# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.1.3 (d sin)^n (a+b sec)^m.txt"

lst:=[

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

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

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

# n>0
[(a+a*sec(c+d*x))*sin(c+d*x)^9,x,5,-a*cos(c+d*x)/d+2*a*cos(c+d*x)^2/d+4/3*a*cos(c+d*x)^3/d-3/2*a*cos(c+d*x)^4/d-6/5*a*cos(c+d*x)^5/d+2/3*a*cos(c+d*x)^6/d+4/7*a*cos(c+d*x)^7/d-1/8*a*cos(c+d*x)^8/d-1/9*a*cos(c+d*x)^9/d-a*log(cos(c+d*x))/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^7,x,5,-a*cos(c+d*x)/d+3/2*a*cos(c+d*x)^2/d+a*cos(c+d*x)^3/d-3/4*a*cos(c+d*x)^4/d-3/5*a*cos(c+d*x)^5/d+1/6*a*cos(c+d*x)^6/d+1/7*a*cos(c+d*x)^7/d-a*log(cos(c+d*x))/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^5,x,5,-a*cos(c+d*x)/d+a*cos(c+d*x)^2/d+2/3*a*cos(c+d*x)^3/d-1/4*a*cos(c+d*x)^4/d-1/5*a*cos(c+d*x)^5/d-a*log(cos(c+d*x))/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^3,x,5,-a*cos(c+d*x)/d+1/2*a*cos(c+d*x)^2/d+1/3*a*cos(c+d*x)^3/d-a*log(cos(c+d*x))/d],
[(a+a*sec(c+d*x))*sin(c+d*x),x,4,-a*cos(c+d*x)/d-a*log(cos(c+d*x))/d],
[csc(c+d*x)*(a+a*sec(c+d*x)),x,6,a*log(1-cos(c+d*x))/d-a*log(cos(c+d*x))/d],
[csc(c+d*x)^3*(a+a*sec(c+d*x)),x,5,-1/2*a^2/(d*(a-a*cos(c+d*x)))+3/4*a*log(1-cos(c+d*x))/d-a*log(cos(c+d*x))/d+1/4*a*log(1+cos(c+d*x))/d],
[csc(c+d*x)^5*(a+a*sec(c+d*x)),x,5,-1/8*a^3/(d*(a-a*cos(c+d*x))^2)-1/2*a^2/(d*(a-a*cos(c+d*x)))-1/8*a^2/(d*(a+a*cos(c+d*x)))+11/16*a*log(1-cos(c+d*x))/d-a*log(cos(c+d*x))/d+5/16*a*log(1+cos(c+d*x))/d],
[csc(c+d*x)^7*(a+a*sec(c+d*x)),x,5,-1/24*a^4/(d*(a-a*cos(c+d*x))^3)-5/32*a^3/(d*(a-a*cos(c+d*x))^2)-1/2*a^2/(d*(a-a*cos(c+d*x)))-1/32*a^3/(d*(a+a*cos(c+d*x))^2)-3/16*a^2/(d*(a+a*cos(c+d*x)))+21/32*a*log(1-cos(c+d*x))/d-a*log(cos(c+d*x))/d+11/32*a*log(1+cos(c+d*x))/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^8,x,11,35/128*a*x+a*arctanh(sin(c+d*x))/d-a*sin(c+d*x)/d-35/128*a*cos(c+d*x)*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d-35/192*a*cos(c+d*x)*sin(c+d*x)^3/d-1/5*a*sin(c+d*x)^5/d-7/48*a*cos(c+d*x)*sin(c+d*x)^5/d-1/7*a*sin(c+d*x)^7/d-1/8*a*cos(c+d*x)*sin(c+d*x)^7/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^6,x,10,5/16*a*x+a*arctanh(sin(c+d*x))/d-a*sin(c+d*x)/d-5/16*a*cos(c+d*x)*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d-5/24*a*cos(c+d*x)*sin(c+d*x)^3/d-1/5*a*sin(c+d*x)^5/d-1/6*a*cos(c+d*x)*sin(c+d*x)^5/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^4,x,9,3/8*a*x+a*arctanh(sin(c+d*x))/d-a*sin(c+d*x)/d-3/8*a*cos(c+d*x)*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d-1/4*a*cos(c+d*x)*sin(c+d*x)^3/d],
[(a+a*sec(c+d*x))*sin(c+d*x)^2,x,7,1/2*a*x+a*arctanh(sin(c+d*x))/d-a*sin(c+d*x)/d-1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[csc(c+d*x)^2*(a+a*sec(c+d*x)),x,7,a*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-a*csc(c+d*x)/d],
[csc(c+d*x)^4*(a+a*sec(c+d*x)),x,8,a*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-1/3*a*cot(c+d*x)^3/d-a*csc(c+d*x)/d-1/3*a*csc(c+d*x)^3/d],
[csc(c+d*x)^6*(a+a*sec(c+d*x)),x,8,a*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-2/3*a*cot(c+d*x)^3/d-1/5*a*cot(c+d*x)^5/d-a*csc(c+d*x)/d-1/3*a*csc(c+d*x)^3/d-1/5*a*csc(c+d*x)^5/d],
[csc(c+d*x)^8*(a+a*sec(c+d*x)),x,8,a*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-a*cot(c+d*x)^3/d-3/5*a*cot(c+d*x)^5/d-1/7*a*cot(c+d*x)^7/d-a*csc(c+d*x)/d-1/3*a*csc(c+d*x)^3/d-1/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d],
[csc(c+d*x)^10*(a+a*sec(c+d*x)),x,8,a*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-4/3*a*cot(c+d*x)^3/d-6/5*a*cot(c+d*x)^5/d-4/7*a*cot(c+d*x)^7/d-1/9*a*cot(c+d*x)^9/d-a*csc(c+d*x)/d-1/3*a*csc(c+d*x)^3/d-1/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d-1/9*a*csc(c+d*x)^9/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^9,x,5,3*a^2*cos(c+d*x)/d+4*a^2*cos(c+d*x)^2/d-2/3*a^2*cos(c+d*x)^3/d-3*a^2*cos(c+d*x)^4/d-2/5*a^2*cos(c+d*x)^5/d+4/3*a^2*cos(c+d*x)^6/d+3/7*a^2*cos(c+d*x)^7/d-1/4*a^2*cos(c+d*x)^8/d-1/9*a^2*cos(c+d*x)^9/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^7,x,5,2*a^2*cos(c+d*x)/d+3*a^2*cos(c+d*x)^2/d-3/2*a^2*cos(c+d*x)^4/d-2/5*a^2*cos(c+d*x)^5/d+1/3*a^2*cos(c+d*x)^6/d+1/7*a^2*cos(c+d*x)^7/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^5,x,5,a^2*cos(c+d*x)/d+2*a^2*cos(c+d*x)^2/d+1/3*a^2*cos(c+d*x)^3/d-1/2*a^2*cos(c+d*x)^4/d-1/5*a^2*cos(c+d*x)^5/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^3,x,5,a^2*cos(c+d*x)^2/d+1/3*a^2*cos(c+d*x)^3/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x),x,5,-a^2*cos(c+d*x)/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[csc(c+d*x)*(a+a*sec(c+d*x))^2,x,5,2*a^2*log(1-cos(c+d*x))/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[csc(c+d*x)^3*(a+a*sec(c+d*x))^2,x,5,-a^3/(d*(a-a*cos(c+d*x)))+2*a^2*log(1-cos(c+d*x))/d-2*a^2*log(cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[csc(c+d*x)^5*(a+a*sec(c+d*x))^2,x,5,-1/4*a^4/(d*(a-a*cos(c+d*x))^2)-5/4*a^3/(d*(a-a*cos(c+d*x)))+17/8*a^2*log(1-cos(c+d*x))/d-2*a^2*log(cos(c+d*x))/d-1/8*a^2*log(1+cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[csc(c+d*x)^7*(a+a*sec(c+d*x))^2,x,5,-1/12*a^5/(d*(a-a*cos(c+d*x))^3)-3/8*a^4/(d*(a-a*cos(c+d*x))^2)-23/16*a^3/(d*(a-a*cos(c+d*x)))+1/16*a^3/(d*(a+a*cos(c+d*x)))+9/4*a^2*log(1-cos(c+d*x))/d-2*a^2*log(cos(c+d*x))/d-1/4*a^2*log(1+cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[csc(c+d*x)^9*(a+a*sec(c+d*x))^2,x,5,-1/32*a^6/(d*(a-a*cos(c+d*x))^4)-7/48*a^5/(d*(a-a*cos(c+d*x))^3)-15/32*a^4/(d*(a-a*cos(c+d*x))^2)-51/32*a^3/(d*(a-a*cos(c+d*x)))+1/64*a^4/(d*(a+a*cos(c+d*x))^2)+9/64*a^3/(d*(a+a*cos(c+d*x)))+303/128*a^2*log(1-cos(c+d*x))/d-2*a^2*log(cos(c+d*x))/d-47/128*a^2*log(1+cos(c+d*x))/d+a^2*sec(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^8,x,27,-245/128*a^2*x+2*a^2*arctanh(sin(c+d*x))/d-2*a^2*sin(c+d*x)/d+139/128*a^2*cos(c+d*x)*sin(c+d*x)/d+11/192*a^2*cos(c+d*x)^3*sin(c+d*x)/d-17/48*a^2*cos(c+d*x)^5*sin(c+d*x)/d+1/8*a^2*cos(c+d*x)^7*sin(c+d*x)/d-2/3*a^2*sin(c+d*x)^3/d-2/5*a^2*sin(c+d*x)^5/d-2/7*a^2*sin(c+d*x)^7/d+a^2*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^6,x,18,-25/16*a^2*x+2*a^2*arctanh(sin(c+d*x))/d-2*a^2*sin(c+d*x)/d+7/16*a^2*cos(c+d*x)*sin(c+d*x)/d+7/24*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a^2*cos(c+d*x)^5*sin(c+d*x)/d-2/3*a^2*sin(c+d*x)^3/d-2/5*a^2*sin(c+d*x)^5/d+a^2*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^4,x,14,-9/8*a^2*x+2*a^2*arctanh(sin(c+d*x))/d-2*a^2*sin(c+d*x)/d-1/8*a^2*cos(c+d*x)*sin(c+d*x)/d+1/4*a^2*cos(c+d*x)^3*sin(c+d*x)/d-2/3*a^2*sin(c+d*x)^3/d+a^2*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sin(c+d*x)^2,x,9,-1/2*a^2*x+2*a^2*arctanh(sin(c+d*x))/d-2*a^2*sin(c+d*x)/d-1/2*a^2*cos(c+d*x)*sin(c+d*x)/d+a^2*tan(c+d*x)/d],
[csc(c+d*x)^2*(a+a*sec(c+d*x))^2,x,11,2*a^2*arctanh(sin(c+d*x))/d-2*a^2*cot(c+d*x)/d-2*a^2*csc(c+d*x)/d+a^2*tan(c+d*x)/d],
[csc(c+d*x)^4*(a+a*sec(c+d*x))^2,x,8,2*a^2*arctanh(sin(c+d*x))/d+10/3*a^2*tan(c+d*x)/d-2*a^2*tan(c+d*x)/(d*(1-cos(c+d*x)))-1/3*a^4*tan(c+d*x)/(d*(a-a*cos(c+d*x))^2)],
[csc(c+d*x)^6*(a+a*sec(c+d*x))^2,x,12,2*a^2*arctanh(sin(c+d*x))/d-4*a^2*cot(c+d*x)/d-5/3*a^2*cot(c+d*x)^3/d-2/5*a^2*cot(c+d*x)^5/d-2*a^2*csc(c+d*x)/d-2/3*a^2*csc(c+d*x)^3/d-2/5*a^2*csc(c+d*x)^5/d+a^2*tan(c+d*x)/d],
[csc(c+d*x)^8*(a+a*sec(c+d*x))^2,x,12,2*a^2*arctanh(sin(c+d*x))/d-5*a^2*cot(c+d*x)/d-3*a^2*cot(c+d*x)^3/d-7/5*a^2*cot(c+d*x)^5/d-2/7*a^2*cot(c+d*x)^7/d-2*a^2*csc(c+d*x)/d-2/3*a^2*csc(c+d*x)^3/d-2/5*a^2*csc(c+d*x)^5/d-2/7*a^2*csc(c+d*x)^7/d+a^2*tan(c+d*x)/d],
[csc(c+d*x)^10*(a+a*sec(c+d*x))^2,x,12,2*a^2*arctanh(sin(c+d*x))/d-6*a^2*cot(c+d*x)/d-14/3*a^2*cot(c+d*x)^3/d-16/5*a^2*cot(c+d*x)^5/d-9/7*a^2*cot(c+d*x)^7/d-2/9*a^2*cot(c+d*x)^9/d-2*a^2*csc(c+d*x)/d-2/3*a^2*csc(c+d*x)^3/d-2/5*a^2*csc(c+d*x)^5/d-2/7*a^2*csc(c+d*x)^7/d-2/9*a^2*csc(c+d*x)^9/d+a^2*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^9,x,5,11*a^3*cos(c+d*x)/d+3*a^3*cos(c+d*x)^2/d-14/3*a^3*cos(c+d*x)^3/d-7/2*a^3*cos(c+d*x)^4/d+6/5*a^3*cos(c+d*x)^5/d+11/6*a^3*cos(c+d*x)^6/d+1/7*a^3*cos(c+d*x)^7/d-3/8*a^3*cos(c+d*x)^8/d-1/9*a^3*cos(c+d*x)^9/d+a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^7,x,5,8*a^3*cos(c+d*x)/d+3*a^3*cos(c+d*x)^2/d-2*a^3*cos(c+d*x)^3/d-2*a^3*cos(c+d*x)^4/d+1/2*a^3*cos(c+d*x)^6/d+1/7*a^3*cos(c+d*x)^7/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^5,x,5,5*a^3*cos(c+d*x)/d+5/2*a^3*cos(c+d*x)^2/d-1/3*a^3*cos(c+d*x)^3/d-3/4*a^3*cos(c+d*x)^4/d-1/5*a^3*cos(c+d*x)^5/d-a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^3,x,4,2*a^3*cos(c+d*x)/d+3/2*a^3*cos(c+d*x)^2/d+1/3*a^3*cos(c+d*x)^3/d-2*a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x),x,5,-a^3*cos(c+d*x)/d-3*a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[csc(c+d*x)*(a+a*sec(c+d*x))^3,x,5,4*a^3*log(1-cos(c+d*x))/d-4*a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[csc(c+d*x)^3*(a+a*sec(c+d*x))^3,x,5,-2*a^4/(d*(a-a*cos(c+d*x)))+5*a^3*log(1-cos(c+d*x))/d-5*a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[csc(c+d*x)^5*(a+a*sec(c+d*x))^3,x,5,-1/2*a^5/(d*(a-a*cos(c+d*x))^2)-3*a^4/(d*(a-a*cos(c+d*x)))+6*a^3*log(1-cos(c+d*x))/d-6*a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[csc(c+d*x)^7*(a+a*sec(c+d*x))^3,x,5,-1/6*a^6/(d*(a-a*cos(c+d*x))^3)-7/8*a^5/(d*(a-a*cos(c+d*x))^2)-31/8*a^4/(d*(a-a*cos(c+d*x)))+111/16*a^3*log(1-cos(c+d*x))/d-7*a^3*log(cos(c+d*x))/d+1/16*a^3*log(1+cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[csc(c+d*x)^9*(a+a*sec(c+d*x))^3,x,5,-1/16*a^7/(d*(a-a*cos(c+d*x))^4)-1/3*a^6/(d*(a-a*cos(c+d*x))^3)-39/32*a^5/(d*(a-a*cos(c+d*x))^2)-75/16*a^4/(d*(a-a*cos(c+d*x)))-1/32*a^4/(d*(a+a*cos(c+d*x)))+501/64*a^3*log(1-cos(c+d*x))/d-8*a^3*log(cos(c+d*x))/d+11/64*a^3*log(1+cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^8,x,29,-805/128*a^3*x-1/2*a^3*arctanh(sin(c+d*x))/d+603/128*a^3*cos(c+d*x)*sin(c+d*x)/d-293/192*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/48*a^3*cos(c+d*x)^5*sin(c+d*x)/d+1/8*a^3*cos(c+d*x)^7*sin(c+d*x)/d-1/3*a^3*sin(c+d*x)^3/d-2/5*a^3*sin(c+d*x)^5/d-3/7*a^3*sin(c+d*x)^7/d+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^6,x,18,-85/16*a^3*x+1/2*a^3*arctanh(sin(c+d*x))/d-a^3*sin(c+d*x)/d+43/16*a^3*cos(c+d*x)*sin(c+d*x)/d-5/24*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a^3*cos(c+d*x)^5*sin(c+d*x)/d-2/3*a^3*sin(c+d*x)^3/d-3/5*a^3*sin(c+d*x)^5/d+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^4,x,16,-33/8*a^3*x+3/2*a^3*arctanh(sin(c+d*x))/d-2*a^3*sin(c+d*x)/d+7/8*a^3*cos(c+d*x)*sin(c+d*x)/d+1/4*a^3*cos(c+d*x)^3*sin(c+d*x)/d-a^3*sin(c+d*x)^3/d+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^3*sin(c+d*x)^2,x,11,-5/2*a^3*x+5/2*a^3*arctanh(sin(c+d*x))/d-3*a^3*sin(c+d*x)/d-1/2*a^3*cos(c+d*x)*sin(c+d*x)/d+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[csc(c+d*x)^2*(a+a*sec(c+d*x))^3,x,9,9/2*a^3*arctanh(sin(c+d*x))/d-4*a^3*sin(c+d*x)/(d*(1-cos(c+d*x)))+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[csc(c+d*x)^4*(a+a*sec(c+d*x))^3,x,11,11/2*a^3*arctanh(sin(c+d*x))/d-2/3*a^3*sin(c+d*x)/(d*(1-cos(c+d*x))^2)-17/3*a^3*sin(c+d*x)/(d*(1-cos(c+d*x)))+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[csc(c+d*x)^6*(a+a*sec(c+d*x))^3,x,10,13/2*a^3*arctanh(sin(c+d*x))/d+152/15*a^3*tan(c+d*x)/d+13/2*a^3*sec(c+d*x)*tan(c+d*x)/d-1/5*a^6*sec(c+d*x)*tan(c+d*x)/(d*(a-a*cos(c+d*x))^3)-11/15*a^5*sec(c+d*x)*tan(c+d*x)/(d*(a-a*cos(c+d*x))^2)-76/15*a^6*sec(c+d*x)*tan(c+d*x)/(d*(a^3-a^3*cos(c+d*x)))],
[csc(c+d*x)^8*(a+a*sec(c+d*x))^3,x,17,15/2*a^3*arctanh(sin(c+d*x))/d-13*a^3*cot(c+d*x)/d-7*a^3*cot(c+d*x)^3/d-3*a^3*cot(c+d*x)^5/d-4/7*a^3*cot(c+d*x)^7/d-15/2*a^3*csc(c+d*x)/d-5/2*a^3*csc(c+d*x)^3/d-3/2*a^3*csc(c+d*x)^5/d-15/14*a^3*csc(c+d*x)^7/d+1/2*a^3*csc(c+d*x)^7*sec(c+d*x)^2/d+3*a^3*tan(c+d*x)/d],
[csc(c+d*x)^10*(a+a*sec(c+d*x))^3,x,17,17/2*a^3*arctanh(sin(c+d*x))/d-16*a^3*cot(c+d*x)/d-34/3*a^3*cot(c+d*x)^3/d-36/5*a^3*cot(c+d*x)^5/d-19/7*a^3*cot(c+d*x)^7/d-4/9*a^3*cot(c+d*x)^9/d-17/2*a^3*csc(c+d*x)/d-17/6*a^3*csc(c+d*x)^3/d-17/10*a^3*csc(c+d*x)^5/d-17/14*a^3*csc(c+d*x)^7/d-17/18*a^3*csc(c+d*x)^9/d+1/2*a^3*csc(c+d*x)^9*sec(c+d*x)^2/d+3*a^3*tan(c+d*x)/d],

