*cot(c+d*x)/(a+b*sin(c+d*x)),x,33,-1/4*(e+f*x)^4/(b*f)-2*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)+3*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-3*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)-6*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+6*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)-6*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)+6*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)-I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)+I*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)-3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)+3*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)-6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^3)+6*I*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^3)+6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^4)-6*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^4)],
[(e+f*x)^2*cos(c+d*x)*cot(c+d*x)/(a+b*sin(c+d*x)),x,27,-1/3*(e+f*x)^3/(b*f)-2*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)+2*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-2*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)-2*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+2*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)-I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)+I*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)-2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)+2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)-2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^3)+2*I*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^3)],
[(e+f*x)*cos(c+d*x)*cot(c+d*x)/(a+b*sin(c+d*x)),x,21,-e*x/b-1/2*f*x^2/b-2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)+I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)-I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)+I*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d)-f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)+f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))*sqrt(a^2-b^2)/(a*b*d^2)],
[cos(c+d*x)*cot(c+d*x)/(a+b*sin(c+d*x)),x,6,-x/b-arctanh(cos(c+d*x))/(a*d)+2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a*b*d)],
[(e+f*x)^3*cos(c+d*x)^2*cot(c+d*x)/(a+b*sin(c+d*x)),x,34,-1/4*I*(e+f*x)^4/(a*f)-1/4*I*(a^2-b^2)*(e+f*x)^4/(a*b^2*f)+6*f^3*cos(c+d*x)/(b*d^4)-3*f*(e+f*x)^2*cos(c+d*x)/(b*d^2)+(e+f*x)^3*log(1-exp(2*I*(c+d*x)))/(a*d)+(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d)+(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d)-3/2*I*f*(e+f*x)^2*polylog(2,exp(2*I*(c+d*x)))/(a*d^2)-3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^2)-3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^2)+3/2*f^2*(e+f*x)*polylog(3,exp(2*I*(c+d*x)))/(a*d^3)+6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^3)+6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^3)+3/4*I*f^3*polylog(4,exp(2*I*(c+d*x)))/(a*d^4)+6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^4)+6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^4)+6*f^2*(e+f*x)*sin(c+d*x)/(b*d^3)-(e+f*x)^3*sin(c+d*x)/(b*d)],
[(e+f*x)^2*cos(c+d*x)^2*cot(c+d*x)/(a+b*sin(c+d*x)),x,26,-1/3*I*(e+f*x)^3/(a*f)-1/3*I*(a^2-b^2)*(e+f*x)^3/(a*b^2*f)-2*f*(e+f*x)*cos(c+d*x)/(b*d^2)+(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a*d)+(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d)+(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d)-I*f*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a*d^2)-2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^2)-2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^2)+1/2*f^2*polylog(3,exp(2*I*(c+d*x)))/(a*d^3)+2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^3)+2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^3)+2*f^2*sin(c+d*x)/(b*d^3)-(e+f*x)^2*sin(c+d*x)/(b*d)],
[(e+f*x)*cos(c+d*x)^2*cot(c+d*x)/(a+b*sin(c+d*x)),x,22,-1/2*I*(e+f*x)^2/(a*f)-1/2*I*(a^2-b^2)*(e+f*x)^2/(a*b^2*f)-f*cos(c+d*x)/(b*d^2)+(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a*d)+(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d)+(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d)-1/2*I*f*polylog(2,exp(2*I*(c+d*x)))/(a*d^2)-I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^2*d^2)-I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^2*d^2)-(e+f*x)*sin(c+d*x)/(b*d)],
[cos(c+d*x)^2*cot(c+d*x)/(a+b*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)+(a^2-b^2)*log(a+b*sin(c+d*x))/(a*b^2*d)-sin(c+d*x)/(b*d)],
[(e+f*x)^3*cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x)),x,53,3/4*e*f^2*x/(b*d^2)+3/8*f^3*x^2/(b*d^2)-1/8*(e+f*x)^4/(b*f)+1/4*(a^2-b^2)*(e+f*x)^4/(b^3*f)-2*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a*d)-6*f^2*(e+f*x)*cos(c+d*x)/(a*d^3)-6*(a^2-b^2)*f^2*(e+f*x)*cos(c+d*x)/(a*b^2*d^3)+(e+f*x)^3*cos(c+d*x)/(a*d)+(a^2-b^2)*(e+f*x)^3*cos(c+d*x)/(a*b^2*d)+3/8*f^3*cos(c+d*x)^2/(b*d^4)-3/4*f*(e+f*x)^2*cos(c+d*x)^2/(b*d^2)+I*(a^2-b^2)^(3/2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d)-I*(a^2-b^2)^(3/2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d)+3*I*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-3*I*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a*d^2)+3*(a^2-b^2)^(3/2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^2)-3*(a^2-b^2)^(3/2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^2)-6*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+6*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a*d^3)+6*I*(a^2-b^2)^(3/2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^3)-6*I*(a^2-b^2)^(3/2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^3)-6*I*f^3*polylog(4,-exp(I*(c+d*x)))/(a*d^4)+6*I*f^3*polylog(4,exp(I*(c+d*x)))/(a*d^4)-6*(a^2-b^2)^(3/2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^4)+6*(a^2-b^2)^(3/2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^4)+6*f^3*sin(c+d*x)/(a*d^4)+6*(a^2-b^2)*f^3*sin(c+d*x)/(a*b^2*d^4)-3*f*(e+f*x)^2*sin(c+d*x)/(a*d^2)-3*(a^2-b^2)*f*(e+f*x)^2*sin(c+d*x)/(a*b^2*d^2)+3/4*f^2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(b*d^3)-1/2*(e+f*x)^3*cos(c+d*x)*sin(c+d*x)/(b*d)],