# n<0
[sin(c+d*x)^9/(a+a*sec(c+d*x)),x,7,1/3*cos(c+d*x)^3/(a*d)-3/5*cos(c+d*x)^5/(a*d)+3/7*cos(c+d*x)^7/(a*d)-1/9*cos(c+d*x)^9/(a*d)+1/8*sin(c+d*x)^8/(a*d)],
[sin(c+d*x)^7/(a+a*sec(c+d*x)),x,7,1/3*cos(c+d*x)^3/(a*d)-2/5*cos(c+d*x)^5/(a*d)+1/7*cos(c+d*x)^7/(a*d)+1/6*sin(c+d*x)^6/(a*d)],
[sin(c+d*x)^5/(a+a*sec(c+d*x)),x,7,1/3*cos(c+d*x)^3/(a*d)-1/5*cos(c+d*x)^5/(a*d)+1/4*sin(c+d*x)^4/(a*d)],
[sin(c+d*x)^3/(a+a*sec(c+d*x)),x,6,1/3*cos(c+d*x)^3/(a*d)+1/2*sin(c+d*x)^2/(a*d)],
[sin(c+d*x)/(a+a*sec(c+d*x)),x,5,-cos(c+d*x)/(a*d)+log(1+cos(c+d*x))/(a*d)],
[csc(c+d*x)/(a+a*sec(c+d*x)),x,6,-1/2*arctanh(cos(c+d*x))/(a*d)+1/2*cot(c+d*x)*csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)],
[csc(c+d*x)^3/(a+a*sec(c+d*x)),x,7,-1/8*arctanh(cos(c+d*x))/(a*d)-1/8*cot(c+d*x)*csc(c+d*x)/(a*d)+1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)-1/4*csc(c+d*x)^4/(a*d)],
[csc(c+d*x)^5/(a+a*sec(c+d*x)),x,8,-1/16*arctanh(cos(c+d*x))/(a*d)-1/16*cot(c+d*x)*csc(c+d*x)/(a*d)-1/24*cot(c+d*x)*csc(c+d*x)^3/(a*d)+1/6*cot(c+d*x)*csc(c+d*x)^5/(a*d)-1/6*csc(c+d*x)^6/(a*d)],
[sin(c+d*x)^8/(a+a*sec(c+d*x)),x,9,-5/128*x/a-5/128*cos(c+d*x)*sin(c+d*x)/(a*d)+5/64*cos(c+d*x)^3*sin(c+d*x)/(a*d)+5/48*cos(c+d*x)^3*sin(c+d*x)^3/(a*d)+1/8*cos(c+d*x)^3*sin(c+d*x)^5/(a*d)+1/7*sin(c+d*x)^7/(a*d)],
[sin(c+d*x)^6/(a+a*sec(c+d*x)),x,8,-1/16*x/a-1/16*cos(c+d*x)*sin(c+d*x)/(a*d)+1/8*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/6*cos(c+d*x)^3*sin(c+d*x)^3/(a*d)+1/5*sin(c+d*x)^5/(a*d)],
[sin(c+d*x)^4/(a+a*sec(c+d*x)),x,7,-1/8*x/a-1/8*cos(c+d*x)*sin(c+d*x)/(a*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/3*sin(c+d*x)^3/(a*d)],
[sin(c+d*x)^2/(a+a*sec(c+d*x)),x,5,-1/2*x/a+sin(c+d*x)/(a*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[csc(c+d*x)^2/(a+a*sec(c+d*x)),x,6,1/3*cot(c+d*x)^3/(a*d)-1/3*csc(c+d*x)^3/(a*d)],
[csc(c+d*x)^4/(a+a*sec(c+d*x)),x,7,1/3*cot(c+d*x)^3/(a*d)+1/5*cot(c+d*x)^5/(a*d)-1/5*csc(c+d*x)^5/(a*d)],
[csc(c+d*x)^6/(a+a*sec(c+d*x)),x,7,1/3*cot(c+d*x)^3/(a*d)+2/5*cot(c+d*x)^5/(a*d)+1/7*cot(c+d*x)^7/(a*d)-1/7*csc(c+d*x)^7/(a*d)],
[csc(c+d*x)^8/(a+a*sec(c+d*x)),x,7,1/3*cot(c+d*x)^3/(a*d)+3/5*cot(c+d*x)^5/(a*d)+3/7*cot(c+d*x)^7/(a*d)+1/9*cot(c+d*x)^9/(a*d)-1/9*csc(c+d*x)^9/(a*d)],
[csc(c+d*x)^10/(a+a*sec(c+d*x)),x,7,1/3*cot(c+d*x)^3/(a*d)+4/5*cot(c+d*x)^5/(a*d)+6/7*cot(c+d*x)^7/(a*d)+4/9*cot(c+d*x)^9/(a*d)+1/11*cot(c+d*x)^11/(a*d)-1/11*csc(c+d*x)^11/(a*d)],
[sin(c+d*x)^11/(a+a*sec(c+d*x))^2,x,5,4/3*(a-a*cos(c+d*x))^6/(a^8*d)-4*(a-a*cos(c+d*x))^7/(a^9*d)+19/4*(a-a*cos(c+d*x))^8/(a^10*d)-25/9*(a-a*cos(c+d*x))^9/(a^11*d)+4/5*(a-a*cos(c+d*x))^10/(a^12*d)-1/11*(a-a*cos(c+d*x))^11/(a^13*d)],
[sin(c+d*x)^9/(a+a*sec(c+d*x))^2,x,5,4/5*(a-a*cos(c+d*x))^5/(a^7*d)-2*(a-a*cos(c+d*x))^6/(a^8*d)+13/7*(a-a*cos(c+d*x))^7/(a^9*d)-3/4*(a-a*cos(c+d*x))^8/(a^10*d)+1/9*(a-a*cos(c+d*x))^9/(a^11*d)],
[sin(c+d*x)^7/(a+a*sec(c+d*x))^2,x,5,-1/3*cos(c+d*x)^3/(a^2*d)+1/2*cos(c+d*x)^4/(a^2*d)-1/3*cos(c+d*x)^6/(a^2*d)+1/7*cos(c+d*x)^7/(a^2*d)],
[sin(c+d*x)^5/(a+a*sec(c+d*x))^2,x,5,-1/3*cos(c+d*x)^3/(a^2*d)+1/2*cos(c+d*x)^4/(a^2*d)-1/5*cos(c+d*x)^5/(a^2*d)],
[sin(c+d*x)^3/(a+a*sec(c+d*x))^2,x,5,2*cos(c+d*x)/(a^2*d)-cos(c+d*x)^2/(a^2*d)+1/3*cos(c+d*x)^3/(a^2*d)-2*log(1+cos(c+d*x))/(a^2*d)],
[sin(c+d*x)/(a+a*sec(c+d*x))^2,x,5,-cos(c+d*x)/(a^2*d)+1/(d*(a^2+a^2*cos(c+d*x)))+2*log(1+cos(c+d*x))/(a^2*d)],
[csc(c+d*x)/(a+a*sec(c+d*x))^2,x,6,-1/4*arctanh(cos(c+d*x))/(a^2*d)+1/4/(d*(a+a*cos(c+d*x))^2)+(-3/4)/(d*(a^2+a^2*cos(c+d*x)))],
[csc(c+d*x)^3/(a+a*sec(c+d*x))^2,x,4,1/6*(-a-2*a*cos(c+d*x))/(d*(1-cos(c+d*x))*(a+a*cos(c+d*x))^3)],
[csc(c+d*x)^5/(a+a*sec(c+d*x))^2,x,6,1/64*arctanh(cos(c+d*x))/(a^2*d)+(-1/64)/(d*(a-a*cos(c+d*x))^2)+1/32*a^2/(d*(a+a*cos(c+d*x))^4)-1/48*a/(d*(a+a*cos(c+d*x))^3)+(-1/32)/(d*(a+a*cos(c+d*x))^2)+(-1/64)/(d*(a^2-a^2*cos(c+d*x)))+(-1/32)/(d*(a^2+a^2*cos(c+d*x)))],
[sin(c+d*x)^8/(a+a*sec(c+d*x))^2,x,16,11/128*x/a^2+11/128*cos(c+d*x)*sin(c+d*x)/(a^2*d)-7/64*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-1/16*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)-1/6*cos(c+d*x)^3*sin(c+d*x)^3/(a^2*d)-1/8*cos(c+d*x)^5*sin(c+d*x)^3/(a^2*d)-2/5*sin(c+d*x)^5/(a^2*d)+2/7*sin(c+d*x)^7/(a^2*d)],
[sin(c+d*x)^6/(a+a*sec(c+d*x))^2,x,7,3/16*x/a^2-3/16*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/8*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)-1/6*(a-a*cos(c+d*x))^3*sin(c+d*x)^3/(a^5*d)-1/10*sin(c+d*x)^5/(a^2*d)],
[sin(c+d*x)^4/(a+a*sec(c+d*x))^2,x,11,7/8*x/a^2-2*sin(c+d*x)/(a^2*d)+7/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+2/3*sin(c+d*x)^3/(a^2*d)],
[sin(c+d*x)^2/(a+a*sec(c+d*x))^2,x,9,-5/2*x/a^2+2*sin(c+d*x)/(a^2*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)+2*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))],
[csc(c+d*x)^2/(a+a*sec(c+d*x))^2,x,11,-1/3*cot(c+d*x)^3/(a^2*d)-2/5*cot(c+d*x)^5/(a^2*d)-2/3*csc(c+d*x)^3/(a^2*d)+2/5*csc(c+d*x)^5/(a^2*d)],
[csc(c+d*x)^4/(a+a*sec(c+d*x))^2,x,13,-1/3*cot(c+d*x)^3/(a^2*d)-3/5*cot(c+d*x)^5/(a^2*d)-2/7*cot(c+d*x)^7/(a^2*d)-2/5*csc(c+d*x)^5/(a^2*d)+2/7*csc(c+d*x)^7/(a^2*d)],
[csc(c+d*x)^6/(a+a*sec(c+d*x))^2,x,13,-1/3*cot(c+d*x)^3/(a^2*d)-4/5*cot(c+d*x)^5/(a^2*d)-5/7*cot(c+d*x)^7/(a^2*d)-2/9*cot(c+d*x)^9/(a^2*d)-2/7*csc(c+d*x)^7/(a^2*d)+2/9*csc(c+d*x)^9/(a^2*d)],
[csc(c+d*x)^8/(a+a*sec(c+d*x))^2,x,13,-1/3*cot(c+d*x)^3/(a^2*d)-cot(c+d*x)^5/(a^2*d)-9/7*cot(c+d*x)^7/(a^2*d)-7/9*cot(c+d*x)^9/(a^2*d)-2/11*cot(c+d*x)^11/(a^2*d)-2/9*csc(c+d*x)^9/(a^2*d)+2/11*csc(c+d*x)^11/(a^2*d)],
[sin(c+d*x)^11/(a+a*sec(c+d*x))^3,x,5,2/3*(a-a*cos(c+d*x))^6/(a^9*d)-16/7*(a-a*cos(c+d*x))^7/(a^10*d)+25/8*(a-a*cos(c+d*x))^8/(a^11*d)-19/9*(a-a*cos(c+d*x))^9/(a^12*d)+7/10*(a-a*cos(c+d*x))^10/(a^13*d)-1/11*(a-a*cos(c+d*x))^11/(a^14*d)],
[sin(c+d*x)^9/(a+a*sec(c+d*x))^3,x,5,-1/4*cos(c+d*x)^4/(a^3*d)+3/5*cos(c+d*x)^5/(a^3*d)-1/3*cos(c+d*x)^6/(a^3*d)-2/7*cos(c+d*x)^7/(a^3*d)+3/8*cos(c+d*x)^8/(a^3*d)-1/9*cos(c+d*x)^9/(a^3*d)],
[sin(c+d*x)^7/(a+a*sec(c+d*x))^3,x,5,-1/4*cos(c+d*x)^4/(a^3*d)+3/5*cos(c+d*x)^5/(a^3*d)-1/2*cos(c+d*x)^6/(a^3*d)+1/7*cos(c+d*x)^7/(a^3*d)],
[sin(c+d*x)^5/(a+a*sec(c+d*x))^3,x,5,-4*cos(c+d*x)/(a^3*d)+2*cos(c+d*x)^2/(a^3*d)-4/3*cos(c+d*x)^3/(a^3*d)+3/4*cos(c+d*x)^4/(a^3*d)-1/5*cos(c+d*x)^5/(a^3*d)+4*log(1+cos(c+d*x))/(a^3*d)],
[sin(c+d*x)^3/(a+a*sec(c+d*x))^3,x,5,5*cos(c+d*x)/(a^3*d)-3/2*cos(c+d*x)^2/(a^3*d)+1/3*cos(c+d*x)^3/(a^3*d)+(-2)/(d*(a^3+a^3*cos(c+d*x)))-7*log(1+cos(c+d*x))/(a^3*d)],
[sin(c+d*x)/(a+a*sec(c+d*x))^3,x,5,-cos(c+d*x)/(a^3*d)+(-1/2)/(a*d*(a+a*cos(c+d*x))^2)+3/(d*(a^3+a^3*cos(c+d*x)))+3*log(1+cos(c+d*x))/(a^3*d)],
[csc(c+d*x)/(a+a*sec(c+d*x))^3,x,6,-1/8*arctanh(cos(c+d*x))/(a^3*d)+(-1/6)/(d*(a+a*cos(c+d*x))^3)+5/8/(a*d*(a+a*cos(c+d*x))^2)+(-7/8)/(d*(a^3+a^3*cos(c+d*x)))],
[csc(c+d*x)^3/(a+a*sec(c+d*x))^3,x,5,1/32*arctanh(cos(c+d*x))/(a^3*d)-1/16*a/(d*(a+a*cos(c+d*x))^4)+1/6/(d*(a+a*cos(c+d*x))^3)+(-3/32)/(a*d*(a+a*cos(c+d*x))^2)+(-1/32)/(d*(a^3-a^3*cos(c+d*x)))+(-1/16)/(d*(a^3+a^3*cos(c+d*x)))],
[csc(c+d*x)^5/(a+a*sec(c+d*x))^3,x,6,3/128*arctanh(cos(c+d*x))/(a^3*d)+(-1/128)/(a*d*(a-a*cos(c+d*x))^2)-1/40*a^2/(d*(a+a*cos(c+d*x))^5)+3/64*a/(d*(a+a*cos(c+d*x))^4)+(-1/64)/(a*d*(a+a*cos(c+d*x))^2)+(-3/128)/(d*(a^3+a^3*cos(c+d*x)))],
[sin(c+d*x)^8/(a+a*sec(c+d*x))^3,x,19,-29/128*x/a^3-29/128*cos(c+d*x)*sin(c+d*x)/(a^3*d)-29/192*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)+23/48*cos(c+d*x)^5*sin(c+d*x)/(a^3*d)+1/8*cos(c+d*x)^7*sin(c+d*x)/(a^3*d)+4/3*sin(c+d*x)^3/(a^3*d)-7/5*sin(c+d*x)^5/(a^3*d)+3/7*sin(c+d*x)^7/(a^3*d)],
[sin(c+d*x)^6/(a+a*sec(c+d*x))^3,x,15,-23/16*x/a^3+4*sin(c+d*x)/(a^3*d)-23/16*cos(c+d*x)*sin(c+d*x)/(a^3*d)-23/24*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)-1/6*cos(c+d*x)^5*sin(c+d*x)/(a^3*d)-7/3*sin(c+d*x)^3/(a^3*d)+3/5*sin(c+d*x)^5/(a^3*d)],
[sin(c+d*x)^4/(a+a*sec(c+d*x))^3,x,13,51/8*x/a^3-7*sin(c+d*x)/(a^3*d)+19/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)-4*sin(c+d*x)/(a^3*d*(1+cos(c+d*x)))+sin(c+d*x)^3/(a^3*d)],
[sin(c+d*x)^2/(a+a*sec(c+d*x))^3,x,10,-11/2*x/a^3+3*sin(c+d*x)/(a^3*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)-2/3*sin(c+d*x)/(a^3*d*(1+cos(c+d*x))^2)+19/3*sin(c+d*x)/(a^3*d*(1+cos(c+d*x)))],
[csc(c+d*x)^2/(a+a*sec(c+d*x))^3,x,15,3/5*cot(c+d*x)^5/(a^3*d)+4/7*cot(c+d*x)^7/(a^3*d)-csc(c+d*x)^3/(a^3*d)+7/5*csc(c+d*x)^5/(a^3*d)-4/7*csc(c+d*x)^7/(a^3*d)],
[csc(c+d*x)^4/(a+a*sec(c+d*x))^3,x,16,3/5*cot(c+d*x)^5/(a^3*d)+cot(c+d*x)^7/(a^3*d)+4/9*cot(c+d*x)^9/(a^3*d)-3/5*csc(c+d*x)^5/(a^3*d)+csc(c+d*x)^7/(a^3*d)-4/9*csc(c+d*x)^9/(a^3*d)],
[csc(c+d*x)^6/(a+a*sec(c+d*x))^3,x,16,3/5*cot(c+d*x)^5/(a^3*d)+10/7*cot(c+d*x)^7/(a^3*d)+11/9*cot(c+d*x)^9/(a^3*d)+4/11*cot(c+d*x)^11/(a^3*d)-3/7*csc(c+d*x)^7/(a^3*d)+7/9*csc(c+d*x)^9/(a^3*d)-4/11*csc(c+d*x)^11/(a^3*d)],
[csc(c+d*x)^8/(a+a*sec(c+d*x))^3,x,16,3/5*cot(c+d*x)^5/(a^3*d)+13/7*cot(c+d*x)^7/(a^3*d)+7/3*cot(c+d*x)^9/(a^3*d)+15/11*cot(c+d*x)^11/(a^3*d)+4/13*cot(c+d*x)^13/(a^3*d)-1/3*csc(c+d*x)^9/(a^3*d)+7/11*csc(c+d*x)^11/(a^3*d)-4/13*csc(c+d*x)^13/(a^3*d)],