[(e+f*x)^2*cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x)),x,41,1/4*f^2*x/(b*d^2)-1/6*(e+f*x)^3/(b*f)+1/3*(a^2-b^2)*(e+f*x)^3/(b^3*f)-2*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d)-2*f^2*cos(c+d*x)/(a*d^3)-2*(a^2-b^2)*f^2*cos(c+d*x)/(a*b^2*d^3)+(e+f*x)^2*cos(c+d*x)/(a*d)+(a^2-b^2)*(e+f*x)^2*cos(c+d*x)/(a*b^2*d)-1/2*f*(e+f*x)*cos(c+d*x)^2/(b*d^2)+I*(a^2-b^2)^(3/2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d)-I*(a^2-b^2)^(3/2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d)+2*I*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-2*I*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^2)+2*(a^2-b^2)^(3/2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^2)-2*(a^2-b^2)^(3/2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^2)-2*f^2*polylog(3,-exp(I*(c+d*x)))/(a*d^3)+2*f^2*polylog(3,exp(I*(c+d*x)))/(a*d^3)+2*I*(a^2-b^2)^(3/2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^3)-2*I*(a^2-b^2)^(3/2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^3)-2*f*(e+f*x)*sin(c+d*x)/(a*d^2)-2*(a^2-b^2)*f*(e+f*x)*sin(c+d*x)/(a*b^2*d^2)+1/4*f^2*cos(c+d*x)*sin(c+d*x)/(b*d^3)-1/2*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(b*d)],
[(e+f*x)*cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x)),x,31,-1/2*e*x/b+(a^2-b^2)*e*x/b^3-1/4*f*x^2/b+1/2*(a^2-b^2)*f*x^2/b^3-2*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d)+(e+f*x)*cos(c+d*x)/(a*d)+(a^2-b^2)*(e+f*x)*cos(c+d*x)/(a*b^2*d)-1/4*f*cos(c+d*x)^2/(b*d^2)+I*(a^2-b^2)^(3/2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d)-I*(a^2-b^2)^(3/2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d)+I*f*polylog(2,-exp(I*(c+d*x)))/(a*d^2)-I*f*polylog(2,exp(I*(c+d*x)))/(a*d^2)+(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a*b^3*d^2)-(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a*b^3*d^2)-f*sin(c+d*x)/(a*d^2)-(a^2-b^2)*f*sin(c+d*x)/(a*b^2*d^2)-1/2*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(b*d)],
[cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x)),x,6,1/2*(2*a^2-3*b^2)*x/b^3-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*b^3*d)-arctanh(cos(c+d*x))/(a*d)+a*cos(c+d*x)/(b^2*d)-1/2*cos(c+d*x)*sin(c+d*x)/(b*d)],

# Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^2 / (a+b Sin[c+d x])
[(e+f*x)^3*cos(c+d*x)*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,48,1/4*I*b*(e+f*x)^4/(a^2*f)+1/4*I*(a^2-b^2)*(e+f*x)^4/(a^2*b*f)-6*f*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d^2)-(e+f*x)^3*csc(c+d*x)/(a*d)-b*(e+f*x)^3*log(1-exp(2*I*(c+d*x)))/(a^2*d)-(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d)-(a^2-b^2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d)+6*I*f^2*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^3)-6*I*f^2*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^3)+3/2*I*b*f*(e+f*x)^2*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)+3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^2)+3*I*(a^2-b^2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^2)-6*f^3*polylog(3,-exp(I*(c+d*x)))/(a*d^4)+6*f^3*polylog(3,exp(I*(c+d*x)))/(a*d^4)-3/2*b*f^2*(e+f*x)*polylog(3,exp(2*I*(c+d*x)))/(a^2*d^3)-6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^3)-6*(a^2-b^2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^3)-3/4*I*b*f^3*polylog(4,exp(2*I*(c+d*x)))/(a^2*d^4)-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^4)-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^4)],
[(e+f*x)^2*cos(c+d*x)*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,37,1/3*I*b*(e+f*x)^3/(a^2*f)+1/3*I*(a^2-b^2)*(e+f*x)^3/(a^2*b*f)-4*f*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d^2)-(e+f*x)^2*csc(c+d*x)/(a*d)-b*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a^2*d)-(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d)-(a^2-b^2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d)+2*I*f^2*polylog(2,-exp(I*(c+d*x)))/(a*d^3)-2*I*f^2*polylog(2,exp(I*(c+d*x)))/(a*d^3)+I*b*f*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)+2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^2)+2*I*(a^2-b^2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^2)-1/2*b*f^2*polylog(3,exp(2*I*(c+d*x)))/(a^2*d^3)-2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^3)-2*(a^2-b^2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^3)],
[(e+f*x)*cos(c+d*x)*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,28,1/2*I*b*(e+f*x)^2/(a^2*f)+1/2*I*(a^2-b^2)*(e+f*x)^2/(a^2*b*f)-f*arctanh(cos(c+d*x))/(a*d^2)-(e+f*x)*csc(c+d*x)/(a*d)-b*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a^2*d)-(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d)-(a^2-b^2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d)+1/2*I*b*f*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)+I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b*d^2)+I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b*d^2)],
[cos(c+d*x)*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)-(1-b^2/a^2)*log(a+b*sin(c+d*x))/(b*d)],
[(e+f*x)^3*cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,66,-I*(e+f*x)^3/(a*d)-1/4*(e+f*x)^4/(a*f)-1/4*(a^2-b^2)*(e+f*x)^4/(a*b^2*f)+2*b*(e+f*x)^3*arctanh(exp(I*(c+d*x)))/(a^2*d)+6*b*f^2*(e+f*x)*cos(c+d*x)/(a^2*d^3)+6*(a^2-b^2)*f^2*(e+f*x)*cos(c+d*x)/(a^2*b*d^3)-b*(e+f*x)^3*cos(c+d*x)/(a^2*d)-(a^2-b^2)*(e+f*x)^3*cos(c+d*x)/(a^2*b*d)-(e+f*x)^3*cot(c+d*x)/(a*d)+3*f*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a*d^2)-I*(a^2-b^2)^(3/2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d)+I*(a^2-b^2)^(3/2)*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d)-3*I*b*f*(e+f*x)^2*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+3*I*b*f*(e+f*x)^2*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-3*I*f^2*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)-3*(a^2-b^2)^(3/2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^2)+3*(a^2-b^2)^(3/2)*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^2)+6*b*f^2*(e+f*x)*polylog(3,-exp(I*(c+d*x)))/(a^2*d^3)-6*b*f^2*(e+f*x)*polylog(3,exp(I*(c+d*x)))/(a^2*d^3)+3/2*f^3*polylog(3,exp(2*I*(c+d*x)))/(a*d^4)-6*I*(a^2-b^2)^(3/2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^3)+6*I*(a^2-b^2)^(3/2)*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^3)+6*I*b*f^3*polylog(4,-exp(I*(c+d*x)))/(a^2*d^4)-6*I*b*f^3*polylog(4,exp(I*(c+d*x)))/(a^2*d^4)+6*(a^2-b^2)^(3/2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^4)-6*(a^2-b^2)^(3/2)*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^4)-6*b*f^3*sin(c+d*x)/(a^2*d^4)-6*(a^2-b^2)*f^3*sin(c+d*x)/(a^2*b*d^4)+3*b*f*(e+f*x)^2*sin(c+d*x)/(a^2*d^2)+3*(a^2-b^2)*f*(e+f*x)^2*sin(c+d*x)/(a^2*b*d^2)],