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

# n>0
[(a+a*sec(c+d*x))*(e*sin(c+d*x))^(5/2),x,11,-a*e^(5/2)*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/d+a*e^(5/2)*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/d-2/3*a*e*(e*sin(c+d*x))^(3/2)/d-2/5*a*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/d+6/5*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))*(e*sin(c+d*x))^(3/2),x,11,a*e^(3/2)*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/d+a*e^(3/2)*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/d+2/3*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-2*a*e*sqrt(e*sin(c+d*x))/d-2/3*a*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+a*sec(c+d*x))*sqrt(e*sin(c+d*x)),x,9,-a*arctan(sqrt(e*sin(c+d*x))/sqrt(e))*sqrt(e)/d+a*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))*sqrt(e)/d+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))/sqrt(e*sin(c+d*x)),x,9,a*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*sqrt(e))+a*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*sqrt(e))+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))],
[(a+a*sec(c+d*x))/(e*sin(c+d*x))^(3/2),x,11,-a*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(3/2))+a*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(3/2))-2*a/(d*e*sqrt(e*sin(c+d*x)))-2*a*cos(c+d*x)/(d*e*sqrt(e*sin(c+d*x)))-2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^2*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))/(e*sin(c+d*x))^(5/2),x,11,a*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(5/2))+a*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(5/2))-2/3*a/(d*e*(e*sin(c+d*x))^(3/2))-2/3*a*cos(c+d*x)/(d*e*(e*sin(c+d*x))^(3/2))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^2*sqrt(e*sin(c+d*x)))],
[(a+a*sec(c+d*x))^2*(e*sin(c+d*x))^(5/2),x,15,-2*a^2*e^(5/2)*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/d+2*a^2*e^(5/2)*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/d-4/3*a^2*e*(e*sin(c+d*x))^(3/2)/d-2/5*a^2*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/d+a^2*e*sec(c+d*x)*(e*sin(c+d*x))^(3/2)/d-9/5*a^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))^2*(e*sin(c+d*x))^(3/2),x,15,2*a^2*e^(3/2)*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/d+2*a^2*e^(3/2)*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/d-1/3*a^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-4*a^2*e*sqrt(e*sin(c+d*x))/d-2/3*a^2*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/d+a^2*e*sec(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+a*sec(c+d*x))^2*sqrt(e*sin(c+d*x)),x,13,a^2*sec(c+d*x)*(e*sin(c+d*x))^(3/2)/(d*e)-2*a^2*arctan(sqrt(e*sin(c+d*x))/sqrt(e))*sqrt(e)/d+2*a^2*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))*sqrt(e)/d+a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))^2/sqrt(e*sin(c+d*x)),x,13,2*a^2*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*sqrt(e))+2*a^2*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*sqrt(e))+3*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))+a^2*sec(c+d*x)*sqrt(e*sin(c+d*x))/(d*e)],
[(a+a*sec(c+d*x))^2/(e*sin(c+d*x))^(3/2),x,16,-2*a^2*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(3/2))+2*a^2*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(3/2))+3*a^2*sec(c+d*x)*(e*sin(c+d*x))^(3/2)/(d*e^3)-4*a^2/(d*e*sqrt(e*sin(c+d*x)))-2*a^2*cos(c+d*x)/(d*e*sqrt(e*sin(c+d*x)))-2*a^2*sec(c+d*x)/(d*e*sqrt(e*sin(c+d*x)))-5*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^2*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))^2/(e*sin(c+d*x))^(5/2),x,16,2*a^2*arctan(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(5/2))+2*a^2*arctanh(sqrt(e*sin(c+d*x))/sqrt(e))/(d*e^(5/2))-4/3*a^2/(d*e*(e*sin(c+d*x))^(3/2))-2/3*a^2*cos(c+d*x)/(d*e*(e*sin(c+d*x))^(3/2))-2/3*a^2*sec(c+d*x)/(d*e*(e*sin(c+d*x))^(3/2))+7/3*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^2*sqrt(e*sin(c+d*x)))+5/3*a^2*sec(c+d*x)*sqrt(e*sin(c+d*x))/(d*e^3)],

# n<0
[(e*sin(c+d*x))^(7/2)/(a+a*sec(c+d*x)),x,8,2/5*e*(e*sin(c+d*x))^(5/2)/(a*d)-4/21*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*sqrt(e*sin(c+d*x)))-2/21*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a*d)+2/7*e^3*cos(c+d*x)^3*sqrt(e*sin(c+d*x))/(a*d)],
[(e*sin(c+d*x))^(5/2)/(a+a*sec(c+d*x)),x,7,2/3*e*(e*sin(c+d*x))^(3/2)/(a*d)-2/5*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/(a*d)-4/5*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+a*sec(c+d*x)),x,7,-4/3*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*sqrt(e*sin(c+d*x)))+2*e*sqrt(e*sin(c+d*x))/(a*d)-2/3*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a*d)],
[sqrt(e*sin(c+d*x))/(a+a*sec(c+d*x)),x,7,-2*e/(a*d*sqrt(e*sin(c+d*x)))+2*e*cos(c+d*x)/(a*d*sqrt(e*sin(c+d*x)))+4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a*d*sqrt(sin(c+d*x)))],
[1/((a+a*sec(c+d*x))*sqrt(e*sin(c+d*x))),x,7,-2/3*e/(a*d*(e*sin(c+d*x))^(3/2))+2/3*e*cos(c+d*x)/(a*d*(e*sin(c+d*x))^(3/2))+4/3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*sqrt(e*sin(c+d*x)))],
[1/((a+a*sec(c+d*x))*(e*sin(c+d*x))^(3/2)),x,8,-2/5*e/(a*d*(e*sin(c+d*x))^(5/2))+2/5*e*cos(c+d*x)/(a*d*(e*sin(c+d*x))^(5/2))-4/5*cos(c+d*x)/(a*d*e*sqrt(e*sin(c+d*x)))-4/5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+a*sec(c+d*x))*(e*sin(c+d*x))^(5/2)),x,8,-2/7*e/(a*d*(e*sin(c+d*x))^(7/2))+2/7*e*cos(c+d*x)/(a*d*(e*sin(c+d*x))^(7/2))-4/21*cos(c+d*x)/(a*d*e*(e*sin(c+d*x))^(3/2))+4/21*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*e^2*sqrt(e*sin(c+d*x)))],
[(e*sin(c+d*x))^(7/2)/(a+a*sec(c+d*x))^2,x,14,4/5*e*(e*sin(c+d*x))^(5/2)/(a^2*d)+52/21*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))-4*e^3*sqrt(e*sin(c+d*x))/(a^2*d)+26/21*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a^2*d)+2/7*e^3*cos(c+d*x)^3*sqrt(e*sin(c+d*x))/(a^2*d)],
[(e*sin(c+d*x))^(5/2)/(a+a*sec(c+d*x))^2,x,14,4/3*e*(e*sin(c+d*x))^(3/2)/(a^2*d)-12/5*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/(a^2*d)+4*e^3/(a^2*d*sqrt(e*sin(c+d*x)))-2*e^3*cos(c+d*x)/(a^2*d*sqrt(e*sin(c+d*x)))-2*e^3*cos(c+d*x)^3/(a^2*d*sqrt(e*sin(c+d*x)))-44/5*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+a*sec(c+d*x))^2,x,14,4/3*e^3/(a^2*d*(e*sin(c+d*x))^(3/2))-2/3*e^3*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(3/2))-2/3*e^3*cos(c+d*x)^3/(a^2*d*(e*sin(c+d*x))^(3/2))-4*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))+4*e*sqrt(e*sin(c+d*x))/(a^2*d)-4/3*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a^2*d)],
[sqrt(e*sin(c+d*x))/(a+a*sec(c+d*x))^2,x,15,4/5*e^3/(a^2*d*(e*sin(c+d*x))^(5/2))-2/5*e^3*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(5/2))-2/5*e^3*cos(c+d*x)^3/(a^2*d*(e*sin(c+d*x))^(5/2))-4*e/(a^2*d*sqrt(e*sin(c+d*x)))+16/5*e*cos(c+d*x)/(a^2*d*sqrt(e*sin(c+d*x)))+28/5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*sqrt(sin(c+d*x)))],
[1/((a+a*sec(c+d*x))^2*sqrt(e*sin(c+d*x))),x,15,4/7*e^3/(a^2*d*(e*sin(c+d*x))^(7/2))-2/7*e^3*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(7/2))-2/7*e^3*cos(c+d*x)^3/(a^2*d*(e*sin(c+d*x))^(7/2))-4/3*e/(a^2*d*(e*sin(c+d*x))^(3/2))+16/21*e*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(3/2))+20/21*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))],
[1/((a+a*sec(c+d*x))^2*(e*sin(c+d*x))^(3/2)),x,17,4/9*e^3/(a^2*d*(e*sin(c+d*x))^(9/2))-2/9*e^3*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(9/2))-2/9*e^3*cos(c+d*x)^3/(a^2*d*(e*sin(c+d*x))^(9/2))-4/5*e/(a^2*d*(e*sin(c+d*x))^(5/2))+16/45*e*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(5/2))-4/15*cos(c+d*x)/(a^2*d*e*sqrt(e*sin(c+d*x)))-4/15*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+a*sec(c+d*x))^2*(e*sin(c+d*x))^(5/2)),x,17,4/11*e^3/(a^2*d*(e*sin(c+d*x))^(11/2))-2/11*e^3*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(11/2))-2/11*e^3*cos(c+d*x)^3/(a^2*d*(e*sin(c+d*x))^(11/2))-4/7*e/(a^2*d*(e*sin(c+d*x))^(7/2))+16/77*e*cos(c+d*x)/(a^2*d*(e*sin(c+d*x))^(7/2))-4/231*cos(c+d*x)/(a^2*d*e*(e*sin(c+d*x))^(3/2))+4/231*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*e^2*sqrt(e*sin(c+d*x)))],

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

# Integrands of the form (e Sin[e+f x])^m (a+a Sec[e+f x])^n with m symbolic
[(a+a*sec(c+d*x))^3*(e*sin(c+d*x))^m,x,9,3*a^3*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a^3*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a^3*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(cos(c+d*x)^2))+3*a^3*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*sin(c+d*x))^(1+m)*sqrt(cos(c+d*x)^2)/(d*e*(1+m))],
[(a+a*sec(c+d*x))^2*(e*sin(c+d*x))^m,x,7,2*a^2*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a^2*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(cos(c+d*x)^2))+a^2*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*sin(c+d*x))^(1+m)*sqrt(cos(c+d*x)^2)/(d*e*(1+m))],
[(a+a*sec(c+d*x))*(e*sin(c+d*x))^m,x,5,a*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+a*sec(c+d*x)),x,5,-e*(e*sin(c+d*x))^(-1+m)/(a*d*(1-m))+e*cos(c+d*x)*hypergeom([-1/2,1/2*(-1+m)],[1/2*(1+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(-1+m)/(a*d*(1-m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+a*sec(c+d*x))^2,x,9,2*e^3*(e*sin(c+d*x))^(-3+m)/(a^2*d*(3-m))-2*e*(e*sin(c+d*x))^(-1+m)/(a^2*d*(1-m))-e^3*cos(c+d*x)*hypergeom([-3/2,1/2*(-3+m)],[1/2*(-1+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(-3+m)/(a^2*d*(3-m)*sqrt(cos(c+d*x)^2))-e^3*cos(c+d*x)*hypergeom([-1/2,1/2*(-3+m)],[1/2*(-1+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(-3+m)/(a^2*d*(3-m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+a*sec(c+d*x))^3,x,12,-4*e^5*(e*sin(c+d*x))^(-5+m)/(a^3*d*(5-m))+7*e^3*(e*sin(c+d*x))^(-3+m)/(a^3*d*(3-m))-3*e*(e*sin(c+d*x))^(-1+m)/(a^3*d*(1-m))+e^5*cos(c+d*x)*hypergeom([-5/2,1/2*(-5+m)],[1/2*(-3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(-5+m)/(a^3*d*(5-m)*sqrt(cos(c+d*x)^2))+3*e^5*cos(c+d*x)*hypergeom([-3/2,1/2*(-5+m)],[1/2*(-3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(-5+m)/(a^3*d*(5-m)*sqrt(cos(c+d*x)^2))],
[(a+a*sec(c+d*x))^(3/2)*(e*sin(c+d*x))^m,x,5,2*a*e*AppellF1(-1/2,1/2*(1-m),1/2*(-2-m),1/2,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)*sqrt(a+a*sec(c+d*x))/(d*(1+cos(c+d*x))^(1/2*m))],
[(a+a*sec(c+d*x))^(1/2)*(e*sin(c+d*x))^m,x,5,-2*e*AppellF1(1/2,1/2*(1-m),-1/2*m,3/2,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/2*(1-m))*cos(c+d*x)*(e*sin(c+d*x))^(-1+m)*sqrt(a+a*sec(c+d*x))/(d*(1+cos(c+d*x))^(1/2*m))],
[(e*sin(c+d*x))^m/(a+a*sec(c+d*x))^(1/2),x,5,-2/3*e*AppellF1(3/2,1/2*(1-m),1/2*(2-m),5/2,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/2*(1-m))*cos(c+d*x)*(1+cos(c+d*x))^(1-1/2*m)*(e*sin(c+d*x))^(-1+m)/(d*sqrt(a+a*sec(c+d*x)))],
[(e*sin(c+d*x))^m/(a+a*sec(c+d*x))^(3/2),x,5,-2/5*e*AppellF1(5/2,1/2*(1-m),1/2*(4-m),7/2,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/2*(1-m))*cos(c+d*x)^2*(1+cos(c+d*x))^(1-1/2*m)*(e*sin(c+d*x))^(-1+m)/(a*d*sqrt(a+a*sec(c+d*x)))],

# Integrands of the form (e Sin[e+f x])^m (a+a Sec[e+f x])^n with n symbolic
[(a+a*sec(c+d*x))^n*(e*sin(c+d*x))^m,x,5,-e*AppellF1(1-n,1/2*(1-m),1/2*(1-m-2*n),2-n,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/2*(1-m))*cos(c+d*x)*(1+cos(c+d*x))^(1/2*(1-m-2*n))*(a+a*sec(c+d*x))^n*(e*sin(c+d*x))^(-1+m)/(d*(1-n))],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^7,x,4,-1/42*(3-n)*(8-n)*(16-n)*hypergeom([6,4+n],[5+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(4+n)/(a^4*d*(4-3*n-n^2))-cos(c+d*x)^7*(1-sec(c+d*x))^2*(a+a*sec(c+d*x))^(4+n)/(a^4*d*(1-n))+1/42*cos(c+d*x)^7*(a+a*sec(c+d*x))^(4+n)*(6*(8-n)-(108-25*n+n^2)*sec(c+d*x))/(a^4*d*(1-n))],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^5,x,4,1/20*(12-n)*cos(c+d*x)^4*(a+a*sec(c+d*x))^(3+n)/(a^3*d)-1/5*cos(c+d*x)^5*(a+a*sec(c+d*x))^(3+n)/(a^3*d)+1/20*(32-13*n+n^2)*hypergeom([4,3+n],[4+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(3+n)/(a^3*d*(3+n))],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^3,x,3,1/3*cos(c+d*x)^3*(a+a*sec(c+d*x))^(2+n)/(a^2*d)-1/3*(4-n)*hypergeom([3,2+n],[3+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(2+n)/(a^2*d*(2+n))],
[(a+a*sec(c+d*x))^n*sin(c+d*x),x,2,hypergeom([2,1+n],[2+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(1+n)/(a*d*(1+n))],
[csc(c+d*x)*(a+a*sec(c+d*x))^n,x,2,-1/2*hypergeom([1,n],[1+n],1/2*(1+sec(c+d*x)))*(a+a*sec(c+d*x))^n/(d*n)],
[csc(c+d*x)^3*(a+a*sec(c+d*x))^n,x,4,-1/4*a*(2-n)*(a+a*sec(c+d*x))^(-1+n)/(d*(1-n))+1/2*a*(a+a*sec(c+d*x))^(-1+n)/(d*(1-sec(c+d*x)))-1/8*(2+n)*hypergeom([1,n],[1+n],1/2*(1+sec(c+d*x)))*(a+a*sec(c+d*x))^n/(d*n)],
[csc(c+d*x)^5*(a+a*sec(c+d*x))^n,x,5,1/16*a^2*(12+9*n+n^2)*hypergeom([1,-2+n],[-1+n],1/2*(1+sec(c+d*x)))*(a+a*sec(c+d*x))^(-2+n)/(d*(2-n))+1/4*a^2*(3+n)*sec(c+d*x)^2*(a+a*sec(c+d*x))^(-2+n)/(d*(1-n)*(1-sec(c+d*x))^2)-a^2*sec(c+d*x)^3*(a+a*sec(c+d*x))^(-2+n)/(d*(1-n)*(1-sec(c+d*x))^2)-1/8*a^2*(a+a*sec(c+d*x))^(-2+n)*(12+4*n-7*n^2-n^3-2*(1-n)*(6+n)*sec(c+d*x))/(d*(2-3*n+n^2)*(1-sec(c+d*x)))],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^4,x,11,-cos(c+d*x)*(a+a*sec(c+d*x))^n*sin(c+d*x)/d+2^(1/2+n)*AppellF1(1/2,-4+n,1/2-n,3/2,1-cos(c+d*x),1/2*(1-cos(c+d*x)))*cos(c+d*x)^n*(1+cos(c+d*x))^(-1/2-n)*(a+a*sec(c+d*x))^n*sin(c+d*x)/d-AppellF1(1-n,-1/2,1/2-n,2-n,cos(c+d*x),-cos(c+d*x))*(1+cos(c+d*x))^(1/2-n)*(n-n*cos(c+d*x))*cot(c+d*x)*(a+a*sec(c+d*x))^n/(d*(1-n)*sqrt(1-cos(c+d*x)))],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^2,x,6,-AppellF1(1-n,-1/2,-1/2-n,2-n,cos(c+d*x),-cos(c+d*x))*(1+cos(c+d*x))^(1/2-n)*cot(c+d*x)*(a+a*sec(c+d*x))^n*sqrt(1-cos(c+d*x))/(d*(1-n))],
[csc(c+d*x)^2*(a+a*sec(c+d*x))^n,x,4,-cot(c+d*x)*(a+a*sec(c+d*x))^n/d+2^(-1/2+n)*n*hypergeom([1/2,3/2-n],[3/2],1/2*(1-sec(c+d*x)))*(1+sec(c+d*x))^(-1/2-n)*(a+a*sec(c+d*x))^n*tan(c+d*x)/d],
[csc(c+d*x)^4*(a+a*sec(c+d*x))^n,x,7,(2-n+n^2)*cos(c+d*x)*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*(1-4*n^2)*(1-cos(c+d*x))^2)-a^4*cos(c+d*x)*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*(a-a*cos(c+d*x))^2*(a+a*cos(c+d*x))^2)-a^3*(4-n)*cos(c+d*x)*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*(3-8*n+4*n^2)*(a-a*cos(c+d*x))^2*(a+a*cos(c+d*x)))+n*(7-3*n-n^2)*cos(c+d*x)*((1+cos(c+d*x))/(1-cos(c+d*x)))^(-1/2-n)*hypergeom([-1/2-n,1-n],[2-n],-2*cos(c+d*x)/(1-cos(c+d*x)))*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*(3-5*n-10*n^2+20*n^3-8*n^4)*(1-cos(c+d*x))^2)],
[(a+a*sec(c+d*x))^n*sin(c+d*x)^(3/2),x,5,-AppellF1(1-n,-1/4,-1/4-n,2-n,cos(c+d*x),-cos(c+d*x))*cos(c+d*x)*(1+cos(c+d*x))^(-1/4-n)*(a+a*sec(c+d*x))^n*sqrt(sin(c+d*x))/(d*(1-n)*(1-cos(c+d*x))^(1/4))],
[(a+a*sec(c+d*x))^n*sqrt(sin(c+d*x)),x,5,-AppellF1(1-n,1/4,1/4-n,2-n,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(1/4)*cos(c+d*x)*(1+cos(c+d*x))^(1/4-n)*(a+a*sec(c+d*x))^n/(d*(1-n)*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))^n/sqrt(sin(c+d*x)),x,5,-AppellF1(1-n,3/4,3/4-n,2-n,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(3/4)*cos(c+d*x)*(1+cos(c+d*x))^(3/4-n)*(a+a*sec(c+d*x))^n/(d*(1-n)*sin(c+d*x)^(3/2))],
[(a+a*sec(c+d*x))^n/sin(c+d*x)^(3/2),x,5,-AppellF1(1-n,5/4,5/4-n,2-n,cos(c+d*x),-cos(c+d*x))*(1-cos(c+d*x))^(5/4)*cos(c+d*x)*(1+cos(c+d*x))^(5/4-n)*(a+a*sec(c+d*x))^n/(d*(1-n)*sin(c+d*x)^(5/2))],

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

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

# n>0
[(a+b*sec(c+d*x))*sin(c+d*x)^7,x,5,-a*cos(c+d*x)/d+3/2*b*cos(c+d*x)^2/d+a*cos(c+d*x)^3/d-3/4*b*cos(c+d*x)^4/d-3/5*a*cos(c+d*x)^5/d+1/6*b*cos(c+d*x)^6/d+1/7*a*cos(c+d*x)^7/d-b*log(cos(c+d*x))/d],
[(a+b*sec(c+d*x))*sin(c+d*x)^5,x,5,-a*cos(c+d*x)/d+b*cos(c+d*x)^2/d+2/3*a*cos(c+d*x)^3/d-1/4*b*cos(c+d*x)^4/d-1/5*a*cos(c+d*x)^5/d-b*log(cos(c+d*x))/d],
[(a+b*sec(c+d*x))*sin(c+d*x)^3,x,5,-a*cos(c+d*x)/d+1/2*b*cos(c+d*x)^2/d+1/3*a*cos(c+d*x)^3/d-b*log(cos(c+d*x))/d],
[(a+b*sec(c+d*x))*sin(c+d*x),x,4,-a*cos(c+d*x)/d-b*log(cos(c+d*x))/d],
[csc(c+d*x)*(a+b*sec(c+d*x)),x,5,-a*arctanh(cos(c+d*x))/d+b*log(tan(c+d*x))/d],
[csc(c+d*x)^3*(a+b*sec(c+d*x)),x,7,-1/2*a*arctanh(cos(c+d*x))/d-1/2*b*cot(c+d*x)^2/d-1/2*a*cot(c+d*x)*csc(c+d*x)/d+b*log(tan(c+d*x))/d],
[csc(c+d*x)^5*(a+b*sec(c+d*x)),x,9,-3/8*a*arctanh(cos(c+d*x))/d-b*cot(c+d*x)^2/d-1/4*b*cot(c+d*x)^4/d-3/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)*csc(c+d*x)^3/d+b*log(tan(c+d*x))/d],
[csc(c+d*x)^7*(a+b*sec(c+d*x)),x,10,-5/16*a*arctanh(cos(c+d*x))/d-3/2*b*cot(c+d*x)^2/d-3/4*b*cot(c+d*x)^4/d-1/6*b*cot(c+d*x)^6/d-5/16*a*cot(c+d*x)*csc(c+d*x)/d-5/24*a*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a*cot(c+d*x)*csc(c+d*x)^5/d+b*log(tan(c+d*x))/d],
[(a+b*sec(c+d*x))*sin(c+d*x)^6,x,10,5/16*a*x+b*arctanh(sin(c+d*x))/d-b*sin(c+d*x)/d-5/16*a*cos(c+d*x)*sin(c+d*x)/d-1/3*b*sin(c+d*x)^3/d-5/24*a*cos(c+d*x)*sin(c+d*x)^3/d-1/5*b*sin(c+d*x)^5/d-1/6*a*cos(c+d*x)*sin(c+d*x)^5/d],
[(a+b*sec(c+d*x))*sin(c+d*x)^4,x,9,3/8*a*x+b*arctanh(sin(c+d*x))/d-b*sin(c+d*x)/d-3/8*a*cos(c+d*x)*sin(c+d*x)/d-1/3*b*sin(c+d*x)^3/d-1/4*a*cos(c+d*x)*sin(c+d*x)^3/d],
[(a+b*sec(c+d*x))*sin(c+d*x)^2,x,7,1/2*a*x+b*arctanh(sin(c+d*x))/d-b*sin(c+d*x)/d-1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[csc(c+d*x)^2*(a+b*sec(c+d*x)),x,7,b*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-b*csc(c+d*x)/d],
[csc(c+d*x)^4*(a+b*sec(c+d*x)),x,8,b*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-1/3*a*cot(c+d*x)^3/d-b*csc(c+d*x)/d-1/3*b*csc(c+d*x)^3/d],
[csc(c+d*x)^6*(a+b*sec(c+d*x)),x,8,b*arctanh(sin(c+d*x))/d-a*cot(c+d*x)/d-2/3*a*cot(c+d*x)^3/d-1/5*a*cot(c+d*x)^5/d-b*csc(c+d*x)/d-1/3*b*csc(c+d*x)^3/d-1/5*b*csc(c+d*x)^5/d],
[(a+b*sec(c+d*x))^2*sin(c+d*x)^5,x,5,-(a^2-2*b^2)*cos(c+d*x)/d+2*a*b*cos(c+d*x)^2/d+1/3*(2*a^2-b^2)*cos(c+d*x)^3/d-1/2*a*b*cos(c+d*x)^4/d-1/5*a^2*cos(c+d*x)^5/d-2*a*b*log(cos(c+d*x))/d+b^2*sec(c+d*x)/d],
[(a+b*sec(c+d*x))^2*sin(c+d*x)^3,x,5,-(a^2-b^2)*cos(c+d*x)/d+a*b*cos(c+d*x)^2/d+1/3*a^2*cos(c+d*x)^3/d-2*a*b*log(cos(c+d*x))/d+b^2*sec(c+d*x)/d],
[(a+b*sec(c+d*x))^2*sin(c+d*x),x,5,-a^2*cos(c+d*x)/d-2*a*b*log(cos(c+d*x))/d+b^2*sec(c+d*x)/d],
[csc(c+d*x)*(a+b*sec(c+d*x))^2,x,5,1/2*(a+b)^2*log(1-cos(c+d*x))/d-2*a*b*log(cos(c+d*x))/d-1/2*(a-b)^2*log(1+cos(c+d*x))/d+b^2*sec(c+d*x)/d],
[csc(c+d*x)^3*(a+b*sec(c+d*x))^2,x,6,-1/2*(2*a*b+(a^2+b^2)*cos(c+d*x))*csc(c+d*x)^2/d+1/4*(a+b)*(a+3*b)*log(1-cos(c+d*x))/d-2*a*b*log(cos(c+d*x))/d-1/4*(a-3*b)*(a-b)*log(1+cos(c+d*x))/d+b^2*sec(c+d*x)/d],
[(a+b*sec(c+d*x))^2*sin(c+d*x)^6,x,12,5/16*(a^2-6*b^2)*x+2*a*b*arctanh(sin(c+d*x))/d-2*a*b*sin(c+d*x)/d-1/16*(11*a^2-18*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*(13*a^2-6*b^2)*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a^2*cos(c+d*x)^5*sin(c+d*x)/d-2/3*a*b*sin(c+d*x)^3/d-2/5*a*b*sin(c+d*x)^5/d+b^2*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^2*sin(c+d*x)^4,x,7,3/8*(a^2-4*b^2)*x+2*a*b*arctanh(sin(c+d*x))/d-1/6*b*(28*a^2+b^2)*sin(c+d*x)/(a*d)-1/24*(39*a^2+2*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/12*(12*a^2+b^2)*(b+a*cos(c+d*x))^2*sin(c+d*x)/(a*b*d)+1/4*(b+a*cos(c+d*x))^3*sin(c+d*x)/(a*d)+(b+a*cos(c+d*x))^3*tan(c+d*x)/(b*d)],
[(a+b*sec(c+d*x))^2*sin(c+d*x)^2,x,10,1/2*a^2*x-b^2*x+2*a*b*arctanh(sin(c+d*x))/d-2*a*b*sin(c+d*x)/d-1/2*a^2*cos(c+d*x)*sin(c+d*x)/d+b^2*tan(c+d*x)/d],
[csc(c+d*x)^2*(a+b*sec(c+d*x))^2,x,8,2*a*b*arctanh(sin(c+d*x))/d-(a^2+b^2)*cot(c+d*x)/d-2*a*b*csc(c+d*x)/d+b^2*tan(c+d*x)/d],
[csc(c+d*x)^4*(a+b*sec(c+d*x))^2,x,9,2*a*b*arctanh(sin(c+d*x))/d-(a^2+2*b^2)*cot(c+d*x)/d-1/3*(a^2+b^2)*cot(c+d*x)^3/d-2*a*b*csc(c+d*x)/d-2/3*a*b*csc(c+d*x)^3/d+b^2*tan(c+d*x)/d],
[csc(c+d*x)^6*(a+b*sec(c+d*x))^2,x,9,2*a*b*arctanh(sin(c+d*x))/d-(a^2+3*b^2)*cot(c+d*x)/d-1/3*(2*a^2+3*b^2)*cot(c+d*x)^3/d-1/5*(a^2+b^2)*cot(c+d*x)^5/d-2*a*b*csc(c+d*x)/d-2/3*a*b*csc(c+d*x)^3/d-2/5*a*b*csc(c+d*x)^5/d+b^2*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^3*sin(c+d*x)^5,x,5,-a*(a^2-6*b^2)*cos(c+d*x)/d+1/2*b*(6*a^2-b^2)*cos(c+d*x)^2/d+1/3*a*(2*a^2-3*b^2)*cos(c+d*x)^3/d-3/4*a^2*b*cos(c+d*x)^4/d-1/5*a^3*cos(c+d*x)^5/d-b*(3*a^2-2*b^2)*log(cos(c+d*x))/d+3*a*b^2*sec(c+d*x)/d+1/2*b^3*sec(c+d*x)^2/d],
[(a+b*sec(c+d*x))^3*sin(c+d*x)^3,x,4,-a*(a^2-3*b^2)*cos(c+d*x)/d+3/2*a^2*b*cos(c+d*x)^2/d+1/3*a^3*cos(c+d*x)^3/d-b*(3*a^2-b^2)*log(cos(c+d*x))/d+3*a*b^2*sec(c+d*x)/d+1/2*b^3*sec(c+d*x)^2/d],
[(a+b*sec(c+d*x))^3*sin(c+d*x),x,5,-a^3*cos(c+d*x)/d-3*a^2*b*log(cos(c+d*x))/d+3*a*b^2*sec(c+d*x)/d+1/2*b^3*sec(c+d*x)^2/d],
[csc(c+d*x)*(a+b*sec(c+d*x))^3,x,5,1/2*(a+b)^3*log(1-cos(c+d*x))/d-b*(3*a^2+b^2)*log(cos(c+d*x))/d-1/2*(a-b)^3*log(1+cos(c+d*x))/d+3*a*b^2*sec(c+d*x)/d+1/2*b^3*sec(c+d*x)^2/d],
[csc(c+d*x)^3*(a+b*sec(c+d*x))^3,x,6,-1/2*a^2*(b*(3+b^2/a^2)+a*(1+3*b^2/a^2)*cos(c+d*x))*csc(c+d*x)^2/d+1/4*(a+b)^2*(a+4*b)*log(1-cos(c+d*x))/d-b*(3*a^2+2*b^2)*log(cos(c+d*x))/d-1/4*(a-4*b)*(a-b)^2*log(1+cos(c+d*x))/d+3*a*b^2*sec(c+d*x)/d+1/2*b^3*sec(c+d*x)^2/d],
[(a+b*sec(c+d*x))^3*sin(c+d*x)^6,x,21,5/16*a^3*x-45/8*a*b^2*x+3*a^2*b*arctanh(sin(c+d*x))/d-5/2*b^3*arctanh(sin(c+d*x))/d-3*a^2*b*sin(c+d*x)/d+5/2*b^3*sin(c+d*x)/d-5/16*a^3*cos(c+d*x)*sin(c+d*x)/d-a^2*b*sin(c+d*x)^3/d+5/6*b^3*sin(c+d*x)^3/d-5/24*a^3*cos(c+d*x)*sin(c+d*x)^3/d-3/5*a^2*b*sin(c+d*x)^5/d-1/6*a^3*cos(c+d*x)*sin(c+d*x)^5/d+45/8*a*b^2*tan(c+d*x)/d-15/8*a*b^2*sin(c+d*x)^2*tan(c+d*x)/d-3/4*a*b^2*sin(c+d*x)^4*tan(c+d*x)/d+1/2*b^3*sin(c+d*x)^3*tan(c+d*x)^2/d],
[(a+b*sec(c+d*x))^3*sin(c+d*x)^4,x,8,3/8*a*(a^2-12*b^2)*x+3/2*b*(2*a^2-b^2)*arctanh(sin(c+d*x))/d-1/2*b*(17*a^2-b^2)*sin(c+d*x)/d-1/8*a*(21*a^2-2*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/4*(6*a^2-b^2)*(b+a*cos(c+d*x))^2*sin(c+d*x)/(b*d)-1/4*(4*a^2-b^2)*(b+a*cos(c+d*x))^3*sin(c+d*x)/(b^2*d)+a*(b+a*cos(c+d*x))^4*tan(c+d*x)/(b^2*d)+1/2*(b+a*cos(c+d*x))^4*sec(c+d*x)*tan(c+d*x)/(b*d)],
[(a+b*sec(c+d*x))^3*sin(c+d*x)^2,x,8,1/2*a*(a^2-6*b^2)*x+1/2*b*(6*a^2-b^2)*arctanh(sin(c+d*x))/d-15/2*a^2*b*sin(c+d*x)/d-5/2*a^3*cos(c+d*x)*sin(c+d*x)/d+3/2*a*(b+a*cos(c+d*x))^2*tan(c+d*x)/d+1/2*(b+a*cos(c+d*x))^3*sec(c+d*x)*tan(c+d*x)/d],
[csc(c+d*x)^2*(a+b*sec(c+d*x))^3,x,15,3*a^2*b*arctanh(sin(c+d*x))/d+3/2*b^3*arctanh(sin(c+d*x))/d-a^3*cot(c+d*x)/d-3*a*b^2*cot(c+d*x)/d-3*a^2*b*csc(c+d*x)/d-3/2*b^3*csc(c+d*x)/d+1/2*b^3*csc(c+d*x)*sec(c+d*x)^2/d+3*a*b^2*tan(c+d*x)/d],
[csc(c+d*x)^4*(a+b*sec(c+d*x))^3,x,17,3*a^2*b*arctanh(sin(c+d*x))/d+5/2*b^3*arctanh(sin(c+d*x))/d-a^3*cot(c+d*x)/d-6*a*b^2*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-a*b^2*cot(c+d*x)^3/d-3*a^2*b*csc(c+d*x)/d-5/2*b^3*csc(c+d*x)/d-a^2*b*csc(c+d*x)^3/d-5/6*b^3*csc(c+d*x)^3/d+1/2*b^3*csc(c+d*x)^3*sec(c+d*x)^2/d+3*a*b^2*tan(c+d*x)/d],
[csc(c+d*x)^6*(a+b*sec(c+d*x))^3,x,17,3*a^2*b*arctanh(sin(c+d*x))/d+7/2*b^3*arctanh(sin(c+d*x))/d-a^3*cot(c+d*x)/d-9*a*b^2*cot(c+d*x)/d-2/3*a^3*cot(c+d*x)^3/d-3*a*b^2*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d-3/5*a*b^2*cot(c+d*x)^5/d-3*a^2*b*csc(c+d*x)/d-7/2*b^3*csc(c+d*x)/d-a^2*b*csc(c+d*x)^3/d-7/6*b^3*csc(c+d*x)^3/d-3/5*a^2*b*csc(c+d*x)^5/d-7/10*b^3*csc(c+d*x)^5/d+1/2*b^3*csc(c+d*x)^5*sec(c+d*x)^2/d+3*a*b^2*tan(c+d*x)/d],

# n<0
[sin(c+d*x)^7/(a+b*sec(c+d*x)),x,5,-(a^2-b^2)^3*cos(c+d*x)/(a^7*d)-1/2*b*(3*a^4-3*a^2*b^2+b^4)*cos(c+d*x)^2/(a^6*d)+1/3*(3*a^4-3*a^2*b^2+b^4)*cos(c+d*x)^3/(a^5*d)+1/4*b*(3*a^2-b^2)*cos(c+d*x)^4/(a^4*d)-1/5*(3*a^2-b^2)*cos(c+d*x)^5/(a^3*d)-1/6*b*cos(c+d*x)^6/(a^2*d)+1/7*cos(c+d*x)^7/(a*d)+b*(a^2-b^2)^3*log(b+a*cos(c+d*x))/(a^8*d)],
[sin(c+d*x)^5/(a+b*sec(c+d*x)),x,5,-(a^2-b^2)^2*cos(c+d*x)/(a^5*d)-1/2*b*(2*a^2-b^2)*cos(c+d*x)^2/(a^4*d)+1/3*(2*a^2-b^2)*cos(c+d*x)^3/(a^3*d)+1/4*b*cos(c+d*x)^4/(a^2*d)-1/5*cos(c+d*x)^5/(a*d)+b*(a^2-b^2)^2*log(b+a*cos(c+d*x))/(a^6*d)],
[sin(c+d*x)^3/(a+b*sec(c+d*x)),x,5,-(a^2-b^2)*cos(c+d*x)/(a^3*d)-1/2*b*cos(c+d*x)^2/(a^2*d)+1/3*cos(c+d*x)^3/(a*d)+b*(a^2-b^2)*log(b+a*cos(c+d*x))/(a^4*d)],
[sin(c+d*x)/(a+b*sec(c+d*x)),x,5,-cos(c+d*x)/(a*d)+b*log(b+a*cos(c+d*x))/(a^2*d)],
[csc(c+d*x)/(a+b*sec(c+d*x)),x,4,1/2*log(1-cos(c+d*x))/((a+b)*d)-1/2*log(1+cos(c+d*x))/((a-b)*d)+b*log(b+a*cos(c+d*x))/((a^2-b^2)*d)],
[csc(c+d*x)^3/(a+b*sec(c+d*x)),x,6,1/2*(b-a*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)*d)+1/4*a*log(1-cos(c+d*x))/((a+b)^2*d)-1/4*a*log(1+cos(c+d*x))/((a-b)^2*d)+a^2*b*log(b+a*cos(c+d*x))/((a^2-b^2)^2*d)],
[csc(c+d*x)^5/(a+b*sec(c+d*x)),x,7,1/8*(4*a^2*b-a*(3*a^2+b^2)*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)^2*d)+1/4*(b-a*cos(c+d*x))*csc(c+d*x)^4/((a^2-b^2)*d)+1/16*a*(3*a+b)*log(1-cos(c+d*x))/((a+b)^3*d)-1/16*a*(3*a-b)*log(1+cos(c+d*x))/((a-b)^3*d)+a^4*b*log(b+a*cos(c+d*x))/((a^2-b^2)^3*d)],
[sin(c+d*x)^6/(a+b*sec(c+d*x)),x,7,1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*x/a^7-2*(a-b)^(5/2)*b*(a+b)^(5/2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^7*d)+1/16*(16*b*(a^2-b^2)^2-a*(5*a^4-14*a^2*b^2+8*b^4)*cos(c+d*x))*sin(c+d*x)/(a^6*d)+1/24*(8*b*(a^2-b^2)-a*(5*a^2-6*b^2)*cos(c+d*x))*sin(c+d*x)^3/(a^4*d)+1/30*(6*b-5*a*cos(c+d*x))*sin(c+d*x)^5/(a^2*d)],
[sin(c+d*x)^4/(a+b*sec(c+d*x)),x,6,1/8*(3*a^4-12*a^2*b^2+8*b^4)*x/a^5-2*(a-b)^(3/2)*b*(a+b)^(3/2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*d)+1/8*(8*b*(a^2-b^2)-a*(3*a^2-4*b^2)*cos(c+d*x))*sin(c+d*x)/(a^4*d)+1/12*(4*b-3*a*cos(c+d*x))*sin(c+d*x)^3/(a^2*d)],
[sin(c+d*x)^2/(a+b*sec(c+d*x)),x,5,1/2*(a^2-2*b^2)*x/a^3+1/2*(2*b-a*cos(c+d*x))*sin(c+d*x)/(a^2*d)-2*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/(a^3*d)],
[csc(c+d*x)^2/(a+b*sec(c+d*x)),x,5,-2*a*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)+(b-a*cos(c+d*x))*csc(c+d*x)/((a^2-b^2)*d)],
[csc(c+d*x)^4/(a+b*sec(c+d*x)),x,6,-2*a^3*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)+1/3*(3*a^2*b-a*(2*a^2+b^2)*cos(c+d*x))*csc(c+d*x)/((a^2-b^2)^2*d)+1/3*(b-a*cos(c+d*x))*csc(c+d*x)^3/((a^2-b^2)*d)],
[csc(c+d*x)^6/(a+b*sec(c+d*x)),x,7,-2*a^5*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)+1/15*(15*a^4*b-a*(8*a^4+9*a^2*b^2-2*b^4)*cos(c+d*x))*csc(c+d*x)/((a^2-b^2)^3*d)+1/15*(5*a^2*b-a*(4*a^2+b^2)*cos(c+d*x))*csc(c+d*x)^3/((a^2-b^2)^2*d)+1/5*(b-a*cos(c+d*x))*csc(c+d*x)^5/((a^2-b^2)*d)],
[sin(c+d*x)^7/(a+b*sec(c+d*x))^2,x,5,-(a^2-7*b^2)*(a^2-b^2)^2*cos(c+d*x)/(a^8*d)-3*b*(a^2-b^2)^2*cos(c+d*x)^2/(a^7*d)+1/3*(3*a^4-9*a^2*b^2+5*b^4)*cos(c+d*x)^3/(a^6*d)+1/2*b*(3*a^2-2*b^2)*cos(c+d*x)^4/(a^5*d)-3/5*(a^2-b^2)*cos(c+d*x)^5/(a^4*d)-1/3*b*cos(c+d*x)^6/(a^3*d)+1/7*cos(c+d*x)^7/(a^2*d)+b^2*(a^2-b^2)^3/(a^9*d*(b+a*cos(c+d*x)))+2*b*(a^2-4*b^2)*(a^2-b^2)^2*log(b+a*cos(c+d*x))/(a^9*d)],
[sin(c+d*x)^5/(a+b*sec(c+d*x))^2,x,5,-(a^4-6*a^2*b^2+5*b^4)*cos(c+d*x)/(a^6*d)-2*b*(a^2-b^2)*cos(c+d*x)^2/(a^5*d)+1/3*(2*a^2-3*b^2)*cos(c+d*x)^3/(a^4*d)+1/2*b*cos(c+d*x)^4/(a^3*d)-1/5*cos(c+d*x)^5/(a^2*d)+b^2*(a^2-b^2)^2/(a^7*d*(b+a*cos(c+d*x)))+2*b*(a^4-4*a^2*b^2+3*b^4)*log(b+a*cos(c+d*x))/(a^7*d)],
[sin(c+d*x)^3/(a+b*sec(c+d*x))^2,x,5,-(a^2-3*b^2)*cos(c+d*x)/(a^4*d)-b*cos(c+d*x)^2/(a^3*d)+1/3*cos(c+d*x)^3/(a^2*d)+b^2*(a^2-b^2)/(a^5*d*(b+a*cos(c+d*x)))+2*b*(a^2-2*b^2)*log(b+a*cos(c+d*x))/(a^5*d)],
[sin(c+d*x)/(a+b*sec(c+d*x))^2,x,5,-cos(c+d*x)/(a^2*d)+b^2/(a^3*d*(b+a*cos(c+d*x)))+2*b*log(b+a*cos(c+d*x))/(a^3*d)],
[csc(c+d*x)/(a+b*sec(c+d*x))^2,x,5,b^2/(a*(a^2-b^2)*d*(b+a*cos(c+d*x)))+1/2*log(1-cos(c+d*x))/((a+b)^2*d)-1/2*log(1+cos(c+d*x))/((a-b)^2*d)+2*a*b*log(b+a*cos(c+d*x))/((a^2-b^2)^2*d)],
[csc(c+d*x)^3/(a+b*sec(c+d*x))^2,x,6,a*b^2/((a^2-b^2)^2*d*(b+a*cos(c+d*x)))+1/2*(2*a*b-(a^2+b^2)*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)^2*d)+1/4*(a-b)*log(1-cos(c+d*x))/((a+b)^3*d)-1/4*(a+b)*log(1+cos(c+d*x))/((a-b)^3*d)+2*a*b*(a^2+b^2)*log(b+a*cos(c+d*x))/((a^2-b^2)^3*d)],
[csc(c+d*x)^5/(a+b*sec(c+d*x))^2,x,7,a^3*b^2/((a^2-b^2)^3*d*(b+a*cos(c+d*x)))+1/8*(8*a*b*(a^2+b^2)-(3*a^4+12*a^2*b^2+b^4)*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)^3*d)+1/4*(2*a*b-(a^2+b^2)*cos(c+d*x))*csc(c+d*x)^4/((a^2-b^2)^2*d)+1/16*(3*a^2-4*a*b-b^2)*log(1-cos(c+d*x))/((a+b)^4*d)-1/16*(3*a^2+4*a*b-b^2)*log(1+cos(c+d*x))/((a-b)^4*d)+2*a^3*b*(a^2+2*b^2)*log(b+a*cos(c+d*x))/((a^2-b^2)^4*d)],
[sin(c+d*x)^6/(a+b*sec(c+d*x))^2,x,10,1/16*(5*a^6-90*a^4*b^2+200*a^2*b^4-112*b^6)*x/a^8-2*(a-b)^(3/2)*b*(a+b)^(3/2)*(2*a^2-7*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^8*d)+1/15*b*(61*a^4-170*a^2*b^2+105*b^4)*sin(c+d*x)/(a^7*d)-1/16*(27*a^4-86*a^2*b^2+56*b^4)*cos(c+d*x)*sin(c+d*x)/(a^6*d)+1/15*(15*a^4-52*a^2*b^2+35*b^4)*cos(c+d*x)^2*sin(c+d*x)/(a^5*b*d)-1/24*(16*a^4-61*a^2*b^2+42*b^4)*cos(c+d*x)^3*sin(c+d*x)/(a^4*b^2*d)-1/3*cos(c+d*x)^3*sin(c+d*x)/(b*d*(b+a*cos(c+d*x)))+1/6*a*cos(c+d*x)^4*sin(c+d*x)/(b^2*d*(b+a*cos(c+d*x)))+1/10*(5*a^4-20*a^2*b^2+14*b^4)*cos(c+d*x)^4*sin(c+d*x)/(a^3*b^2*d*(b+a*cos(c+d*x)))+7/30*b*cos(c+d*x)^5*sin(c+d*x)/(a^2*d*(b+a*cos(c+d*x)))-1/6*cos(c+d*x)^6*sin(c+d*x)/(a*d*(b+a*cos(c+d*x)))],
[sin(c+d*x)^4/(a+b*sec(c+d*x))^2,x,8,1/8*(3*a^4-36*a^2*b^2+40*b^4)*x/a^6+1/3*b*(11*a^2-15*b^2)*sin(c+d*x)/(a^5*d)-1/8*(13*a^2-20*b^2)*cos(c+d*x)*sin(c+d*x)/(a^4*d)+1/3*(3*a^2-5*b^2)*cos(c+d*x)^2*sin(c+d*x)/(a^3*b*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-(a^2-b^2)*cos(c+d*x)^3*sin(c+d*x)/(a^2*b*d*(b+a*cos(c+d*x)))-2*b*(2*a^2-5*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/(a^6*d)],
[sin(c+d*x)^2/(a+b*sec(c+d*x))^2,x,8,1/2*(a^2-6*b^2)*x/a^4+3*b*sin(c+d*x)/(a^3*d)-3/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)+cos(c+d*x)^2*sin(c+d*x)/(a*d*(b+a*cos(c+d*x)))-2*b*(2*a^2-3*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*d*sqrt(a-b)*sqrt(a+b))],
[csc(c+d*x)^2/(a+b*sec(c+d*x))^2,x,11,-4*a^2*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-2*b^3*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-1/2*sin(c+d*x)/((a+b)^2*d*(1-cos(c+d*x)))+1/2*sin(c+d*x)/((a-b)^2*d*(1+cos(c+d*x)))+a*b^2*sin(c+d*x)/((a^2-b^2)^2*d*(b+a*cos(c+d*x)))],
[csc(c+d*x)^4/(a+b*sec(c+d*x))^2,x,15,-2*a^2*b^3*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-4*a^2*b*(a^2+b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/12*sin(c+d*x)/((a+b)^2*d*(1-cos(c+d*x))^2)-1/4*(a-b)*sin(c+d*x)/((a+b)^3*d*(1-cos(c+d*x)))-1/12*sin(c+d*x)/((a+b)^2*d*(1-cos(c+d*x)))+1/12*sin(c+d*x)/((a-b)^2*d*(1+cos(c+d*x))^2)+1/12*sin(c+d*x)/((a-b)^2*d*(1+cos(c+d*x)))+1/4*(a+b)*sin(c+d*x)/((a-b)^3*d*(1+cos(c+d*x)))+a^3*b^2*sin(c+d*x)/((a^2-b^2)^3*d*(b+a*cos(c+d*x)))],
[sin(c+d*x)^7/(a+b*sec(c+d*x))^3,x,5,-(a^6-18*a^4*b^2+45*a^2*b^4-28*b^6)*cos(c+d*x)/(a^9*d)-3/2*b*(3*a^4-10*a^2*b^2+7*b^4)*cos(c+d*x)^2/(a^8*d)+(a^4-6*a^2*b^2+5*b^4)*cos(c+d*x)^3/(a^7*d)+1/4*b*(9*a^2-10*b^2)*cos(c+d*x)^4/(a^6*d)-3/5*(a^2-2*b^2)*cos(c+d*x)^5/(a^5*d)-1/2*b*cos(c+d*x)^6/(a^4*d)+1/7*cos(c+d*x)^7/(a^3*d)-1/2*b^3*(a^2-b^2)^3/(a^10*d*(b+a*cos(c+d*x))^2)+3*b^2*(a^2-3*b^2)*(a^2-b^2)^2/(a^10*d*(b+a*cos(c+d*x)))+3*b*(a^2-b^2)*(a^4-9*a^2*b^2+12*b^4)*log(b+a*cos(c+d*x))/(a^10*d)],
[sin(c+d*x)^5/(a+b*sec(c+d*x))^3,x,5,-(a^4-12*a^2*b^2+15*b^4)*cos(c+d*x)/(a^7*d)-b*(3*a^2-5*b^2)*cos(c+d*x)^2/(a^6*d)+2/3*(a^2-3*b^2)*cos(c+d*x)^3/(a^5*d)+3/4*b*cos(c+d*x)^4/(a^4*d)-1/5*cos(c+d*x)^5/(a^3*d)-1/2*b^3*(a^2-b^2)^2/(a^8*d*(b+a*cos(c+d*x))^2)+b^2*(3*a^4-10*a^2*b^2+7*b^4)/(a^8*d*(b+a*cos(c+d*x)))+b*(3*a^4-20*a^2*b^2+21*b^4)*log(b+a*cos(c+d*x))/(a^8*d)],
[sin(c+d*x)^3/(a+b*sec(c+d*x))^3,x,5,-(a^2-6*b^2)*cos(c+d*x)/(a^5*d)-3/2*b*cos(c+d*x)^2/(a^4*d)+1/3*cos(c+d*x)^3/(a^3*d)-1/2*b^3*(a^2-b^2)/(a^6*d*(b+a*cos(c+d*x))^2)+b^2*(3*a^2-5*b^2)/(a^6*d*(b+a*cos(c+d*x)))+b*(3*a^2-10*b^2)*log(b+a*cos(c+d*x))/(a^6*d)],
[sin(c+d*x)/(a+b*sec(c+d*x))^3,x,5,-cos(c+d*x)/(a^3*d)-1/2*b^3/(a^4*d*(b+a*cos(c+d*x))^2)+3*b^2/(a^4*d*(b+a*cos(c+d*x)))+3*b*log(b+a*cos(c+d*x))/(a^4*d)],
[csc(c+d*x)/(a+b*sec(c+d*x))^3,x,5,-1/2*b^3/(a^2*(a^2-b^2)*d*(b+a*cos(c+d*x))^2)+b^2*(3*a^2-b^2)/(a^2*(a^2-b^2)^2*d*(b+a*cos(c+d*x)))+1/2*log(1-cos(c+d*x))/((a+b)^3*d)-1/2*log(1+cos(c+d*x))/((a-b)^3*d)+b*(3*a^2+b^2)*log(b+a*cos(c+d*x))/((a^2-b^2)^3*d)],
[csc(c+d*x)^3/(a+b*sec(c+d*x))^3,x,5,-1/2*b^3/((a^2-b^2)^2*d*(b+a*cos(c+d*x))^2)+b^2*(3*a^2+b^2)/((a^2-b^2)^3*d*(b+a*cos(c+d*x)))+1/2*(b*(3*a^2+b^2)-a*(a^2+3*b^2)*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)^3*d)+1/4*(a-2*b)*log(1-cos(c+d*x))/((a+b)^4*d)-1/4*(a+2*b)*log(1+cos(c+d*x))/((a-b)^4*d)+b*(3*a^4+8*a^2*b^2+b^4)*log(b+a*cos(c+d*x))/((a^2-b^2)^4*d)],
[csc(c+d*x)^5/(a+b*sec(c+d*x))^3,x,7,-1/2*a^2*b^3/((a^2-b^2)^3*d*(b+a*cos(c+d*x))^2)+3*a^2*b^2*(a^2+b^2)/((a^2-b^2)^4*d*(b+a*cos(c+d*x)))+1/8*(4*b*(3*a^4+8*a^2*b^2+b^4)-3*a*(a^4+10*a^2*b^2+5*b^4)*cos(c+d*x))*csc(c+d*x)^2/((a^2-b^2)^4*d)+1/4*(b*(3*a^2+b^2)-a*(a^2+3*b^2)*cos(c+d*x))*csc(c+d*x)^4/((a^2-b^2)^3*d)+3/16*a*(a-3*b)*log(1-cos(c+d*x))/((a+b)^5*d)-3/16*a*(a+3*b)*log(1+cos(c+d*x))/((a-b)^5*d)+3*a^2*b*(a^4+5*a^2*b^2+2*b^4)*log(b+a*cos(c+d*x))/((a^2-b^2)^5*d)],
[sin(c+d*x)^6/(a+b*sec(c+d*x))^3,x,11,1/16*(5*a^6-180*a^4*b^2+600*a^2*b^4-448*b^6)*x/a^9+1/30*b*(213*a^4-985*a^2*b^2+840*b^4)*sin(c+d*x)/(a^8*d)-1/16*(43*a^4-244*a^2*b^2+224*b^4)*cos(c+d*x)*sin(c+d*x)/(a^7*d)+1/30*(45*a^4-291*a^2*b^2+280*b^4)*cos(c+d*x)^2*sin(c+d*x)/(a^6*b*d)-1/24*(24*a^4-169*a^2*b^2+168*b^4)*cos(c+d*x)^3*sin(c+d*x)/(a^5*b^2*d)-1/4*cos(c+d*x)^4*sin(c+d*x)/(b*d*(b+a*cos(c+d*x))^2)+1/10*a*cos(c+d*x)^5*sin(c+d*x)/(b^2*d*(b+a*cos(c+d*x))^2)+1/60*(9*a^4-60*a^2*b^2+56*b^4)*cos(c+d*x)^5*sin(c+d*x)/(a^3*b^2*d*(b+a*cos(c+d*x))^2)+4/15*b*cos(c+d*x)^6*sin(c+d*x)/(a^2*d*(b+a*cos(c+d*x))^2)-1/6*cos(c+d*x)^7*sin(c+d*x)/(a*d*(b+a*cos(c+d*x))^2)+1/20*(15*a^4-110*a^2*b^2+112*b^4)*cos(c+d*x)^4*sin(c+d*x)/(a^4*b^2*d*(b+a*cos(c+d*x)))-b*(6*a^4-47*a^2*b^2+56*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/(a^9*d)],
[sin(c+d*x)^4/(a+b*sec(c+d*x))^3,x,9,3/8*(a^4-24*a^2*b^2+40*b^4)*x/a^7+1/2*b*(13*a^2-30*b^2)*sin(c+d*x)/(a^6*d)-3/8*(7*a^2-20*b^2)*cos(c+d*x)*sin(c+d*x)/(a^5*d)+1/2*(3*a^2-10*b^2)*cos(c+d*x)^2*sin(c+d*x)/(a^4*b*d)-1/4*(4*a^2-15*b^2)*cos(c+d*x)^3*sin(c+d*x)/(a^3*b^2*d)-1/2*(a^2-b^2)*cos(c+d*x)^4*sin(c+d*x)/(a^2*b*d*(b+a*cos(c+d*x))^2)+1/2*(2*a^2-7*b^2)*cos(c+d*x)^4*sin(c+d*x)/(a^2*b^2*d*(b+a*cos(c+d*x)))-3*b*(2*a^4-11*a^2*b^2+10*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^7*d*sqrt(a-b)*sqrt(a+b))],
[sin(c+d*x)^2/(a+b*sec(c+d*x))^3,x,9,1/2*(a^2-12*b^2)*x/a^5-b*(6*a^4-19*a^2*b^2+12*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(3/2)*(a+b)^(3/2)*d)+1/2*b*(11*a^2-12*b^2)*sin(c+d*x)/(a^4*(a^2-b^2)*d)-1/2*(5*a^2-6*b^2)*cos(c+d*x)*sin(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*cos(c+d*x)^3*sin(c+d*x)/(a*d*(b+a*cos(c+d*x))^2)+1/2*(3*a^2-4*b^2)*cos(c+d*x)^2*sin(c+d*x)/(a^2*(a^2-b^2)*d*(b+a*cos(c+d*x)))],
[csc(c+d*x)^2/(a+b*sec(c+d*x))^3,x,16,-2*b^3*(3*a^2-b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*(a-b)^(7/2)*(a+b)^(7/2)*d)-2*a*b*(3*a^2+b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-b^3*(a^2+2*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*(a-b)^(7/2)*(a+b)^(7/2)*d)-1/2*sin(c+d*x)/((a+b)^3*d*(1-cos(c+d*x)))+1/2*sin(c+d*x)/((a-b)^3*d*(1+cos(c+d*x)))-1/2*b^3*sin(c+d*x)/((a^2-b^2)^2*d*(b+a*cos(c+d*x))^2)+3/2*b^4*sin(c+d*x)/((a^2-b^2)^3*d*(b+a*cos(c+d*x)))+b^2*(3*a^2-b^2)*sin(c+d*x)/((a^2-b^2)^3*d*(b+a*cos(c+d*x)))],
[csc(c+d*x)^4/(a+b*sec(c+d*x))^3,x,20,-2*a*b^3*(3*a^2+b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(9/2)*(a+b)^(9/2)*d)-a*b^3*(a^2+2*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(9/2)*(a+b)^(9/2)*d)-2*a*b*(3*a^4+8*a^2*b^2+b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(9/2)*(a+b)^(9/2)*d)-1/12*sin(c+d*x)/((a+b)^3*d*(1-cos(c+d*x))^2)-1/4*(a-2*b)*sin(c+d*x)/((a+b)^4*d*(1-cos(c+d*x)))-1/12*sin(c+d*x)/((a+b)^3*d*(1-cos(c+d*x)))+1/12*sin(c+d*x)/((a-b)^3*d*(1+cos(c+d*x))^2)+1/12*sin(c+d*x)/((a-b)^3*d*(1+cos(c+d*x)))+1/4*(a+2*b)*sin(c+d*x)/((a-b)^4*d*(1+cos(c+d*x)))-1/2*a^2*b^3*sin(c+d*x)/((a^2-b^2)^3*d*(b+a*cos(c+d*x))^2)+3/2*a^2*b^4*sin(c+d*x)/((a^2-b^2)^4*d*(b+a*cos(c+d*x)))+a^2*b^2*(3*a^2+b^2)*sin(c+d*x)/((a^2-b^2)^4*d*(b+a*cos(c+d*x)))],

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

# n>0

# n<0
[(e*sin(c+d*x))^(7/2)/(a+b*sec(c+d*x)),x,15,-b*(a^2-b^2)^(5/4)*e^(7/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*d)-b*(a^2-b^2)^(5/4)*e^(7/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*d)+2/35*e*(7*b-5*a*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(a^2*d)+2/21*(5*a^4-28*a^2*b^2+21*b^4)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*sqrt(e*sin(c+d*x)))+b^2*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+b^2*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2/21*e^3*(21*b*(a^2-b^2)-a*(5*a^2-7*b^2)*cos(c+d*x))*sqrt(e*sin(c+d*x))/(a^4*d)],
[(e*sin(c+d*x))^(5/2)/(a+b*sec(c+d*x)),x,14,b*(a^2-b^2)^(3/4)*e^(5/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*d)-b*(a^2-b^2)^(3/4)*e^(5/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*d)+2/15*e*(5*b-3*a*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(a^2*d)-b^2*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-b^2*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2/5*(3*a^2-5*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^3*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+b*sec(c+d*x)),x,14,-b*(a^2-b^2)^(1/4)*e^(3/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(5/2)*d)-b*(a^2-b^2)^(1/4)*e^(3/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(5/2)*d)+2/3*(a^2-3*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*d*sqrt(e*sin(c+d*x)))+b^2*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+b^2*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2/3*e*(3*b-a*cos(c+d*x))*sqrt(e*sin(c+d*x))/(a^2*d)],
[(e*sin(c+d*x))^(1/2)/(a+b*sec(c+d*x)),x,13,b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(3/2)*(a^2-b^2)^(1/4)*d)-b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(3/2)*(a^2-b^2)^(1/4)*d)-b^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-b^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a*d*sqrt(sin(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*sin(c+d*x))^(1/2)),x,13,-b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(3/4)*d*sqrt(a)*sqrt(e))-b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(3/4)*d*sqrt(a)*sqrt(e))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*sqrt(e*sin(c+d*x)))+b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*sin(c+d*x))^(3/2)),x,14,b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(5/4)*d*e^(3/2))-b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(5/4)*d*e^(3/2))+2*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*sqrt(e*sin(c+d*x)))-b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*sin(c+d*x))^(5/2)),x,14,-a^(3/2)*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(7/4)*d*e^(5/2))-a^(3/2)*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(7/4)*d*e^(5/2))+2/3*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*(e*sin(c+d*x))^(3/2))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*sqrt(e*sin(c+d*x)))+a*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+a*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*sin(c+d*x))^(7/2)),x,15,a^(5/2)*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(9/4)*d*e^(7/2))-a^(5/2)*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(9/4)*d*e^(7/2))+2/5*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*(e*sin(c+d*x))^(5/2))+2/5*(5*a^2*b-a*(3*a^2+2*b^2)*cos(c+d*x))/((a^2-b^2)^2*d*e^3*sqrt(e*sin(c+d*x)))-a^2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^3*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-a^2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^3*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2/5*a*(3*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^2*d*e^4*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(9/2)/(a+b*sec(c+d*x))^2,x,35,-7/2*b^3*(a^2-b^2)^(3/4)*e^(9/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(13/2)*d)+2*b*(a^2-b^2)^(7/4)*e^(9/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(13/2)*d)+7/2*b^3*(a^2-b^2)^(3/4)*e^(9/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(13/2)*d)-2*b*(a^2-b^2)^(7/4)*e^(9/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(13/2)*d)-14/45*e^3*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/(a^2*d)-7/15*b^2*e^3*(5*b-3*a*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(a^5*d)+4/15*b*e^3*(5*(a^2-b^2)+3*a*b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(a^5*d)+4/7*b*e*(e*sin(c+d*x))^(7/2)/(a^3*d)-2/9*e*cos(c+d*x)*(e*sin(c+d*x))^(7/2)/(a^2*d)+b^2*e*(e*sin(c+d*x))^(7/2)/(a^3*d*(b+a*cos(c+d*x)))+7/2*b^4*(a^2-b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^7*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^7*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+7/2*b^4*(a^2-b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^7*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^7*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+14/15*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*sqrt(sin(c+d*x)))-7/5*b^2*(3*a^2-5*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^6*d*sqrt(sin(c+d*x)))-4/5*b^2*(8*a^2-5*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^6*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(7/2)/(a+b*sec(c+d*x))^2,x,35,5/2*b^3*(a^2-b^2)^(1/4)*e^(7/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(11/2)*d)-2*b*(a^2-b^2)^(5/4)*e^(7/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(11/2)*d)+5/2*b^3*(a^2-b^2)^(1/4)*e^(7/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(11/2)*d)-2*b*(a^2-b^2)^(5/4)*e^(7/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(11/2)*d)+4/5*b*e*(e*sin(c+d*x))^(5/2)/(a^3*d)-2/7*e*cos(c+d*x)*(e*sin(c+d*x))^(5/2)/(a^2*d)+b^2*e*(e*sin(c+d*x))^(5/2)/(a^3*d*(b+a*cos(c+d*x)))+10/21*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))-5/3*b^2*(a^2-3*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*sqrt(e*sin(c+d*x)))-4/3*b^2*(4*a^2-3*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*sqrt(e*sin(c+d*x)))-5/2*b^4*(a^2-b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-5/2*b^4*(a^2-b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^6*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-10/21*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a^2*d)-5/3*b^2*e^3*(3*b-a*cos(c+d*x))*sqrt(e*sin(c+d*x))/(a^5*d)+4/3*b*e^3*(3*(a^2-b^2)+a*b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(a^5*d)],
[(e*sin(c+d*x))^(5/2)/(a+b*sec(c+d*x))^2,x,32,-3/2*b^3*e^(5/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*(a^2-b^2)^(1/4)*d)+2*b*(a^2-b^2)^(3/4)*e^(5/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*d)+3/2*b^3*e^(5/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*(a^2-b^2)^(1/4)*d)-2*b*(a^2-b^2)^(3/4)*e^(5/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(9/2)*d)+4/3*b*e*(e*sin(c+d*x))^(3/2)/(a^3*d)-2/5*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/(a^2*d)+b^2*e*(e*sin(c+d*x))^(3/2)/(a^3*d*(b+a*cos(c+d*x)))+3/2*b^4*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+3/2*b^4*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^5*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+6/5*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*sqrt(sin(c+d*x)))-7*b^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^4*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+b*sec(c+d*x))^2,x,32,1/2*b^3*e^(3/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*(a^2-b^2)^(3/4)*d)-2*b*(a^2-b^2)^(1/4)*e^(3/2)*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*d)+1/2*b^3*e^(3/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*(a^2-b^2)^(3/4)*d)-2*b*(a^2-b^2)^(1/4)*e^(3/2)*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(7/2)*d)+2/3*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))-5*b^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*sqrt(e*sin(c+d*x)))-1/2*b^4*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-1/2*b^4*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^4*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+4*b*e*sqrt(e*sin(c+d*x))/(a^3*d)-2/3*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/(a^2*d)+b^2*e*sqrt(e*sin(c+d*x))/(a^3*d*(b+a*cos(c+d*x)))],
[(e*sin(c+d*x))^(1/2)/(a+b*sec(c+d*x))^2,x,27,b^2*(e*sin(c+d*x))^(3/2)/(a*(a^2-b^2)*d*e*(b+a*cos(c+d*x)))+1/2*b^3*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(5/2)*(a^2-b^2)^(5/4)*d)+2*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(5/2)*(a^2-b^2)^(1/4)*d)-1/2*b^3*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(5/2)*(a^2-b^2)^(5/4)*d)-2*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(e)/(a^(5/2)*(a^2-b^2)^(1/4)*d)-2*b^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-1/2*b^4*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*(a^2-b^2)*d*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-1/2*b^4*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^3*(a^2-b^2)*d*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*sqrt(sin(c+d*x)))-b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(sin(c+d*x)))],
[1/((a+b*sec(c+d*x))^2*(e*sin(c+d*x))^(1/2)),x,27,-3/2*b^3*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(3/2)*(a^2-b^2)^(7/4)*d*sqrt(e))-2*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(3/2)*(a^2-b^2)^(3/4)*d*sqrt(e))-3/2*b^3*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(3/2)*(a^2-b^2)^(7/4)*d*sqrt(e))-2*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/(a^(3/2)*(a^2-b^2)^(3/4)*d*sqrt(e))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*sqrt(e*sin(c+d*x)))+b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(e*sin(c+d*x)))+2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+3/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*(a^2-b^2)*d*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+3/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*(a^2-b^2)*d*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+b^2*sqrt(e*sin(c+d*x))/(a*(a^2-b^2)*d*e*(b+a*cos(c+d*x)))],
[1/((a+b*sec(c+d*x))^2*(e*sin(c+d*x))^(3/2)),x,33,5/2*b^3*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(9/4)*d*e^(3/2)*sqrt(a))+2*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(5/4)*d*e^(3/2)*sqrt(a))-5/2*b^3*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(9/4)*d*e^(3/2)*sqrt(a))-2*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))/((a^2-b^2)^(5/4)*d*e^(3/2)*sqrt(a))-2*cos(c+d*x)/(a^2*d*e*sqrt(e*sin(c+d*x)))+b^2/(a*(a^2-b^2)*d*e*(b+a*cos(c+d*x))*sqrt(e*sin(c+d*x)))+4*b*(a-b*cos(c+d*x))/(a^2*(a^2-b^2)*d*e*sqrt(e*sin(c+d*x)))+b^2*(5*a*b-(3*a^2+2*b^2)*cos(c+d*x))/(a^2*(a^2-b^2)^2*d*e*sqrt(e*sin(c+d*x)))-5/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*(a^2-b^2)^2*d*e*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*(a^2-b^2)*d*e*(a-sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-5/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*(a^2-b^2)^2*d*e*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/(a*(a^2-b^2)*d*e*(a+sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))-2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*d*e^2*sqrt(sin(c+d*x)))-4*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*(a^2-b^2)*d*e^2*sqrt(sin(c+d*x)))-b^2*(3*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(a^2*(a^2-b^2)^2*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+b*sec(c+d*x))^2*(e*sin(c+d*x))^(5/2)),x,33,-2/3*cos(c+d*x)/(a^2*d*e*(e*sin(c+d*x))^(3/2))+b^2/(a*(a^2-b^2)*d*e*(b+a*cos(c+d*x))*(e*sin(c+d*x))^(3/2))+4/3*b*(a-b*cos(c+d*x))/(a^2*(a^2-b^2)*d*e*(e*sin(c+d*x))^(3/2))+1/3*b^2*(7*a*b-(5*a^2+2*b^2)*cos(c+d*x))/(a^2*(a^2-b^2)^2*d*e*(e*sin(c+d*x))^(3/2))-7/2*b^3*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(11/4)*d*e^(5/2))-2*b*arctan(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(7/4)*d*e^(5/2))-7/2*b^3*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(11/4)*d*e^(5/2))-2*b*arctanh(sqrt(a)*sqrt(e*sin(c+d*x))/((a^2-b^2)^(1/4)*sqrt(e)))*sqrt(a)/((a^2-b^2)^(7/4)*d*e^(5/2))+2/3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*d*e^2*sqrt(e*sin(c+d*x)))+4/3*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*(a^2-b^2)*d*e^2*sqrt(e*sin(c+d*x)))+1/3*b^2*(5*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(a^2*(a^2-b^2)^2*d*e^2*sqrt(e*sin(c+d*x)))+7/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a-sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b^2-a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+7/2*b^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))+2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*a/(a+sqrt(a^2-b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b^2+a*sqrt(a^2-b^2))*sqrt(e*sin(c+d*x)))],

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

# n>0
#  {Sin[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]], x, 11, -(((a - b)*Sqrt[a + b]*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(7680*a^5*f*Sqrt[b + a*Cos[e + f*x]])) + (1/(7680*a^5*f*Sqrt[b + a*Cos[e + f*x]]))*Sqrt[a + b]*(10080*a^5 - 3248*a^4*b - 928*a^3*b^2 + 1416*a^2*b^3 + 210*a*b^4 - 315*b^5)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]] - (Sqrt[a + b]*(320*a^6 + 240*a^4*b^2 - 100*a^2*b^4 + 21*b^6)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(512*a^6*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(7680*a^5*f) - ((880*a^4 - 416*a^2*b^2 + 105*b^4)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(1280*a^4*f) - (2*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(b*f) + ((1920*a^4 - 428*a^2*b^2 + 105*b^4)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(960*a^4*b*f) + (8*a*Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*b^2*f) - ((1280*a^4 - 260*a^2*b^2 + 63*b^4)*Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(480*a^3*b^2*f) + (3*b*Cos[e + f*x]^3*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(20*a^2*f) - (Cos[e + f*x]^4*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(6*a*f)}
# {Sin[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]], x, 9, -(((a - b)*Sqrt[a + b]*(68*a^2 - 15*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(192*a^3*f*Sqrt[b + a*Cos[e + f*x]])) + (Sqrt[a + b]*(264*a^3 - 68*a^2*b - 10*a*b^2 + 15*b^3)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(192*a^3*f*Sqrt[b + a*Cos[e + f*x]]) - (Sqrt[a + b]*(48*a^4 + 24*a^2*b^2 - 5*b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^4*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(68*a^2 - 15*b^2)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(192*a^3*f) - (5*(4*a^2 - b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(32*a^2*f) - (5*b*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(24*a^2*f) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f)}

# {Sin[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]], x, 9, -(((a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a*f)) + ((6*a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a*f) - (Sqrt[a + b]*(4*a^2 + b^2)*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f) - (b*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f) - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(2*f)} 
[sqrt(a+b*sec(e+f*x)),x,1,-2*cot(e+f*x)*EllipticPi(sqrt(a+b)/sqrt(a+b*sec(e+f*x)),a/(a+b),sqrt((a-b)/(a+b)))*(a+b*sec(e+f*x))*sqrt(-b*(1-sec(e+f*x))/(a+b*sec(e+f*x)))*sqrt(b*(1+sec(e+f*x))/(a+b*sec(e+f*x)))/(f*sqrt(a+b))],
[csc(e+f*x)^2*sqrt(a+b*sec(e+f*x)),x,2,cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f-cot(e+f*x)*sqrt(a+b*sec(e+f*x))/f],
#  {Csc[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]], x, 0, 0}

# {Csc[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]], x, 0, 0} 
#  {Sin[e + f*x]^4*(a + b*Sec[e + f*x])^(3/2), x, 10, -(((a - b)*Sqrt[a + b]*(236*a^2 + 3*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^2*f*Sqrt[b + a*Cos[e + f*x]])) + (Sqrt[a + b]*(216*a^3 - 236*a^2*b + 2*a*b^2 - 3*b^3)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^2*f*Sqrt[b + a*Cos[e + f*x]]) - (3*Sqrt[a + b]*(16*a^4 - 24*a^2*b^2 + b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^3*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(236*a^2 + 3*b^2)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(64*a^2*f) - (3*(28*a^2 + b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(32*a*f) - ((16*a^2 + b^2)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(8*a*b*f) + (2*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(b*f) + (Cos[e + f*x]*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f)}

# {Sin[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2), x, 10, -((13*(a - b)*Sqrt[a + b]*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*f*Sqrt[b + a*Cos[e + f*x]])) + ((14*a - 13*b)*Sqrt[a + b]*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*f*Sqrt[b + a*Cos[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 3*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*a*f*Sqrt[b + a*Cos[e + f*x]]) - (13*b*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*f) - (5*a*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(2*f) + (2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/f} 
[(a+b*sec(e+f*x))^(3/2),x,5,-2*(a-b)*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f+2*(2*a-b)*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f-2*a*cot(e+f*x)*EllipticPi(sqrt(a+b*sec(e+f*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f],
[csc(e+f*x)^2*(a+b*sec(e+f*x))^(3/2),x,4,-cot(e+f*x)*(a+b*sec(e+f*x))^(3/2)/f-3*(a-b)*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f+3*(a-b)*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/f],
#  {Csc[e + f*x]^4*(a + b*Sec[e + f*x])^(3/2), x, 0, 0}

# {Csc[e + f*x]^6*(a + b*Sec[e + f*x])^(3/2), x, 0, 0} 

# n<0
#  {Sin[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]], x, 9, ((a - b)*Sqrt[a + b]*(188*a^2 - 105*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(120*a^3 - 188*a^2*b - 70*a*b^2 + 105*b^3)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(48*a^4 - 72*a^2*b^2 + 35*b^4)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(12*a^2 - 7*b^2)*(b + a*Cos[e + f*x])*Sin[e + f*x])/(96*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (7*b*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(4*a*f*Sqrt[a + b*Sec[e + f*x]]) + (b*(188*a^2 - 105*b^2)*(b + a*Cos[e + f*x])*Tan[e + f*x])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]])}

# {Sin[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]], x, 9, (3*(a - b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) - ((2*a - 3*b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 3*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - ((b + a*Cos[e + f*x])*Sin[e + f*x])/(2*a*f*Sqrt[a + b*Sec[e + f*x]]) + (3*b*(b + a*Cos[e + f*x])*Tan[e + f*x])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]])} 
[1/sqrt(a+b*sec(e+f*x)),x,1,-2*cot(e+f*x)*EllipticPi(sqrt(a+b*sec(e+f*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(a*f)],
[csc(e+f*x)^2/sqrt(a+b*sec(e+f*x)),x,6,cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(f*sqrt(a+b))-cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(f*sqrt(a+b))-cot(e+f*x)/(f*sqrt(a+b*sec(e+f*x)))+b^2*tan(e+f*x)/((a^2-b^2)*f*sqrt(a+b*sec(e+f*x)))],
#  {Csc[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]], x, 0, 0}

# {Csc[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]], x, 0, 0} 
#  {Sin[e + f*x]^4/(a + b*Sec[e + f*x])^(3/2), x, 10, (3*(a - b)*Sqrt[a + b]*(92*a^2 - 105*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (3*Sqrt[a + b]*(56*a^3 - 92*a^2*b - 70*a*b^2 + 105*b^3)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (3*Sqrt[a + b]*(16*a^4 - 120*a^2*b^2 + 105*b^4)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^6*f*Sqrt[a + b*Sec[e + f*x]]) - (2*(a^2 - b^2)*Cos[e + f*x]^2*Sin[e + f*x])/(a^2*b*f*Sqrt[a + b*Sec[e + f*x]]) - (21*(4*a^2 - 5*b^2)*(b + a*Cos[e + f*x])*Sin[e + f*x])/(32*a^4*f*Sqrt[a + b*Sec[e + f*x]]) + ((16*a^2 - 21*b^2)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sin[e + f*x])/(8*a^3*b*f*Sqrt[a + b*Sec[e + f*x]]) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (3*b*(92*a^2 - 105*b^2)*(b + a*Cos[e + f*x])*Tan[e + f*x])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]])}

# {Sin[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2), x, 10, (15*(a - b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(2*a - 3*b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 15*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^4*f*Sqrt[a + b*Sec[e + f*x]]) + (2*Cos[e + f*x]*Sin[e + f*x])/(a*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(b + a*Cos[e + f*x])*Sin[e + f*x])/(2*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (15*b*(b + a*Cos[e + f*x])*Tan[e + f*x])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]])} 
[1/(a+b*sec(e+f*x))^(3/2),x,6,2*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(a*f*sqrt(a+b))-2*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(a*f*sqrt(a+b))-2*cot(e+f*x)*EllipticPi(sqrt(a+b*sec(e+f*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(a^2*f)+2*b^2*tan(e+f*x)/(a*(a^2-b^2)*f*sqrt(a+b*sec(e+f*x)))],
[csc(e+f*x)^2/(a+b*sec(e+f*x))^(3/2),x,6,-cot(e+f*x)/(f*(a+b*sec(e+f*x))^(3/2))+4*a*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/((a-b)*(a+b)^(3/2)*f)-(3*a-b)*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/((a-b)*(a+b)^(3/2)*f)+b^2*tan(e+f*x)/((a^2-b^2)*f*(a+b*sec(e+f*x))^(3/2))+4*a*b^2*tan(e+f*x)/((a^2-b^2)^2*f*sqrt(a+b*sec(e+f*x)))],
#  {Csc[e + f*x]^4/(a + b*Sec[e + f*x])^(3/2), x, 0, 0}

# {Csc[e + f*x]^6/(a + b*Sec[e + f*x])^(3/2), x, 0, 0} 

# Integrands of the form (e Sin[e+f x])^m (a+b Sec[e+f x])^n with m symbolic
[(a+b*sec(c+d*x))^3*(e*sin(c+d*x))^m,x,9,3*a^2*b*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+b^3*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a^3*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(cos(c+d*x)^2))+3*a*b^2*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*sin(c+d*x))^(1+m)*sqrt(cos(c+d*x)^2)/(d*e*(1+m))],
[(a+b*sec(c+d*x))^2*(e*sin(c+d*x))^m,x,9,2*a*b*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a^2*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*sin(c+d*x)*(e*sin(c+d*x))^m/(d*(1+m)*sqrt(cos(c+d*x)^2))+b^2*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^m*sqrt(cos(c+d*x)^2)*tan(c+d*x)/(d*(1+m))],
[(a+b*sec(c+d*x))*(e*sin(c+d*x))^m,x,5,b*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m))+a*cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+b*sec(c+d*x)),x,4,-b*e*AppellF1(1-m,1/2*(1-m),1/2*(1-m),2-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^2*d*(1-m))+cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(a*d*e*(1+m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+b*sec(c+d*x))^2,x,6,-2*b*e*AppellF1(1-m,1/2*(1-m),1/2*(1-m),2-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^3*d*(1-m))+b^2*e*AppellF1(2-m,1/2*(1-m),1/2*(1-m),3-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^3*d*(2-m)*(b+a*cos(c+d*x)))+cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(a^2*d*e*(1+m)*sqrt(cos(c+d*x)^2))],
[(e*sin(c+d*x))^m/(a+b*sec(c+d*x))^3,x,7,-3*b*e*AppellF1(1-m,1/2*(1-m),1/2*(1-m),2-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^4*d*(1-m))-b^3*e*AppellF1(3-m,1/2*(1-m),1/2*(1-m),4-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^4*d*(3-m)*(b+a*cos(c+d*x))^2)+3*b^2*e*AppellF1(2-m,1/2*(1-m),1/2*(1-m),3-m,(-a+b)/(b+a*cos(c+d*x)),(a+b)/(b+a*cos(c+d*x)))*(-a*(1-cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(a*(1+cos(c+d*x))/(b+a*cos(c+d*x)))^(1/2*(1-m))*(e*sin(c+d*x))^(-1+m)/(a^4*d*(2-m)*(b+a*cos(c+d*x)))+cos(c+d*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(c+d*x)^2)*(e*sin(c+d*x))^(1+m)/(a^3*d*e*(1+m)*sqrt(cos(c+d*x)^2))],
[(a+b*sec(c+d*x))^(3/2)*(e*sin(c+d*x))^m,x,0,Unintegrable((a+b*sec(c+d*x))^(3/2)*(e*sin(c+d*x))^m,x)],
[(a+b*sec(c+d*x))^(1/2)*(e*sin(c+d*x))^m,x,0,Unintegrable((e*sin(c+d*x))^m*sqrt(a+b*sec(c+d*x)),x)],
[(e*sin(c+d*x))^m/(a+b*sec(c+d*x))^(1/2),x,0,Unintegrable((e*sin(c+d*x))^m/sqrt(a+b*sec(c+d*x)),x)],
[(e*sin(c+d*x))^m/(a+b*sec(c+d*x))^(3/2),x,0,Unintegrable((e*sin(c+d*x))^m/(a+b*sec(c+d*x))^(3/2),x)],

# Integrands of the form (e Sin[e+f x])^m (a+b Sec[e+f x])^n with n symbolic
[(a+b*sec(c+d*x))^n*(e*sin(c+d*x))^m,x,0,Unintegrable((a+b*sec(c+d*x))^n*(e*sin(c+d*x))^m,x)],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^5,x,6,b*hypergeom([2,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a^2*d*(1+n))-2*b^3*hypergeom([4,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a^4*d*(1+n))+b^5*hypergeom([6,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a^6*d*(1+n))],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^3,x,3,1/6*b*(6*a^2-b^2*(2-3*n+n^2))*hypergeom([2,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a^4*d*(1+n))+1/6*cos(c+d*x)^3*(a+b*sec(c+d*x))^(1+n)*(2*a-b*(2-n)*sec(c+d*x))/(a^2*d)],
[(a+b*sec(c+d*x))^n*sin(c+d*x),x,2,b*hypergeom([2,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a^2*d*(1+n))],
[csc(c+d*x)*(a+b*sec(c+d*x))^n,x,4,1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)*d*(1+n))-1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)*d*(1+n))],
[csc(c+d*x)^3*(a+b*sec(c+d*x))^n,x,9,1/4*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)*d*(1+n))-1/4*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)*d*(1+n))+1/4*b*hypergeom([2,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)^2*d*(1+n))+1/4*b*hypergeom([2,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)^2*d*(1+n))],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^4,x,0,Unintegrable((a+b*sec(c+d*x))^n*sin(c+d*x)^4,x)],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^2,x,0,Unintegrable((a+b*sec(c+d*x))^n*sin(c+d*x)^2,x)],
[csc(c+d*x)^2*(a+b*sec(c+d*x))^n,x,4,-cot(c+d*x)*(a+b*sec(c+d*x))^n/d+b*n*AppellF1(1/2,1/2,1-n,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^n*sqrt(2)*tan(c+d*x)/((a+b)*d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(1+sec(c+d*x)))],
[csc(c+d*x)^4*(a+b*sec(c+d*x))^n,x,-1,-1/6*AppellF1(-3/2,5/2,-n,-1/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*cot(c+d*x)^3*(1+sec(c+d*x))^(3/2)*(a+b*sec(c+d*x))^n/(d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(2))-3/2*AppellF1(-1/2,5/2,-n,1/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*cot(c+d*x)*(a+b*sec(c+d*x))^n*sqrt(1+sec(c+d*x))/(d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(2))+AppellF1(1/2,3/2,-n,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^n*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(2)*sqrt(1+sec(c+d*x)))+1/2*AppellF1(1/2,5/2,-n,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^n*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(2)*sqrt(1+sec(c+d*x)))],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^(3/2),x,0,Unintegrable((a+b*sec(c+d*x))^n*sin(c+d*x)^(3/2),x)],
[(a+b*sec(c+d*x))^n*sin(c+d*x)^(1/2),x,0,Unintegrable((a+b*sec(c+d*x))^n*sqrt(sin(c+d*x)),x)],
[(a+b*sec(c+d*x))^n/sin(c+d*x)^(1/2),x,0,Unintegrable((a+b*sec(c+d*x))^n/sqrt(sin(c+d*x)),x)],
[(a+b*sec(c+d*x))^n/sin(c+d*x)^(3/2),x,0,Unintegrable((a+b*sec(c+d*x))^n/sin(c+d*x)^(3/2),x)],

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

# Integrands of the form (d Csc[e+f x])^n (a+b Sec[e+f x])^m when a^2-b^2=0

# Integrands of the form (e Csc[e+f x])^(m/2) (a+a Sec[e+f x])^n

# n>0
[(e*csc(c+d*x))^(5/2)*(a+a*sec(c+d*x)),x,11,-2/3*a*e^2*cot(c+d*x)*sqrt(e*csc(c+d*x))/d-2/3*a*e^2*csc(c+d*x)*sqrt(e*csc(c+d*x))/d+a*e^2*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+a*e^2*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+2/3*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d],
[(e*csc(c+d*x))^(3/2)*(a+a*sec(c+d*x)),x,11,-2*a*e*sqrt(e*csc(c+d*x))/d-2*a*e*cos(c+d*x)*sqrt(e*csc(c+d*x))/d-a*e*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+a*e*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d-2*a*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d],
[(a+a*sec(c+d*x))*sqrt(e*csc(c+d*x)),x,9,a*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+a*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d],
[(a+a*sec(c+d*x))/sqrt(e*csc(c+d*x)),x,9,-a*arctan(sqrt(sin(c+d*x)))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a*arctanh(sqrt(sin(c+d*x)))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))/(e*csc(c+d*x))^(3/2),x,11,-2*a/(d*e*sqrt(e*csc(c+d*x)))-2/3*a*cos(c+d*x)/(d*e*sqrt(e*csc(c+d*x)))+a*arctan(sqrt(sin(c+d*x)))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a*arctanh(sqrt(sin(c+d*x)))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))/(e*csc(c+d*x))^(5/2),x,11,-2/3*a*sin(c+d*x)/(d*e^2*sqrt(e*csc(c+d*x)))-2/5*a*cos(c+d*x)*sin(c+d*x)/(d*e^2*sqrt(e*csc(c+d*x)))-a*arctan(sqrt(sin(c+d*x)))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a*arctanh(sqrt(sin(c+d*x)))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+6/5*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[(e*csc(c+d*x))^(5/2)*(a+a*sec(c+d*x))^2,x,15,-2/3*a^2*e^2*cot(c+d*x)*sqrt(e*csc(c+d*x))/d-4/3*a^2*e^2*csc(c+d*x)*sqrt(e*csc(c+d*x))/d-2/3*a^2*e^2*csc(c+d*x)*sec(c+d*x)*sqrt(e*csc(c+d*x))/d+2*a^2*e^2*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+2*a^2*e^2*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+7/3*a^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+5/3*a^2*e^2*sqrt(e*csc(c+d*x))*tan(c+d*x)/d],
[(e*csc(c+d*x))^(3/2)*(a+a*sec(c+d*x))^2,x,15,-4*a^2*e*sqrt(e*csc(c+d*x))/d-2*a^2*e*cos(c+d*x)*sqrt(e*csc(c+d*x))/d-2*a^2*e*sec(c+d*x)*sqrt(e*csc(c+d*x))/d-2*a^2*e*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+2*a^2*e*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d-5*a^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+3*a^2*e*sin(c+d*x)*sqrt(e*csc(c+d*x))*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2*sqrt(e*csc(c+d*x)),x,12,2*a^2*arctan(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+2*a^2*arctanh(sqrt(sin(c+d*x)))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+3*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/d+a^2*sqrt(e*csc(c+d*x))*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2/sqrt(e*csc(c+d*x)),x,12,-2*a^2*arctan(sqrt(sin(c+d*x)))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+2*a^2*arctanh(sqrt(sin(c+d*x)))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a^2*tan(c+d*x)/(d*sqrt(e*csc(c+d*x)))],
[(a+a*sec(c+d*x))^2/(e*csc(c+d*x))^(3/2),x,14,-4*a^2/(d*e*sqrt(e*csc(c+d*x)))-2/3*a^2*cos(c+d*x)/(d*e*sqrt(e*csc(c+d*x)))+a^2*sec(c+d*x)/(d*e*sqrt(e*csc(c+d*x)))+2*a^2*arctan(sqrt(sin(c+d*x)))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+2*a^2*arctanh(sqrt(sin(c+d*x)))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))-1/3*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[(a+a*sec(c+d*x))^2/(e*csc(c+d*x))^(5/2),x,14,-4/3*a^2*sin(c+d*x)/(d*e^2*sqrt(e*csc(c+d*x)))-2/5*a^2*cos(c+d*x)*sin(c+d*x)/(d*e^2*sqrt(e*csc(c+d*x)))-2*a^2*arctan(sqrt(sin(c+d*x)))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+2*a^2*arctanh(sqrt(sin(c+d*x)))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))-9/5*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))+a^2*tan(c+d*x)/(d*e^2*sqrt(e*csc(c+d*x)))],

# n<0
[(e*csc(c+d*x))^(5/2)/(a+a*sec(c+d*x)),x,8,-4/21*e^2*cot(c+d*x)*sqrt(e*csc(c+d*x))/(a*d)+2/7*e^2*cot(c+d*x)*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a*d)-2/7*e^2*csc(c+d*x)^3*sqrt(e*csc(c+d*x))/(a*d)+4/21*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a*d)],
[(e*csc(c+d*x))^(3/2)/(a+a*sec(c+d*x)),x,8,-4/5*e*cos(c+d*x)*sqrt(e*csc(c+d*x))/(a*d)+2/5*e*cot(c+d*x)*csc(c+d*x)*sqrt(e*csc(c+d*x))/(a*d)-2/5*e*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a*d)-4/5*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a*d)],
[sqrt(e*csc(c+d*x))/(a+a*sec(c+d*x)),x,7,2/3*cot(c+d*x)*sqrt(e*csc(c+d*x))/(a*d)-2/3*csc(c+d*x)*sqrt(e*csc(c+d*x))/(a*d)+4/3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a*d)],
[1/((a+a*sec(c+d*x))*sqrt(e*csc(c+d*x))),x,7,2*cot(c+d*x)/(a*d*sqrt(e*csc(c+d*x)))-2*csc(c+d*x)/(a*d*sqrt(e*csc(c+d*x)))+4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a*d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(3/2)*(a+a*sec(c+d*x))),x,7,2/(a*d*e*sqrt(e*csc(c+d*x)))-2/3*cos(c+d*x)/(a*d*e*sqrt(e*csc(c+d*x)))-4/3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a*d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(5/2)*(a+a*sec(c+d*x))),x,7,2/3*sin(c+d*x)/(a*d*e^2*sqrt(e*csc(c+d*x)))-2/5*cos(c+d*x)*sin(c+d*x)/(a*d*e^2*sqrt(e*csc(c+d*x)))-4/5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a*d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(7/2)*(a+a*sec(c+d*x))),x,8,-2/21*cos(c+d*x)/(a*d*e^3*sqrt(e*csc(c+d*x)))+2/7*cos(c+d*x)^3/(a*d*e^3*sqrt(e*csc(c+d*x)))+2/5*sin(c+d*x)^2/(a*d*e^3*sqrt(e*csc(c+d*x)))-4/21*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a*d*e^3*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[(e*csc(c+d*x))^(5/2)/(a+a*sec(c+d*x))^2,x,16,-4/231*e^2*cot(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)+16/77*e^2*cot(c+d*x)*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a^2*d)-2/11*e^2*cot(c+d*x)^3*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a^2*d)-4/7*e^2*csc(c+d*x)^3*sqrt(e*csc(c+d*x))/(a^2*d)-2/11*e^2*cot(c+d*x)*csc(c+d*x)^4*sqrt(e*csc(c+d*x))/(a^2*d)+4/11*e^2*csc(c+d*x)^5*sqrt(e*csc(c+d*x))/(a^2*d)+4/231*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a^2*d)],
[(e*csc(c+d*x))^(3/2)/(a+a*sec(c+d*x))^2,x,16,-4/15*e*cos(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)+16/45*e*cot(c+d*x)*csc(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)-2/9*e*cot(c+d*x)^3*csc(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)-4/5*e*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a^2*d)-2/9*e*cot(c+d*x)*csc(c+d*x)^3*sqrt(e*csc(c+d*x))/(a^2*d)+4/9*e*csc(c+d*x)^4*sqrt(e*csc(c+d*x))/(a^2*d)-4/15*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a^2*d)],
[sqrt(e*csc(c+d*x))/(a+a*sec(c+d*x))^2,x,14,16/21*cot(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)-2/7*cot(c+d*x)^3*sqrt(e*csc(c+d*x))/(a^2*d)-4/3*csc(c+d*x)*sqrt(e*csc(c+d*x))/(a^2*d)-2/7*cot(c+d*x)*csc(c+d*x)^2*sqrt(e*csc(c+d*x))/(a^2*d)+4/7*csc(c+d*x)^3*sqrt(e*csc(c+d*x))/(a^2*d)+20/21*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x))/(a^2*d)],
[1/((a+a*sec(c+d*x))^2*sqrt(e*csc(c+d*x))),x,14,16/5*cot(c+d*x)/(a^2*d*sqrt(e*csc(c+d*x)))-2/5*cot(c+d*x)^3/(a^2*d*sqrt(e*csc(c+d*x)))-4*csc(c+d*x)/(a^2*d*sqrt(e*csc(c+d*x)))-2/5*cot(c+d*x)*csc(c+d*x)^2/(a^2*d*sqrt(e*csc(c+d*x)))+4/5*csc(c+d*x)^3/(a^2*d*sqrt(e*csc(c+d*x)))+28/5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a^2*d*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(3/2)*(a+a*sec(c+d*x))^2),x,13,4/(a^2*d*e*sqrt(e*csc(c+d*x)))-4/3*cos(c+d*x)/(a^2*d*e*sqrt(e*csc(c+d*x)))-2/3*cos(c+d*x)*cot(c+d*x)^2/(a^2*d*e*sqrt(e*csc(c+d*x)))-2/3*cot(c+d*x)*csc(c+d*x)/(a^2*d*e*sqrt(e*csc(c+d*x)))+4/3*csc(c+d*x)^2/(a^2*d*e*sqrt(e*csc(c+d*x)))-4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a^2*d*e*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(5/2)*(a+a*sec(c+d*x))^2),x,13,-2*cot(c+d*x)/(a^2*d*e^2*sqrt(e*csc(c+d*x)))-2*cos(c+d*x)^2*cot(c+d*x)/(a^2*d*e^2*sqrt(e*csc(c+d*x)))+4*csc(c+d*x)/(a^2*d*e^2*sqrt(e*csc(c+d*x)))+4/3*sin(c+d*x)/(a^2*d*e^2*sqrt(e*csc(c+d*x)))-12/5*cos(c+d*x)*sin(c+d*x)/(a^2*d*e^2*sqrt(e*csc(c+d*x)))-44/5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a^2*d*e^2*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))],
[1/((e*csc(c+d*x))^(7/2)*(a+a*sec(c+d*x))^2),x,13,(-4)/(a^2*d*e^3*sqrt(e*csc(c+d*x)))+26/21*cos(c+d*x)/(a^2*d*e^3*sqrt(e*csc(c+d*x)))+2/7*cos(c+d*x)^3/(a^2*d*e^3*sqrt(e*csc(c+d*x)))+4/5*sin(c+d*x)^2/(a^2*d*e^3*sqrt(e*csc(c+d*x)))+52/21*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))/(a^2*d*e^3*sqrt(e*csc(c+d*x))*sqrt(sin(c+d*x)))]]:

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