[(e+f*x)^2*cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,53,-I*(e+f*x)^2/(a*d)-1/3*(e+f*x)^3/(a*f)-1/3*(a^2-b^2)*(e+f*x)^3/(a*b^2*f)+2*b*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a^2*d)+2*b*f^2*cos(c+d*x)/(a^2*d^3)+2*(a^2-b^2)*f^2*cos(c+d*x)/(a^2*b*d^3)-b*(e+f*x)^2*cos(c+d*x)/(a^2*d)-(a^2-b^2)*(e+f*x)^2*cos(c+d*x)/(a^2*b*d)-(e+f*x)^2*cot(c+d*x)/(a*d)+2*f*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a*d^2)-I*(a^2-b^2)^(3/2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d)+I*(a^2-b^2)^(3/2)*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d)-2*I*b*f*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+2*I*b*f*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-I*f^2*polylog(2,exp(2*I*(c+d*x)))/(a*d^3)-2*(a^2-b^2)^(3/2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^2)+2*(a^2-b^2)^(3/2)*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^2)+2*b*f^2*polylog(3,-exp(I*(c+d*x)))/(a^2*d^3)-2*b*f^2*polylog(3,exp(I*(c+d*x)))/(a^2*d^3)-2*I*(a^2-b^2)^(3/2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^3)+2*I*(a^2-b^2)^(3/2)*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^3)+2*b*f*(e+f*x)*sin(c+d*x)/(a^2*d^2)+2*(a^2-b^2)*f*(e+f*x)*sin(c+d*x)/(a^2*b*d^2)],
[(e+f*x)*cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,38,-e*x/a+(1-a^2/b^2)*e*x/a-1/2*f*x^2/a+1/2*(1-a^2/b^2)*f*x^2/a+2*b*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a^2*d)-b*(e+f*x)*cos(c+d*x)/(a^2*d)-(a^2-b^2)*(e+f*x)*cos(c+d*x)/(a^2*b*d)-(e+f*x)*cot(c+d*x)/(a*d)+f*log(sin(c+d*x))/(a*d^2)-I*(a^2-b^2)^(3/2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d)+I*(a^2-b^2)^(3/2)*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d)-I*b*f*polylog(2,-exp(I*(c+d*x)))/(a^2*d^2)+I*b*f*polylog(2,exp(I*(c+d*x)))/(a^2*d^2)-(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^2*d^2)+(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^2*d^2)+b*f*sin(c+d*x)/(a^2*d^2)+(a^2-b^2)*f*sin(c+d*x)/(a^2*b*d^2)],
[cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,6,-a*x/b^2+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*b^2*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cos(c+d*x)/(b*d)-cot(c+d*x)/(a*d)],
[(e+f*x)^3*cos(c+d*x)^3*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,85,3/8*b*f^3*x/(a^2*d^3)+3/8*(a^2-b^2)*f^3*x/(a^2*b*d^3)-1/4*b*(e+f*x)^3/(a^2*d)-1/4*(a^2-b^2)*(e+f*x)^3/(a^2*b*d)+1/4*I*b*(e+f*x)^4/(a^2*f)-1/4*I*(a^2-b^2)^2*(e+f*x)^4/(a^2*b^3*f)-6*f*(e+f*x)^2*arctanh(exp(I*(c+d*x)))/(a*d^2)+6*f^3*cos(c+d*x)/(a*d^4)+6*(a^2-b^2)*f^3*cos(c+d*x)/(a*b^2*d^4)-3*f*(e+f*x)^2*cos(c+d*x)/(a*d^2)-3*(a^2-b^2)*f*(e+f*x)^2*cos(c+d*x)/(a*b^2*d^2)-(e+f*x)^3*csc(c+d*x)/(a*d)-b*(e+f*x)^3*log(1-exp(2*I*(c+d*x)))/(a^2*d)+(a^2-b^2)^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d)+(a^2-b^2)^2*(e+f*x)^3*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d)+6*I*f^2*(e+f*x)*polylog(2,-exp(I*(c+d*x)))/(a*d^3)-6*I*f^2*(e+f*x)*polylog(2,exp(I*(c+d*x)))/(a*d^3)+3/2*I*b*f*(e+f*x)^2*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)-3*I*(a^2-b^2)^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^2)-3*I*(a^2-b^2)^2*f*(e+f*x)^2*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^2)-6*f^3*polylog(3,-exp(I*(c+d*x)))/(a*d^4)+6*f^3*polylog(3,exp(I*(c+d*x)))/(a*d^4)-3/2*b*f^2*(e+f*x)*polylog(3,exp(2*I*(c+d*x)))/(a^2*d^3)+6*(a^2-b^2)^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^3)+6*(a^2-b^2)^2*f^2*(e+f*x)*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^3)-3/4*I*b*f^3*polylog(4,exp(2*I*(c+d*x)))/(a^2*d^4)+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^4)+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^4)+6*f^2*(e+f*x)*sin(c+d*x)/(a*d^3)+6*(a^2-b^2)*f^2*(e+f*x)*sin(c+d*x)/(a*b^2*d^3)-(e+f*x)^3*sin(c+d*x)/(a*d)-(a^2-b^2)*(e+f*x)^3*sin(c+d*x)/(a*b^2*d)-3/8*b*f^3*cos(c+d*x)*sin(c+d*x)/(a^2*d^4)-3/8*(a^2-b^2)*f^3*cos(c+d*x)*sin(c+d*x)/(a^2*b*d^4)+3/4*b*f*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(a^2*d^2)+3/4*(a^2-b^2)*f*(e+f*x)^2*cos(c+d*x)*sin(c+d*x)/(a^2*b*d^2)-3/4*b*f^2*(e+f*x)*sin(c+d*x)^2/(a^2*d^3)-3/4*(a^2-b^2)*f^2*(e+f*x)*sin(c+d*x)^2/(a^2*b*d^3)+1/2*b*(e+f*x)^3*sin(c+d*x)^2/(a^2*d)+1/2*(a^2-b^2)*(e+f*x)^3*sin(c+d*x)^2/(a^2*b*d)],
[(e+f*x)^2*cos(c+d*x)^3*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,60,-1/2*b*e*f*x/(a^2*d)-1/2*(a^2-b^2)*e*f*x/(a^2*b*d)-1/4*b*f^2*x^2/(a^2*d)-1/4*(a^2-b^2)*f^2*x^2/(a^2*b*d)+1/3*I*b*(e+f*x)^3/(a^2*f)-1/3*I*(a^2-b^2)^2*(e+f*x)^3/(a^2*b^3*f)-4*f*(e+f*x)*arctanh(exp(I*(c+d*x)))/(a*d^2)-2*f*(e+f*x)*cos(c+d*x)/(a*d^2)-2*(a^2-b^2)*f*(e+f*x)*cos(c+d*x)/(a*b^2*d^2)-(e+f*x)^2*csc(c+d*x)/(a*d)-b*(e+f*x)^2*log(1-exp(2*I*(c+d*x)))/(a^2*d)+(a^2-b^2)^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d)+(a^2-b^2)^2*(e+f*x)^2*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d)+2*I*f^2*polylog(2,-exp(I*(c+d*x)))/(a*d^3)-2*I*f^2*polylog(2,exp(I*(c+d*x)))/(a*d^3)+I*b*f*(e+f*x)*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)-2*I*(a^2-b^2)^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^2)-2*I*(a^2-b^2)^2*f*(e+f*x)*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^2)-1/2*b*f^2*polylog(3,exp(2*I*(c+d*x)))/(a^2*d^3)+2*(a^2-b^2)^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^3)+2*(a^2-b^2)^2*f^2*polylog(3,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^3)+2*f^2*sin(c+d*x)/(a*d^3)+2*(a^2-b^2)*f^2*sin(c+d*x)/(a*b^2*d^3)-(e+f*x)^2*sin(c+d*x)/(a*d)-(a^2-b^2)*(e+f*x)^2*sin(c+d*x)/(a*b^2*d)+1/2*b*f*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(a^2*d^2)+1/2*(a^2-b^2)*f*(e+f*x)*cos(c+d*x)*sin(c+d*x)/(a^2*b*d^2)-1/4*b*f^2*sin(c+d*x)^2/(a^2*d^3)-1/4*(a^2-b^2)*f^2*sin(c+d*x)^2/(a^2*b*d^3)+1/2*b*(e+f*x)^2*sin(c+d*x)^2/(a^2*d)+1/2*(a^2-b^2)*(e+f*x)^2*sin(c+d*x)^2/(a^2*b*d)],
[(e+f*x)*cos(c+d*x)^3*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,45,-1/4*b*f*x/(a^2*d)-1/4*(a^2-b^2)*f*x/(a^2*b*d)+1/2*I*b*(e+f*x)^2/(a^2*f)-1/2*I*(a^2-b^2)^2*(e+f*x)^2/(a^2*b^3*f)-f*arctanh(cos(c+d*x))/(a*d^2)-f*cos(c+d*x)/(a*d^2)-(a^2-b^2)*f*cos(c+d*x)/(a*b^2*d^2)-(e+f*x)*csc(c+d*x)/(a*d)-b*(e+f*x)*log(1-exp(2*I*(c+d*x)))/(a^2*d)+(a^2-b^2)^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d)+(a^2-b^2)^2*(e+f*x)*log(1-I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d)+1/2*I*b*f*polylog(2,exp(2*I*(c+d*x)))/(a^2*d^2)-I*(a^2-b^2)^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a-sqrt(a^2-b^2)))/(a^2*b^3*d^2)-I*(a^2-b^2)^2*f*polylog(2,I*b*exp(I*(c+d*x))/(a+sqrt(a^2-b^2)))/(a^2*b^3*d^2)-(e+f*x)*sin(c+d*x)/(a*d)-(a^2-b^2)*(e+f*x)*sin(c+d*x)/(a*b^2*d)+1/4*b*f*cos(c+d*x)*sin(c+d*x)/(a^2*d^2)+1/4*(a^2-b^2)*f*cos(c+d*x)*sin(c+d*x)/(a^2*b*d^2)+1/2*b*(e+f*x)*sin(c+d*x)^2/(a^2*d)+1/2*(a^2-b^2)*(e+f*x)*sin(c+d*x)^2/(a^2*b*d)],
[cos(c+d*x)^3*cot(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)+(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^2*b^3*d)-a*sin(c+d*x)/(b^2*d)+1/2*sin(c+d*x)^2/(b*d)]]:
