# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.1.2 (d sec)^m (a+b tan)^n.txt"

lst:=[

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

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

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

# n>0
[sec(c+d*x)^10*(a+I*a*tan(c+d*x)),x,3,1/10*I*a*sec(c+d*x)^10/d+a*tan(c+d*x)/d+4/3*a*tan(c+d*x)^3/d+6/5*a*tan(c+d*x)^5/d+4/7*a*tan(c+d*x)^7/d+1/9*a*tan(c+d*x)^9/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x)),x,3,1/8*I*a*sec(c+d*x)^8/d+a*tan(c+d*x)/d+a*tan(c+d*x)^3/d+3/5*a*tan(c+d*x)^5/d+1/7*a*tan(c+d*x)^7/d],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x)),x,3,1/6*I*a*sec(c+d*x)^6/d+a*tan(c+d*x)/d+2/3*a*tan(c+d*x)^3/d+1/5*a*tan(c+d*x)^5/d],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x)),x,3,1/4*I*a*sec(c+d*x)^4/d+a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x)),x,3,-1/2*I*(a+I*a*tan(c+d*x))^2/(a*d),1/2*I*a*sec(c+d*x)^2/d+a*tan(c+d*x)/d],
[a+I*a*tan(c+d*x),x,2,a*x-I*a*log(cos(c+d*x))/d],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x)),x,3,1/2*a*x-1/2*I*a*cos(c+d*x)^2/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x)),x,4,3/8*a*x-1/4*I*a*cos(c+d*x)^4/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x)),x,5,5/16*a*x-1/6*I*a*cos(c+d*x)^6/d+5/16*a*cos(c+d*x)*sin(c+d*x)/d+5/24*a*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^8*(a+I*a*tan(c+d*x)),x,6,35/128*a*x-1/8*I*a*cos(c+d*x)^8/d+35/128*a*cos(c+d*x)*sin(c+d*x)/d+35/192*a*cos(c+d*x)^3*sin(c+d*x)/d+7/48*a*cos(c+d*x)^5*sin(c+d*x)/d+1/8*a*cos(c+d*x)^7*sin(c+d*x)/d],
[sec(c+d*x)^7*(a+I*a*tan(c+d*x)),x,5,5/16*a*arctanh(sin(c+d*x))/d+1/7*I*a*sec(c+d*x)^7/d+5/16*a*sec(c+d*x)*tan(c+d*x)/d+5/24*a*sec(c+d*x)^3*tan(c+d*x)/d+1/6*a*sec(c+d*x)^5*tan(c+d*x)/d],
[sec(c+d*x)^5*(a+I*a*tan(c+d*x)),x,4,3/8*a*arctanh(sin(c+d*x))/d+1/5*I*a*sec(c+d*x)^5/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x)),x,3,1/2*a*arctanh(sin(c+d*x))/d+1/3*I*a*sec(c+d*x)^3/d+1/2*a*sec(c+d*x)*tan(c+d*x)/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x)),x,2,a*arctanh(sin(c+d*x))/d+I*a*sec(c+d*x)/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x)),x,2,-I*a*cos(c+d*x)/d+a*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x)),x,3,-1/3*I*a*cos(c+d*x)^3/d+a*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x)),x,3,-1/5*I*a*cos(c+d*x)^5/d+a*sin(c+d*x)/d-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x)),x,3,-1/7*I*a*cos(c+d*x)^7/d+a*sin(c+d*x)/d-a*sin(c+d*x)^3/d+3/5*a*sin(c+d*x)^5/d-1/7*a*sin(c+d*x)^7/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^2,x,3,-4/3*I*(a+I*a*tan(c+d*x))^6/(a^4*d)+12/7*I*(a+I*a*tan(c+d*x))^7/(a^5*d)-3/4*I*(a+I*a*tan(c+d*x))^8/(a^6*d)+1/9*I*(a+I*a*tan(c+d*x))^9/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^2,x,3,-4/5*I*(a+I*a*tan(c+d*x))^5/(a^3*d)+2/3*I*(a+I*a*tan(c+d*x))^6/(a^4*d)-1/7*I*(a+I*a*tan(c+d*x))^7/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^2,x,3,-1/2*I*(a+I*a*tan(c+d*x))^4/(a^2*d)+1/5*I*(a+I*a*tan(c+d*x))^5/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^2,x,2,-1/3*I*(a+I*a*tan(c+d*x))^3/(a*d)],
[(a+I*a*tan(c+d*x))^2,x,2,2*a^2*x-2*I*a^2*log(cos(c+d*x))/d-a^2*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^2,x,2,-I*a^3/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^2,x,4,1/4*a^2*x-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2)-1/4*I*a^3/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^2,x,4,1/4*a^2*x-1/12*I*a^5/(d*(a-I*a*tan(c+d*x))^3)-1/8*I*a^4/(d*(a-I*a*tan(c+d*x))^2)-3/16*I*a^3/(d*(a-I*a*tan(c+d*x)))+1/16*I*a^3/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^8*(a+I*a*tan(c+d*x))^2,x,4,15/64*a^2*x-1/32*I*a^6/(d*(a-I*a*tan(c+d*x))^4)-1/16*I*a^5/(d*(a-I*a*tan(c+d*x))^3)-3/32*I*a^4/(d*(a-I*a*tan(c+d*x))^2)-5/32*I*a^3/(d*(a-I*a*tan(c+d*x)))+1/64*I*a^4/(d*(a+I*a*tan(c+d*x))^2)+5/64*I*a^3/(d*(a+I*a*tan(c+d*x)))],
[sec(c+d*x)^5*(a+I*a*tan(c+d*x))^2,x,5,7/16*a^2*arctanh(sin(c+d*x))/d+7/30*I*a^2*sec(c+d*x)^5/d+7/16*a^2*sec(c+d*x)*tan(c+d*x)/d+7/24*a^2*sec(c+d*x)^3*tan(c+d*x)/d+1/6*I*sec(c+d*x)^5*(a^2+I*a^2*tan(c+d*x))/d],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^2,x,4,5/8*a^2*arctanh(sin(c+d*x))/d+5/12*I*a^2*sec(c+d*x)^3/d+5/8*a^2*sec(c+d*x)*tan(c+d*x)/d+1/4*I*sec(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^2,x,3,3/2*a^2*arctanh(sin(c+d*x))/d+3/2*I*a^2*sec(c+d*x)/d+1/2*I*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^2,x,2,-a^2*arctanh(sin(c+d*x))/d-2*I*cos(c+d*x)*(a^2+I*a^2*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^2,x,2,1/3*a^2*sin(c+d*x)/d-2/3*I*cos(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^2,x,3,3/5*a^2*sin(c+d*x)/d-1/5*a^2*sin(c+d*x)^3/d-2/5*I*cos(c+d*x)^5*(a^2+I*a^2*tan(c+d*x))/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^2,x,3,5/7*a^2*sin(c+d*x)/d-10/21*a^2*sin(c+d*x)^3/d+1/7*a^2*sin(c+d*x)^5/d-2/7*I*cos(c+d*x)^7*(a^2+I*a^2*tan(c+d*x))/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^2,x,3,7/9*a^2*sin(c+d*x)/d-7/9*a^2*sin(c+d*x)^3/d+7/15*a^2*sin(c+d*x)^5/d-1/9*a^2*sin(c+d*x)^7/d-2/9*I*cos(c+d*x)^9*(a^2+I*a^2*tan(c+d*x))/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^3,x,3,-8/7*I*(a+I*a*tan(c+d*x))^7/(a^4*d)+3/2*I*(a+I*a*tan(c+d*x))^8/(a^5*d)-2/3*I*(a+I*a*tan(c+d*x))^9/(a^6*d)+1/10*I*(a+I*a*tan(c+d*x))^10/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^3,x,3,-2/3*I*(a+I*a*tan(c+d*x))^6/(a^3*d)+4/7*I*(a+I*a*tan(c+d*x))^7/(a^4*d)-1/8*I*(a+I*a*tan(c+d*x))^8/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^3,x,3,-2/5*I*(a+I*a*tan(c+d*x))^5/(a^2*d)+1/6*I*(a+I*a*tan(c+d*x))^6/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^3,x,2,-1/4*I*(a+I*a*tan(c+d*x))^4/(a*d)],
[(a+I*a*tan(c+d*x))^3,x,3,4*a^3*x-4*I*a^3*log(cos(c+d*x))/d-2*a^3*tan(c+d*x)/d+1/2*I*a*(a+I*a*tan(c+d*x))^2/d],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^3,x,3,-a^3*x+I*a^3*log(cos(c+d*x))/d-2*I*a^4/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^3,x,2,-1/2*I*a^5/(d*(a-I*a*tan(c+d*x))^2)],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^3,x,4,1/8*a^3*x-1/6*I*a^6/(d*(a-I*a*tan(c+d*x))^3)-1/8*I*a^5/(d*(a-I*a*tan(c+d*x))^2)-1/8*I*a^4/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^8*(a+I*a*tan(c+d*x))^3,x,4,5/32*a^3*x-1/16*I*a^7/(d*(a-I*a*tan(c+d*x))^4)-1/12*I*a^6/(d*(a-I*a*tan(c+d*x))^3)-3/32*I*a^5/(d*(a-I*a*tan(c+d*x))^2)-1/8*I*a^4/(d*(a-I*a*tan(c+d*x)))+1/32*I*a^4/(d*(a+I*a*tan(c+d*x)))],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^3,x,5,7/8*a^3*arctanh(sin(c+d*x))/d+7/12*I*a^3*sec(c+d*x)^3/d+7/8*a^3*sec(c+d*x)*tan(c+d*x)/d+1/5*I*a*sec(c+d*x)^3*(a+I*a*tan(c+d*x))^2/d+7/20*I*sec(c+d*x)^3*(a^3+I*a^3*tan(c+d*x))/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^3,x,4,5/2*a^3*arctanh(sin(c+d*x))/d+5/2*I*a^3*sec(c+d*x)/d+1/3*I*a*sec(c+d*x)*(a+I*a*tan(c+d*x))^2/d+5/6*I*sec(c+d*x)*(a^3+I*a^3*tan(c+d*x))/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^3,x,3,-3*a^3*arctanh(sin(c+d*x))/d-3*I*a^3*sec(c+d*x)/d-2*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^2/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^3,x,1,-1/3*I*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^3,x,4,-1/15*I*a^3*cos(c+d*x)^3/d+1/5*a^3*sin(c+d*x)/d-1/15*a^3*sin(c+d*x)^3/d-2/5*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^2/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^3,x,4,-3/35*I*a^3*cos(c+d*x)^5/d+3/7*a^3*sin(c+d*x)/d-2/7*a^3*sin(c+d*x)^3/d+3/35*a^3*sin(c+d*x)^5/d-2/7*I*a*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^2/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^3,x,4,-5/63*I*a^3*cos(c+d*x)^7/d+5/9*a^3*sin(c+d*x)/d-5/9*a^3*sin(c+d*x)^3/d+1/3*a^3*sin(c+d*x)^5/d-5/63*a^3*sin(c+d*x)^7/d-2/9*I*a*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^2/d],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^4,x,6,21/16*a^4*arctanh(sin(c+d*x))/d+7/8*I*a^4*sec(c+d*x)^3/d+21/16*a^4*sec(c+d*x)*tan(c+d*x)/d+1/6*I*a*sec(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d+3/10*I*sec(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))^2/d+21/40*I*sec(c+d*x)^3*(a^4+I*a^4*tan(c+d*x))/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^4,x,5,35/8*a^4*arctanh(sin(c+d*x))/d+35/8*I*a^4*sec(c+d*x)/d+1/4*I*a*sec(c+d*x)*(a+I*a*tan(c+d*x))^3/d+7/12*I*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))^2/d+35/24*I*sec(c+d*x)*(a^4+I*a^4*tan(c+d*x))/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^4,x,4,-15/2*a^4*arctanh(sin(c+d*x))/d-15/2*I*a^4*sec(c+d*x)/d-2*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^3/d-5/2*I*sec(c+d*x)*(a^4+I*a^4*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^4,x,3,a^4*arctanh(sin(c+d*x))/d-2/3*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d+2*I*cos(c+d*x)*(a^4+I*a^4*tan(c+d*x))/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^4,x,2,-1/15*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-1/5*I*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^4/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^4,x,4,3/35*a^4*sin(c+d*x)/d-1/35*a^4*sin(c+d*x)^3/d-2/7*I*a*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^3/d-2/35*I*cos(c+d*x)^5*(a^4+I*a^4*tan(c+d*x))/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^4,x,4,5/21*a^4*sin(c+d*x)/d-10/63*a^4*sin(c+d*x)^3/d+1/21*a^4*sin(c+d*x)^5/d-2/9*I*a*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^3/d-2/21*I*cos(c+d*x)^7*(a^4+I*a^4*tan(c+d*x))/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^5,x,3,-8/9*I*(a+I*a*tan(c+d*x))^9/(a^4*d)+6/5*I*(a+I*a*tan(c+d*x))^10/(a^5*d)-6/11*I*(a+I*a*tan(c+d*x))^11/(a^6*d)+1/12*I*(a+I*a*tan(c+d*x))^12/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^5,x,3,-1/2*I*(a+I*a*tan(c+d*x))^8/(a^3*d)+4/9*I*(a+I*a*tan(c+d*x))^9/(a^4*d)-1/10*I*(a+I*a*tan(c+d*x))^10/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^5,x,3,-2/7*I*(a+I*a*tan(c+d*x))^7/(a^2*d)+1/8*I*(a+I*a*tan(c+d*x))^8/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^5,x,2,-1/6*I*(a+I*a*tan(c+d*x))^6/(a*d)],
[(a+I*a*tan(c+d*x))^5,x,5,16*a^5*x-16*I*a^5*log(cos(c+d*x))/d-8*a^5*tan(c+d*x)/d+2/3*I*a^2*(a+I*a*tan(c+d*x))^3/d+1/4*I*a*(a+I*a*tan(c+d*x))^4/d+2*I*a*(a^2+I*a^2*tan(c+d*x))^2/d],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^5,x,3,-12*a^5*x+12*I*a^5*log(cos(c+d*x))/d+5*a^5*tan(c+d*x)/d+1/2*I*a^5*tan(c+d*x)^2/d-8*I*a^6/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^5,x,3,a^5*x-I*a^5*log(cos(c+d*x))/d-2*I*a^7/(d*(a-I*a*tan(c+d*x))^2)+4*I*a^6/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^5,x,3,-2/3*I*a^8/(d*(a-I*a*tan(c+d*x))^3)+1/2*I*a^7/(d*(a-I*a*tan(c+d*x))^2)],
[cos(c+d*x)^8*(a+I*a*tan(c+d*x))^5,x,2,-1/4*I*a^9/(d*(a-I*a*tan(c+d*x))^4)],
[cos(c+d*x)^10*(a+I*a*tan(c+d*x))^5,x,4,1/32*a^5*x-1/10*I*a^10/(d*(a-I*a*tan(c+d*x))^5)-1/16*I*a^9/(d*(a-I*a*tan(c+d*x))^4)-1/24*I*a^8/(d*(a-I*a*tan(c+d*x))^3)-1/32*I*a^7/(d*(a-I*a*tan(c+d*x))^2)-1/32*I*a^6/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^12*(a+I*a*tan(c+d*x))^5,x,4,7/128*a^5*x-1/24*I*a^11/(d*(a-I*a*tan(c+d*x))^6)-1/20*I*a^10/(d*(a-I*a*tan(c+d*x))^5)-3/64*I*a^9/(d*(a-I*a*tan(c+d*x))^4)-1/24*I*a^8/(d*(a-I*a*tan(c+d*x))^3)-5/128*I*a^7/(d*(a-I*a*tan(c+d*x))^2)-3/64*I*a^6/(d*(a-I*a*tan(c+d*x)))+1/128*I*a^6/(d*(a+I*a*tan(c+d*x)))],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^5,x,6,63/8*a^5*arctanh(sin(c+d*x))/d+63/8*I*a^5*sec(c+d*x)/d+9/20*I*a^2*sec(c+d*x)*(a+I*a*tan(c+d*x))^3/d+1/5*I*a*sec(c+d*x)*(a+I*a*tan(c+d*x))^4/d+21/20*I*a*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))^2/d+21/8*I*sec(c+d*x)*(a^5+I*a^5*tan(c+d*x))/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^5,x,5,-35/2*a^5*arctanh(sin(c+d*x))/d-35/2*I*a^5*sec(c+d*x)/d-7/3*I*a^3*sec(c+d*x)*(a+I*a*tan(c+d*x))^2/d-2*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^4/d-35/6*I*sec(c+d*x)*(a^5+I*a^5*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^5,x,4,5*a^5*arctanh(sin(c+d*x))/d+5*I*a^5*sec(c+d*x)/d+10/3*I*a^3*cos(c+d*x)*(a+I*a*tan(c+d*x))^2/d-2/3*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^4/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^5,x,1,-1/5*I*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^5/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^5,x,3,-2/105*I*a^2*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-2/35*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^4/d-1/7*I*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^5/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^5,x,5,-1/105*I*a^5*cos(c+d*x)^5/d+1/21*a^5*sin(c+d*x)/d-2/63*a^5*sin(c+d*x)^3/d+1/105*a^5*sin(c+d*x)^5/d-2/63*I*a^3*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^2/d-2/9*I*a*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^4/d],
[cos(c+d*x)^11*(a+I*a*tan(c+d*x))^5,x,5,-5/231*I*a^5*cos(c+d*x)^7/d+5/33*a^5*sin(c+d*x)/d-5/33*a^5*sin(c+d*x)^3/d+1/11*a^5*sin(c+d*x)^5/d-5/231*a^5*sin(c+d*x)^7/d-2/33*I*a^3*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^2/d-2/11*I*a*cos(c+d*x)^11*(a+I*a*tan(c+d*x))^4/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^8,x,3,-2/3*I*(a+I*a*tan(c+d*x))^12/(a^4*d)+12/13*I*(a+I*a*tan(c+d*x))^13/(a^5*d)-3/7*I*(a+I*a*tan(c+d*x))^14/(a^6*d)+1/15*I*(a+I*a*tan(c+d*x))^15/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^8,x,3,-4/11*I*(a+I*a*tan(c+d*x))^11/(a^3*d)+1/3*I*(a+I*a*tan(c+d*x))^12/(a^4*d)-1/13*I*(a+I*a*tan(c+d*x))^13/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^8,x,3,-1/5*I*(a+I*a*tan(c+d*x))^10/(a^2*d)+1/11*I*(a+I*a*tan(c+d*x))^11/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^8,x,2,-1/9*I*(a+I*a*tan(c+d*x))^9/(a*d)],
[(a+I*a*tan(c+d*x))^8,x,8,128*a^8*x-128*I*a^8*log(cos(c+d*x))/d-64*a^8*tan(c+d*x)/d+4/5*I*a^3*(a+I*a*tan(c+d*x))^5/d+1/3*I*a^2*(a+I*a*tan(c+d*x))^6/d+1/7*I*a*(a+I*a*tan(c+d*x))^7/d+16/3*I*a^2*(a^2+I*a^2*tan(c+d*x))^3/d+2*I*(a^2+I*a^2*tan(c+d*x))^4/d+16*I*(a^4+I*a^4*tan(c+d*x))^2/d],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^8,x,3,-192*a^8*x+192*I*a^8*log(cos(c+d*x))/d+129*a^8*tan(c+d*x)/d+36*I*a^8*tan(c+d*x)^2/d-10*a^8*tan(c+d*x)^3/d-2*I*a^8*tan(c+d*x)^4/d+1/5*a^8*tan(c+d*x)^5/d-64*I*a^9/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^8,x,3,80*a^8*x-80*I*a^8*log(cos(c+d*x))/d-31*a^8*tan(c+d*x)/d-4*I*a^8*tan(c+d*x)^2/d+1/3*a^8*tan(c+d*x)^3/d-16*I*a^10/(d*(a-I*a*tan(c+d*x))^2)+80*I*a^9/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^8,x,3,-8*a^8*x+8*I*a^8*log(cos(c+d*x))/d+a^8*tan(c+d*x)/d-16/3*I*a^11/(d*(a-I*a*tan(c+d*x))^3)+16*I*a^10/(d*(a-I*a*tan(c+d*x))^2)-24*I*a^9/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^8*(a+I*a*tan(c+d*x))^8,x,2,-1/8*I*(a^3+I*a^3*tan(c+d*x))^4/(d*(a-I*a*tan(c+d*x))^4)],
[cos(c+d*x)^10*(a+I*a*tan(c+d*x))^8,x,3,-4/5*I*a^13/(d*(a-I*a*tan(c+d*x))^5)+I*a^12/(d*(a-I*a*tan(c+d*x))^4)-1/3*I*a^11/(d*(a-I*a*tan(c+d*x))^3)],
[cos(c+d*x)^12*(a+I*a*tan(c+d*x))^8,x,3,-1/3*I*a^14/(d*(a-I*a*tan(c+d*x))^6)+1/5*I*a^13/(d*(a-I*a*tan(c+d*x))^5)],
[cos(c+d*x)^14*(a+I*a*tan(c+d*x))^8,x,2,-1/7*I*a^15/(d*(a-I*a*tan(c+d*x))^7)],
[cos(c+d*x)^16*(a+I*a*tan(c+d*x))^8,x,4,1/256*a^8*x-1/16*I*a^16/(d*(a-I*a*tan(c+d*x))^8)-1/28*I*a^15/(d*(a-I*a*tan(c+d*x))^7)-1/48*I*a^14/(d*(a-I*a*tan(c+d*x))^6)-1/80*I*a^13/(d*(a-I*a*tan(c+d*x))^5)-1/128*I*a^12/(d*(a-I*a*tan(c+d*x))^4)-1/192*I*a^11/(d*(a-I*a*tan(c+d*x))^3)-1/256*I*a^10/(d*(a-I*a*tan(c+d*x))^2)-1/256*I*a^9/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^18*(a+I*a*tan(c+d*x))^8,x,4,5/512*a^8*x-1/36*I*a^17/(d*(a-I*a*tan(c+d*x))^9)-1/32*I*a^16/(d*(a-I*a*tan(c+d*x))^8)-3/112*I*a^15/(d*(a-I*a*tan(c+d*x))^7)-1/48*I*a^14/(d*(a-I*a*tan(c+d*x))^6)-1/64*I*a^13/(d*(a-I*a*tan(c+d*x))^5)-3/256*I*a^12/(d*(a-I*a*tan(c+d*x))^4)-7/768*I*a^11/(d*(a-I*a*tan(c+d*x))^3)-1/128*I*a^10/(d*(a-I*a*tan(c+d*x))^2)-9/1024*I*a^9/(d*(a-I*a*tan(c+d*x)))+1/1024*I*a^9/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^8,x,8,-3003/16*a^8*arctanh(sin(c+d*x))/d-3003/16*I*a^8*sec(c+d*x)/d-13/6*I*a^3*sec(c+d*x)*(a+I*a*tan(c+d*x))^5/d-2*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^7/d-429/40*I*a^2*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))^3/d-143/30*I*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))^4/d-1001/40*I*sec(c+d*x)*(a^4+I*a^4*tan(c+d*x))^2/d-1001/16*I*sec(c+d*x)*(a^8+I*a^8*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^8,x,7,1155/8*a^8*arctanh(sin(c+d*x))/d+1155/8*I*a^8*sec(c+d*x)/d+22/3*I*a^3*cos(c+d*x)*(a+I*a*tan(c+d*x))^5/d-2/3*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^7/d+33/4*I*a^2*sec(c+d*x)*(a^2+I*a^2*tan(c+d*x))^3/d+77/4*I*sec(c+d*x)*(a^4+I*a^4*tan(c+d*x))^2/d+385/8*I*sec(c+d*x)*(a^8+I*a^8*tan(c+d*x))/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^8,x,6,-63/2*a^8*arctanh(sin(c+d*x))/d-63/2*I*a^8*sec(c+d*x)/d+6/5*I*a^3*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^5/d-2/5*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^7/d-42/5*I*a^2*cos(c+d*x)*(a^2+I*a^2*tan(c+d*x))^3/d-21/2*I*sec(c+d*x)*(a^8+I*a^8*tan(c+d*x))/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^8,x,5,a^8*arctanh(sin(c+d*x))/d+2/5*I*a^3*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^5/d-2/7*I*a*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^7/d-2/3*I*a^2*cos(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))^3/d+2*I*cos(c+d*x)*(a^8+I*a^8*tan(c+d*x))/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^8,x,2,-1/63*I*a*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^7/d-1/9*I*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^8/d],
[cos(c+d*x)^11*(a+I*a*tan(c+d*x))^8,x,4,-2/1155*I*a^3*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^5/d-2/231*I*a^2*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^6/d-1/33*I*a*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^7/d-1/11*I*cos(c+d*x)^11*(a+I*a*tan(c+d*x))^8/d],
[cos(c+d*x)^13*(a+I*a*tan(c+d*x))^8,x,6,-20/3003*I*a^3*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^5/d-20/1287*I*a^2*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^6/d-5/143*I*a*cos(c+d*x)^11*(a+I*a*tan(c+d*x))^7/d-1/13*I*cos(c+d*x)^13*(a+I*a*tan(c+d*x))^8/d-8/9009*I*a^2*cos(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))^3/d-8/3003*I*cos(c+d*x)^5*(a^2+I*a^2*tan(c+d*x))^4/d],
[cos(c+d*x)^15*(a+I*a*tan(c+d*x))^8,x,6,7/1287*a^8*sin(c+d*x)/d-7/1287*a^8*sin(c+d*x)^3/d+7/2145*a^8*sin(c+d*x)^5/d-1/1287*a^8*sin(c+d*x)^7/d-2/195*I*a^3*cos(c+d*x)^13*(a+I*a*tan(c+d*x))^5/d-2/15*I*a*cos(c+d*x)^15*(a+I*a*tan(c+d*x))^7/d-2/715*I*a^2*cos(c+d*x)^11*(a^2+I*a^2*tan(c+d*x))^3/d-2/1287*I*cos(c+d*x)^9*(a^8+I*a^8*tan(c+d*x))/d],

# n<0
[sec(c+d*x)^10/(a+I*a*tan(c+d*x)),x,3,8/5*I*(a-I*a*tan(c+d*x))^5/(a^6*d)-2*I*(a-I*a*tan(c+d*x))^6/(a^7*d)+6/7*I*(a-I*a*tan(c+d*x))^7/(a^8*d)-1/8*I*(a-I*a*tan(c+d*x))^8/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x)),x,3,I*(a-I*a*tan(c+d*x))^4/(a^5*d)-4/5*I*(a-I*a*tan(c+d*x))^5/(a^6*d)+1/6*I*(a-I*a*tan(c+d*x))^6/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x)),x,3,2/3*I*(a-I*a*tan(c+d*x))^3/(a^4*d)-1/4*I*(a-I*a*tan(c+d*x))^4/(a^5*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x)),x,2,tan(c+d*x)/(a*d)-1/2*I*tan(c+d*x)^2/(a*d)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x)),x,2,x/a+I*log(cos(c+d*x))/(a*d)],
[1/(a+I*a*tan(c+d*x)),x,2,1/2*x/a+1/2*I/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x)),x,4,3/8*x/a+(-1/8*I)/(d*(a-I*a*tan(c+d*x)))+1/8*I*a/(d*(a+I*a*tan(c+d*x))^2)+1/4*I/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x)),x,4,5/16*x/a-1/32*I*a/(d*(a-I*a*tan(c+d*x))^2)+(-1/8*I)/(d*(a-I*a*tan(c+d*x)))+1/24*I*a^2/(d*(a+I*a*tan(c+d*x))^3)+3/32*I*a/(d*(a+I*a*tan(c+d*x))^2)+3/16*I/(d*(a+I*a*tan(c+d*x)))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x)),x,4,3/8*arctanh(sin(c+d*x))/(a*d)-1/5*I*sec(c+d*x)^5/(a*d)+3/8*sec(c+d*x)*tan(c+d*x)/(a*d)+1/4*sec(c+d*x)^3*tan(c+d*x)/(a*d)],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x)),x,3,1/2*arctanh(sin(c+d*x))/(a*d)-1/3*I*sec(c+d*x)^3/(a*d)+1/2*sec(c+d*x)*tan(c+d*x)/(a*d)],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x)),x,2,arctanh(sin(c+d*x))/(a*d)-I*sec(c+d*x)/(a*d)],
[sec(c+d*x)/(a+I*a*tan(c+d*x)),x,1,I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)/(a+I*a*tan(c+d*x)),x,2,2/3*sin(c+d*x)/(a*d)+1/3*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x)),x,3,4/5*sin(c+d*x)/(a*d)-4/15*sin(c+d*x)^3/(a*d)+1/5*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^5/(a+I*a*tan(c+d*x)),x,3,6/7*sin(c+d*x)/(a*d)-4/7*sin(c+d*x)^3/(a*d)+6/35*sin(c+d*x)^5/(a*d)+1/7*I*cos(c+d*x)^5/(d*(a+I*a*tan(c+d*x)))],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^2,x,3,4/5*I*(a-I*a*tan(c+d*x))^5/(a^7*d)-2/3*I*(a-I*a*tan(c+d*x))^6/(a^8*d)+1/7*I*(a-I*a*tan(c+d*x))^7/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^2,x,3,1/2*I*(a-I*a*tan(c+d*x))^4/(a^6*d)-1/5*I*(a-I*a*tan(c+d*x))^5/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^2,x,2,1/3*I*(a-I*a*tan(c+d*x))^3/(a^5*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^2,x,3,2*x/a^2+2*I*log(cos(c+d*x))/(a^2*d)-tan(c+d*x)/(a^2*d)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^2,x,2,I/(d*(a^2+I*a^2*tan(c+d*x)))],
[1/(a+I*a*tan(c+d*x))^2,x,3,1/4*x/a^2+1/4*I/(d*(a+I*a*tan(c+d*x))^2)+1/4*I/(d*(a^2+I*a^2*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^2,x,4,1/4*x/a^2+1/12*I*a/(d*(a+I*a*tan(c+d*x))^3)+1/8*I/(d*(a+I*a*tan(c+d*x))^2)+(-1/16*I)/(d*(a^2-I*a^2*tan(c+d*x)))+3/16*I/(d*(a^2+I*a^2*tan(c+d*x)))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^2,x,4,15/64*x/a^2+(-1/64*I)/(d*(a-I*a*tan(c+d*x))^2)+1/32*I*a^2/(d*(a+I*a*tan(c+d*x))^4)+1/16*I*a/(d*(a+I*a*tan(c+d*x))^3)+3/32*I/(d*(a+I*a*tan(c+d*x))^2)+(-5/64*I)/(d*(a^2-I*a^2*tan(c+d*x)))+5/32*I/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^2,x,5,7/16*arctanh(sin(c+d*x))/(a^2*d)+7/16*sec(c+d*x)*tan(c+d*x)/(a^2*d)+7/24*sec(c+d*x)^3*tan(c+d*x)/(a^2*d)+7/30*sec(c+d*x)^5*tan(c+d*x)/(a^2*d)-2/5*I*sec(c+d*x)^7/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^2,x,4,5/8*arctanh(sin(c+d*x))/(a^2*d)+5/8*sec(c+d*x)*tan(c+d*x)/(a^2*d)+5/12*sec(c+d*x)^3*tan(c+d*x)/(a^2*d)-2/3*I*sec(c+d*x)^5/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^2,x,3,3/2*arctanh(sin(c+d*x))/(a^2*d)+3/2*sec(c+d*x)*tan(c+d*x)/(a^2*d)-2*I*sec(c+d*x)^3/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^2,x,2,-arctanh(sin(c+d*x))/(a^2*d)+2*I*sec(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^2,x,2,1/3*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^2)+1/3*I*sec(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x)))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^2,x,3,3/5*sin(c+d*x)/(a^2*d)-1/5*sin(c+d*x)^3/(a^2*d)+2/5*I*cos(c+d*x)^3/(d*(a^2+I*a^2*tan(c+d*x)))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^2,x,3,5/7*sin(c+d*x)/(a^2*d)-10/21*sin(c+d*x)^3/(a^2*d)+1/7*sin(c+d*x)^5/(a^2*d)+2/7*I*cos(c+d*x)^5/(d*(a^2+I*a^2*tan(c+d*x)))],
[cos(c+d*x)^5/(a+I*a*tan(c+d*x))^2,x,3,7/9*sin(c+d*x)/(a^2*d)-7/9*sin(c+d*x)^3/(a^2*d)+7/15*sin(c+d*x)^5/(a^2*d)-1/9*sin(c+d*x)^7/(a^2*d)+2/9*I*cos(c+d*x)^7/(d*(a^2+I*a^2*tan(c+d*x)))],
[sec(c+d*x)^14/(a+I*a*tan(c+d*x))^3,x,3,8/7*I*(a-I*a*tan(c+d*x))^7/(a^10*d)-3/2*I*(a-I*a*tan(c+d*x))^8/(a^11*d)+2/3*I*(a-I*a*tan(c+d*x))^9/(a^12*d)-1/10*I*(a-I*a*tan(c+d*x))^10/(a^13*d)],
[sec(c+d*x)^12/(a+I*a*tan(c+d*x))^3,x,3,2/3*I*(a-I*a*tan(c+d*x))^6/(a^9*d)-4/7*I*(a-I*a*tan(c+d*x))^7/(a^10*d)+1/8*I*(a-I*a*tan(c+d*x))^8/(a^11*d)],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^3,x,3,2/5*I*(a-I*a*tan(c+d*x))^5/(a^8*d)-1/6*I*(a-I*a*tan(c+d*x))^6/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^3,x,2,1/4*I*(a-I*a*tan(c+d*x))^4/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^3,x,3,4*x/a^3+4*I*log(cos(c+d*x))/(a^3*d)-3*tan(c+d*x)/(a^3*d)+1/2*I*tan(c+d*x)^2/(a^3*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^3,x,3,-x/a^3-I*log(cos(c+d*x))/(a^3*d)+2*I/(d*(a^3+I*a^3*tan(c+d*x)))],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^3,x,2,1/2*I/(a*d*(a+I*a*tan(c+d*x))^2)],
[1/(a+I*a*tan(c+d*x))^3,x,4,1/8*x/a^3+1/6*I/(d*(a+I*a*tan(c+d*x))^3)+1/8*I/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*I/(d*(a^3+I*a^3*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^3,x,4,5/32*x/a^3+1/16*I*a/(d*(a+I*a*tan(c+d*x))^4)+1/12*I/(d*(a+I*a*tan(c+d*x))^3)+3/32*I/(a*d*(a+I*a*tan(c+d*x))^2)+(-1/32*I)/(d*(a^3-I*a^3*tan(c+d*x)))+1/8*I/(d*(a^3+I*a^3*tan(c+d*x)))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^3,x,4,21/128*x/a^3+(-1/128*I)/(a*d*(a-I*a*tan(c+d*x))^2)+1/40*I*a^2/(d*(a+I*a*tan(c+d*x))^5)+3/64*I*a/(d*(a+I*a*tan(c+d*x))^4)+1/16*I/(d*(a+I*a*tan(c+d*x))^3)+5/64*I/(a*d*(a+I*a*tan(c+d*x))^2)+(-3/64*I)/(d*(a^3-I*a^3*tan(c+d*x)))+15/128*I/(d*(a^3+I*a^3*tan(c+d*x)))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^3,x,5,7/8*arctanh(sin(c+d*x))/(a^3*d)-7/15*I*sec(c+d*x)^5/(a^3*d)+7/8*sec(c+d*x)*tan(c+d*x)/(a^3*d)+7/12*sec(c+d*x)^3*tan(c+d*x)/(a^3*d)-2/3*I*sec(c+d*x)^7/(a*d*(a+I*a*tan(c+d*x))^2)],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^3,x,4,5/2*arctanh(sin(c+d*x))/(a^3*d)-5/3*I*sec(c+d*x)^3/(a^3*d)+5/2*sec(c+d*x)*tan(c+d*x)/(a^3*d)-2*I*sec(c+d*x)^5/(a*d*(a+I*a*tan(c+d*x))^2)],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^3,x,3,-3*arctanh(sin(c+d*x))/(a^3*d)+3*I*sec(c+d*x)/(a^3*d)+2*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^2)],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^3,x,1,1/3*I*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^3)],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^3,x,3,1/5*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^3)+2/15*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^2)+2/15*I*sec(c+d*x)/(d*(a^3+I*a^3*tan(c+d*x)))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^3,x,4,12/35*sin(c+d*x)/(a^3*d)-4/35*sin(c+d*x)^3/(a^3*d)+1/7*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^3)+8/35*I*cos(c+d*x)^3/(d*(a^3+I*a^3*tan(c+d*x)))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^3,x,4,10/21*sin(c+d*x)/(a^3*d)-20/63*sin(c+d*x)^3/(a^3*d)+2/21*sin(c+d*x)^5/(a^3*d)+1/9*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^3)+4/21*I*cos(c+d*x)^5/(d*(a^3+I*a^3*tan(c+d*x)))],
[cos(c+d*x)^5/(a+I*a*tan(c+d*x))^3,x,4,56/99*sin(c+d*x)/(a^3*d)-56/99*sin(c+d*x)^3/(a^3*d)+56/165*sin(c+d*x)^5/(a^3*d)-8/99*sin(c+d*x)^7/(a^3*d)+1/11*I*cos(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^3)+16/99*I*cos(c+d*x)^7/(d*(a^3+I*a^3*tan(c+d*x)))],
[sec(c+d*x)^14/(a+I*a*tan(c+d*x))^4,x,3,4/7*I*(a-I*a*tan(c+d*x))^7/(a^11*d)-1/2*I*(a-I*a*tan(c+d*x))^8/(a^12*d)+1/9*I*(a-I*a*tan(c+d*x))^9/(a^13*d)],
[sec(c+d*x)^12/(a+I*a*tan(c+d*x))^4,x,3,1/3*I*(a-I*a*tan(c+d*x))^6/(a^10*d)-1/7*I*(a-I*a*tan(c+d*x))^7/(a^11*d)],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^4,x,2,1/5*I*(a-I*a*tan(c+d*x))^5/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^4,x,3,8*x/a^4+8*I*log(cos(c+d*x))/(a^4*d)-4*tan(c+d*x)/(a^4*d)-I*(a-I*a*tan(c+d*x))^2/(a^6*d)-1/3*I*(a-I*a*tan(c+d*x))^3/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^4,x,3,-4*x/a^4-4*I*log(cos(c+d*x))/(a^4*d)+tan(c+d*x)/(a^4*d)+4*I/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^4,x,2,tan(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x))^2)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^4,x,2,1/3*I/(a*d*(a+I*a*tan(c+d*x))^3)],
[1/(a+I*a*tan(c+d*x))^4,x,5,1/16*x/a^4+1/8*I/(d*(a+I*a*tan(c+d*x))^4)+1/12*I/(a*d*(a+I*a*tan(c+d*x))^3)+1/16*I/(d*(a^2+I*a^2*tan(c+d*x))^2)+1/16*I/(d*(a^4+I*a^4*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^4,x,4,3/32*x/a^4+1/20*I*a/(d*(a+I*a*tan(c+d*x))^5)+1/16*I/(d*(a+I*a*tan(c+d*x))^4)+1/16*I/(a*d*(a+I*a*tan(c+d*x))^3)+1/16*I/(d*(a^2+I*a^2*tan(c+d*x))^2)+(-1/64*I)/(d*(a^4-I*a^4*tan(c+d*x)))+5/64*I/(d*(a^4+I*a^4*tan(c+d*x)))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^4,x,4,7/64*x/a^4+1/48*I*a^2/(d*(a+I*a*tan(c+d*x))^6)+3/80*I*a/(d*(a+I*a*tan(c+d*x))^5)+3/64*I/(d*(a+I*a*tan(c+d*x))^4)+5/96*I/(a*d*(a+I*a*tan(c+d*x))^3)+(-1/256*I)/(d*(a^2-I*a^2*tan(c+d*x))^2)+15/256*I/(d*(a^2+I*a^2*tan(c+d*x))^2)+(-7/256*I)/(d*(a^4-I*a^4*tan(c+d*x)))+21/256*I/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^4,x,5,35/8*arctanh(sin(c+d*x))/(a^4*d)+35/8*sec(c+d*x)*tan(c+d*x)/(a^4*d)+35/12*sec(c+d*x)^3*tan(c+d*x)/(a^4*d)-2*I*sec(c+d*x)^7/(a*d*(a+I*a*tan(c+d*x))^3)-14/3*I*sec(c+d*x)^5/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^4,x,4,-15/2*arctanh(sin(c+d*x))/(a^4*d)-15/2*sec(c+d*x)*tan(c+d*x)/(a^4*d)+2*I*sec(c+d*x)^5/(a*d*(a+I*a*tan(c+d*x))^3)+10*I*sec(c+d*x)^3/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^4,x,3,arctanh(sin(c+d*x))/(a^4*d)+2/3*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)-2*I*sec(c+d*x)/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^4,x,2,1/5*I*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^4)+1/15*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^4,x,4,1/7*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^4)+3/35*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^3)+2/35*I*sec(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x))^2)+2/35*I*sec(c+d*x)/(d*(a^4+I*a^4*tan(c+d*x)))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^4,x,5,4/21*sin(c+d*x)/(a^4*d)-4/63*sin(c+d*x)^3/(a^4*d)+1/9*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^4)+5/63*I*cos(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^3)+8/63*I*cos(c+d*x)^3/(d*(a^4+I*a^4*tan(c+d*x)))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^4,x,5,10/33*sin(c+d*x)/(a^4*d)-20/99*sin(c+d*x)^3/(a^4*d)+2/33*sin(c+d*x)^5/(a^4*d)+1/11*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^4)+7/99*I*cos(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)+4/33*I*cos(c+d*x)^5/(d*(a^4+I*a^4*tan(c+d*x)))],
[cos(c+d*x)^5/(a+I*a*tan(c+d*x))^4,x,5,56/143*sin(c+d*x)/(a^4*d)-56/143*sin(c+d*x)^3/(a^4*d)+168/715*sin(c+d*x)^5/(a^4*d)-8/143*sin(c+d*x)^7/(a^4*d)+1/13*I*cos(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^4)+9/143*I*cos(c+d*x)^5/(a*d*(a+I*a*tan(c+d*x))^3)+16/143*I*cos(c+d*x)^7/(d*(a^4+I*a^4*tan(c+d*x)))],
[sec(c+d*x)^14/(a+I*a*tan(c+d*x))^8,x,3,-192*x/a^8-192*I*log(cos(c+d*x))/(a^8*d)+129*tan(c+d*x)/(a^8*d)-36*I*tan(c+d*x)^2/(a^8*d)-10*tan(c+d*x)^3/(a^8*d)+2*I*tan(c+d*x)^4/(a^8*d)+1/5*tan(c+d*x)^5/(a^8*d)+64*I/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^12/(a+I*a*tan(c+d*x))^8,x,3,80*x/a^8+80*I*log(cos(c+d*x))/(a^8*d)-31*tan(c+d*x)/(a^8*d)+4*I*tan(c+d*x)^2/(a^8*d)+1/3*tan(c+d*x)^3/(a^8*d)+16*I/(d*(a^4+I*a^4*tan(c+d*x))^2)+(-80*I)/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^8,x,3,-8*x/a^8-8*I*log(cos(c+d*x))/(a^8*d)+tan(c+d*x)/(a^8*d)+16/3*I/(a^5*d*(a+I*a*tan(c+d*x))^3)+(-16*I)/(d*(a^4+I*a^4*tan(c+d*x))^2)+24*I/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^8,x,2,1/8*I*(a-I*a*tan(c+d*x))^4/(d*(a^3+I*a^3*tan(c+d*x))^4)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^8,x,3,4/5*I/(a^3*d*(a+I*a*tan(c+d*x))^5)+1/3*I/(a^5*d*(a+I*a*tan(c+d*x))^3)+(-I)/(d*(a^2+I*a^2*tan(c+d*x))^4)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^8,x,3,1/3*I/(a^2*d*(a+I*a*tan(c+d*x))^6)+(-1/5*I)/(a^3*d*(a+I*a*tan(c+d*x))^5)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^8,x,2,1/7*I/(a*d*(a+I*a*tan(c+d*x))^7)],
[1/(a+I*a*tan(c+d*x))^8,x,9,1/256*x/a^8+1/16*I/(d*(a+I*a*tan(c+d*x))^8)+1/28*I/(a*d*(a+I*a*tan(c+d*x))^7)+1/48*I/(a^2*d*(a+I*a*tan(c+d*x))^6)+1/80*I/(a^3*d*(a+I*a*tan(c+d*x))^5)+1/128*I/(d*(a^2+I*a^2*tan(c+d*x))^4)+1/192*I/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)+1/256*I/(d*(a^4+I*a^4*tan(c+d*x))^2)+1/256*I/(d*(a^8+I*a^8*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^8,x,4,5/512*x/a^8+1/36*I*a/(d*(a+I*a*tan(c+d*x))^9)+1/32*I/(d*(a+I*a*tan(c+d*x))^8)+3/112*I/(a*d*(a+I*a*tan(c+d*x))^7)+1/48*I/(a^2*d*(a+I*a*tan(c+d*x))^6)+1/64*I/(a^3*d*(a+I*a*tan(c+d*x))^5)+7/768*I/(a^5*d*(a+I*a*tan(c+d*x))^3)+3/256*I/(d*(a^2+I*a^2*tan(c+d*x))^4)+1/128*I/(d*(a^4+I*a^4*tan(c+d*x))^2)+(-1/1024*I)/(d*(a^8-I*a^8*tan(c+d*x)))+9/1024*I/(d*(a^8+I*a^8*tan(c+d*x)))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^8,x,4,33/2048*x/a^8+1/80*I*a^2/(d*(a+I*a*tan(c+d*x))^10)+1/48*I*a/(d*(a+I*a*tan(c+d*x))^9)+3/128*I/(d*(a+I*a*tan(c+d*x))^8)+5/224*I/(a*d*(a+I*a*tan(c+d*x))^7)+5/256*I/(a^2*d*(a+I*a*tan(c+d*x))^6)+21/1280*I/(a^3*d*(a+I*a*tan(c+d*x))^5)+3/256*I/(a^5*d*(a+I*a*tan(c+d*x))^3)+7/512*I/(d*(a^2+I*a^2*tan(c+d*x))^4)+(-1/4096*I)/(d*(a^4-I*a^4*tan(c+d*x))^2)+45/4096*I/(d*(a^4+I*a^4*tan(c+d*x))^2)+(-11/4096*I)/(d*(a^8-I*a^8*tan(c+d*x)))+55/4096*I/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^13/(a+I*a*tan(c+d*x))^8,x,7,1155/8*arctanh(sin(c+d*x))/(a^8*d)+1155/8*sec(c+d*x)*tan(c+d*x)/(a^8*d)+385/4*sec(c+d*x)^3*tan(c+d*x)/(a^8*d)+2/3*I*sec(c+d*x)^11/(a*d*(a+I*a*tan(c+d*x))^7)-22/3*I*sec(c+d*x)^9/(a^3*d*(a+I*a*tan(c+d*x))^5)-66*I*sec(c+d*x)^7/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)-154*I*sec(c+d*x)^5/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^11/(a+I*a*tan(c+d*x))^8,x,6,-63/2*arctanh(sin(c+d*x))/(a^8*d)-63/2*sec(c+d*x)*tan(c+d*x)/(a^8*d)+2/5*I*sec(c+d*x)^9/(a*d*(a+I*a*tan(c+d*x))^7)-6/5*I*sec(c+d*x)^7/(a^3*d*(a+I*a*tan(c+d*x))^5)+42/5*I*sec(c+d*x)^5/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)+42*I*sec(c+d*x)^3/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^8,x,5,arctanh(sin(c+d*x))/(a^8*d)+2/7*I*sec(c+d*x)^7/(a*d*(a+I*a*tan(c+d*x))^7)-2/5*I*sec(c+d*x)^5/(a^3*d*(a+I*a*tan(c+d*x))^5)+2/3*I*sec(c+d*x)^3/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)-2*I*sec(c+d*x)/(d*(a^8+I*a^8*tan(c+d*x)))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^8,x,2,1/9*I*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^8)+1/63*I*sec(c+d*x)^7/(a*d*(a+I*a*tan(c+d*x))^7)],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^8,x,4,1/11*I*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^8)+1/33*I*sec(c+d*x)^5/(a*d*(a+I*a*tan(c+d*x))^7)+2/231*I*sec(c+d*x)^5/(a^2*d*(a+I*a*tan(c+d*x))^6)+2/1155*I*sec(c+d*x)^5/(a^3*d*(a+I*a*tan(c+d*x))^5)],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^8,x,6,1/13*I*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^8)+5/143*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^7)+20/1287*I*sec(c+d*x)^3/(a^2*d*(a+I*a*tan(c+d*x))^6)+20/3003*I*sec(c+d*x)^3/(a^3*d*(a+I*a*tan(c+d*x))^5)+8/3003*I*sec(c+d*x)^3/(d*(a^2+I*a^2*tan(c+d*x))^4)+8/9009*I*sec(c+d*x)^3/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^8,x,8,1/15*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^8)+7/195*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^7)+14/715*I*sec(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^6)+14/1287*I*sec(c+d*x)/(a^3*d*(a+I*a*tan(c+d*x))^5)+8/1287*I*sec(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x))^4)+8/2145*I*sec(c+d*x)/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)+16/6435*I*sec(c+d*x)/(d*(a^4+I*a^4*tan(c+d*x))^2)+16/6435*I*sec(c+d*x)/(d*(a^8+I*a^8*tan(c+d*x)))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^8,x,9,192/12155*sin(c+d*x)/(a^8*d)-64/12155*sin(c+d*x)^3/(a^8*d)+1/17*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^8)+3/85*I*cos(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^7)+24/1105*I*cos(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^6)+168/12155*I*cos(c+d*x)/(a^3*d*(a+I*a*tan(c+d*x))^5)+112/12155*I*cos(c+d*x)/(d*(a^2+I*a^2*tan(c+d*x))^4)+16/2431*I*cos(c+d*x)/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)+128/12155*I*cos(c+d*x)^3/(d*(a^8+I*a^8*tan(c+d*x)))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^8,x,9,160/4199*sin(c+d*x)/(a^8*d)-320/12597*sin(c+d*x)^3/(a^8*d)+32/4199*sin(c+d*x)^5/(a^8*d)+1/19*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^8)+11/323*I*cos(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^7)+22/969*I*cos(c+d*x)^3/(a^2*d*(a+I*a*tan(c+d*x))^6)+66/4199*I*cos(c+d*x)^3/(a^3*d*(a+I*a*tan(c+d*x))^5)+48/4199*I*cos(c+d*x)^3/(d*(a^2+I*a^2*tan(c+d*x))^4)+112/12597*I*cos(c+d*x)^3/(a^2*d*(a^2+I*a^2*tan(c+d*x))^3)+64/4199*I*cos(c+d*x)^5/(d*(a^8+I*a^8*tan(c+d*x)))],

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

# n>0
[(e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x)),x,5,2/7*I*a*(e*sec(c+d*x))^(7/2)/d+2/5*a*e*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/d-6/5*a*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+6/5*a*e^3*sin(c+d*x)*sqrt(e*sec(c+d*x))/d],
[(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x)),x,4,2/5*I*a*(e*sec(c+d*x))^(5/2)/d+2/3*a*e*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/3*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x)),x,4,2/3*I*a*(e*sec(c+d*x))^(3/2)/d-2*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2*a*e*sin(c+d*x)*sqrt(e*sec(c+d*x))/d],
[(e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x)),x,3,2*I*a*sqrt(e*sec(c+d*x))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d],
[(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(1/2),x,3,-2*I*a/(d*sqrt(e*sec(c+d*x)))+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(3/2),x,4,-2/3*I*a/(d*(e*sec(c+d*x))^(3/2))+2/3*a*sin(c+d*x)/(d*e*sqrt(e*sec(c+d*x)))+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^2)],
[(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(5/2),x,4,-2/5*I*a/(d*(e*sec(c+d*x))^(5/2))+2/5*a*sin(c+d*x)/(d*e*(e*sec(c+d*x))^(3/2))+6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(7/2),x,5,-2/7*I*a/(d*(e*sec(c+d*x))^(7/2))+2/7*a*sin(c+d*x)/(d*e*(e*sec(c+d*x))^(5/2))+10/21*a*sin(c+d*x)/(d*e^3*sqrt(e*sec(c+d*x)))+10/21*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^4)],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^2,x,5,14/15*I*a^2*(e*sec(c+d*x))^(3/2)/d-14/5*a^2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+14/5*a^2*e*sin(c+d*x)*sqrt(e*sec(c+d*x))/d+2/5*I*(e*sec(c+d*x))^(3/2)*(a^2+I*a^2*tan(c+d*x))/d],
[(e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^2,x,4,10/3*I*a^2*sqrt(e*sec(c+d*x))/d+10/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d+2/3*I*sqrt(e*sec(c+d*x))*(a^2+I*a^2*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(1/2),x,4,6*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-6*a^2*sin(c+d*x)*sqrt(e*sec(c+d*x))/(d*e)-4*I*(a^2+I*a^2*tan(c+d*x))/(d*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(3/2),x,3,-2/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^2)-4/3*I*(a^2+I*a^2*tan(c+d*x))/(d*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(5/2),x,3,2/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-4/5*I*(a^2+I*a^2*tan(c+d*x))/(d*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(7/2),x,4,2/7*a^2*sin(c+d*x)/(d*e^3*sqrt(e*sec(c+d*x)))+2/7*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^4)-4/7*I*(a^2+I*a^2*tan(c+d*x))/(d*(e*sec(c+d*x))^(7/2))],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(9/2),x,4,2/9*a^2*sin(c+d*x)/(d*e^3*(e*sec(c+d*x))^(3/2))+2/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^4*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-4/9*I*(a^2+I*a^2*tan(c+d*x))/(d*(e*sec(c+d*x))^(9/2))],
[(a+I*a*tan(c+d*x))^2/(e*sec(c+d*x))^(11/2),x,5,2/11*a^2*sin(c+d*x)/(d*e^3*(e*sec(c+d*x))^(5/2))+10/33*a^2*sin(c+d*x)/(d*e^5*sqrt(e*sec(c+d*x)))+10/33*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^6)-4/11*I*(a^2+I*a^2*tan(c+d*x))/(d*(e*sec(c+d*x))^(11/2))],
[(e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x))^3,x,7,10/21*I*a^3*(e*sec(c+d*x))^(7/2)/d+2/3*a^3*e*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/d-2*a^3*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2*a^3*e^3*sin(c+d*x)*sqrt(e*sec(c+d*x))/d+2/11*I*a*(e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x))^2/d+10/33*I*(e*sec(c+d*x))^(7/2)*(a^3+I*a^3*tan(c+d*x))/d],
[(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^3,x,6,26/35*I*a^3*(e*sec(c+d*x))^(5/2)/d+26/21*a^3*e*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/d+26/21*a^3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d+2/9*I*a*(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^2/d+26/63*I*(e*sec(c+d*x))^(5/2)*(a^3+I*a^3*tan(c+d*x))/d],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^3,x,6,22/15*I*a^3*(e*sec(c+d*x))^(3/2)/d-22/5*a^3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+22/5*a^3*e*sin(c+d*x)*sqrt(e*sec(c+d*x))/d+2/7*I*a*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^2/d+22/35*I*(e*sec(c+d*x))^(3/2)*(a^3+I*a^3*tan(c+d*x))/d],
[(e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^3,x,5,6*I*a^3*sqrt(e*sec(c+d*x))/d+6*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d+2/5*I*a*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^2/d+6/5*I*sqrt(e*sec(c+d*x))*(a^3+I*a^3*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(1/2),x,5,-26/3*I*a^3/(d*sqrt(e*sec(c+d*x)))+14*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-6*a^3*tan(c+d*x)/(d*sqrt(e*sec(c+d*x)))-2/3*I*a^3*tan(c+d*x)^2/(d*sqrt(e*sec(c+d*x))),14*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-14*a^3*sin(c+d*x)*sqrt(e*sec(c+d*x))/(d*e)+2/3*I*a*(a+I*a*tan(c+d*x))^2/(d*sqrt(e*sec(c+d*x)))-28/3*I*(a^3+I*a^3*tan(c+d*x))/(d*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(3/2),x,4,-10/3*I*a^3*sqrt(e*sec(c+d*x))/(d*e^2)-10/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^2)-4/3*I*a*(a+I*a*tan(c+d*x))^2/(d*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(5/2),x,4,6/5*I*a^3/(d*e^2*sqrt(e*sec(c+d*x)))-6/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-4/5*I*a*(a+I*a*tan(c+d*x))^2/(d*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(7/2),x,4,-2/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^4)-2/7*I*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(7/2))-4/21*I*(a^3+I*a^3*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(9/2),x,4,2/15*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^4*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-2/9*I*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(9/2))-4/15*I*(a^3+I*a^3*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(11/2),x,5,10/77*a^3*sin(c+d*x)/(d*e^5*sqrt(e*sec(c+d*x)))+10/77*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^6)-2/11*I*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(11/2))-20/77*I*(a^3+I*a^3*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(7/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(13/2),x,5,14/117*a^3*sin(c+d*x)/(d*e^5*(e*sec(c+d*x))^(3/2))+14/39*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^6*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-2/13*I*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(13/2))-28/117*I*(a^3+I*a^3*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(9/2))],
[(a+I*a*tan(c+d*x))^3/(e*sec(c+d*x))^(15/2),x,6,6/55*a^3*sin(c+d*x)/(d*e^5*(e*sec(c+d*x))^(5/2))+2/11*a^3*sin(c+d*x)/(d*e^7*sqrt(e*sec(c+d*x)))+2/11*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^8)-2/15*I*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(15/2))-12/55*I*(a^3+I*a^3*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(11/2))],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^4,x,7,22/9*I*a^4*(e*sec(c+d*x))^(3/2)/d-22/3*a^4*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+22/3*a^4*e*sin(c+d*x)*sqrt(e*sec(c+d*x))/d+2/9*I*a*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^3/d+10/21*I*(e*sec(c+d*x))^(3/2)*(a^2+I*a^2*tan(c+d*x))^2/d+22/21*I*(e*sec(c+d*x))^(3/2)*(a^4+I*a^4*tan(c+d*x))/d],
[(e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^4,x,6,78/7*I*a^4*sqrt(e*sec(c+d*x))/d+78/7*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/d+2/7*I*a*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^3/d+26/35*I*sqrt(e*sec(c+d*x))*(a^2+I*a^2*tan(c+d*x))^2/d+78/35*I*sqrt(e*sec(c+d*x))*(a^4+I*a^4*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(1/2),x,6,-154/15*I*a^4*(e*sec(c+d*x))^(3/2)/(d*e^2)+154/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-154/5*a^4*sin(c+d*x)*sqrt(e*sec(c+d*x))/(d*e)-4*I*a*(a+I*a*tan(c+d*x))^3/(d*sqrt(e*sec(c+d*x)))-22/5*I*(e*sec(c+d*x))^(3/2)*(a^4+I*a^4*tan(c+d*x))/(d*e^2)],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(3/2),x,5,-10*I*a^4*sqrt(e*sec(c+d*x))/(d*e^2)-10*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^2)-4/3*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(3/2))-2*I*sqrt(e*sec(c+d*x))*(a^4+I*a^4*tan(c+d*x))/(d*e^2)],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(5/2),x,5,-42/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+42/5*a^4*sin(c+d*x)*sqrt(e*sec(c+d*x))/(d*e^3)-4/5*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(5/2))+28/5*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(7/2),x,4,10/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^4)-4/7*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(7/2))+20/21*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(9/2),x,4,-2/15*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^4*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-4/9*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(9/2))+4/15*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(11/2),x,5,-2/77*a^4*sin(c+d*x)/(d*e^5*sqrt(e*sec(c+d*x)))-2/77*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^6)-4/11*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(11/2))+4/77*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(7/2))],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(13/2),x,5,2/117*a^4*sin(c+d*x)/(d*e^5*(e*sec(c+d*x))^(3/2))+2/39*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*e^6*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-4/13*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(13/2))-4/117*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(9/2))],
[(a+I*a*tan(c+d*x))^4/(e*sec(c+d*x))^(15/2),x,6,2/55*a^4*sin(c+d*x)/(d*e^5*(e*sec(c+d*x))^(5/2))+2/33*a^4*sin(c+d*x)/(d*e^7*sqrt(e*sec(c+d*x)))+2/33*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(d*e^8)-4/15*I*a*(a+I*a*tan(c+d*x))^3/(d*(e*sec(c+d*x))^(15/2))-4/55*I*(a^4+I*a^4*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(11/2))],

# n<0
[(e*sec(c+d*x))^(11/2)/(a+I*a*tan(c+d*x)),x,5,-2/7*I*e^2*(e*sec(c+d*x))^(7/2)/(a*d)+2/5*e^3*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/(a*d)-6/5*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+6/5*e^5*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a*d)],
[(e*sec(c+d*x))^(9/2)/(a+I*a*tan(c+d*x)),x,4,-2/5*I*e^2*(e*sec(c+d*x))^(5/2)/(a*d)+2/3*e^3*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+2/3*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a*d)],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x)),x,4,-2/3*I*e^2*(e*sec(c+d*x))^(3/2)/(a*d)-2*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2*e^3*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a*d)],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x)),x,3,-2*I*e^2*sqrt(e*sec(c+d*x))/(a*d)+2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a*d)],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x)),x,3,2*I*e^2/(a*d*sqrt(e*sec(c+d*x)))+2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))],
[(e*sec(c+d*x))^(1/2)/(a+I*a*tan(c+d*x)),x,3,2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a*d)+2/3*I*sqrt(e*sec(c+d*x))/(d*(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))),x,3,6/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2/5*I/(d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))),x,4,10/21*sin(c+d*x)/(a*d*e*sqrt(e*sec(c+d*x)))+10/21*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a*d*e^2)+2/7*I/(d*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))),x,4,14/45*sin(c+d*x)/(a*d*e*(e*sec(c+d*x))^(3/2))+14/15*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2/9*I/(d*(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x))),x,5,18/77*sin(c+d*x)/(a*d*e*(e*sec(c+d*x))^(5/2))+30/77*sin(c+d*x)/(a*d*e^3*sqrt(e*sec(c+d*x)))+30/77*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a*d*e^4)+2/11*I/(d*(e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^(15/2)/(a+I*a*tan(c+d*x))^2,x,6,22/45*e^5*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/(a^2*d)+22/63*e^3*(e*sec(c+d*x))^(9/2)*sin(c+d*x)/(a^2*d)-22/15*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+22/15*e^7*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^2*d)-4/7*I*e^2*(e*sec(c+d*x))^(11/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(13/2)/(a+I*a*tan(c+d*x))^2,x,5,6/7*e^5*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/(a^2*d)+18/35*e^3*(e*sec(c+d*x))^(7/2)*sin(c+d*x)/(a^2*d)+6/7*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d)-4/5*I*e^2*(e*sec(c+d*x))^(9/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(11/2)/(a+I*a*tan(c+d*x))^2,x,5,14/15*e^3*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/(a^2*d)-14/5*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+14/5*e^5*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^2*d)-4/3*I*e^2*(e*sec(c+d*x))^(7/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(9/2)/(a+I*a*tan(c+d*x))^2,x,4,10/3*e^3*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/(a^2*d)+10/3*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d)-4*I*e^2*(e*sec(c+d*x))^(5/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^2,x,4,6*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-6*e^3*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^2*d)+4*I*e^2*(e*sec(c+d*x))^(3/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^2,x,3,-2/3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d)+4/3*I*e^2*sqrt(e*sec(c+d*x))/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^2,x,3,2/5*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/5*I*e^2/(d*sqrt(e*sec(c+d*x))*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(1/2)/(a+I*a*tan(c+d*x))^2,x,4,2/7*e*sin(c+d*x)/(a^2*d*sqrt(e*sec(c+d*x)))+2/7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d)+4/7*I*e^2/(d*(e*sec(c+d*x))^(3/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^2),x,4,2/9*e*sin(c+d*x)/(a^2*d*(e*sec(c+d*x))^(3/2))+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/9*I*e^2/(d*(e*sec(c+d*x))^(5/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^2),x,5,2/11*e*sin(c+d*x)/(a^2*d*(e*sec(c+d*x))^(5/2))+10/33*sin(c+d*x)/(a^2*d*e*sqrt(e*sec(c+d*x)))+10/33*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d*e^2)+4/11*I*e^2/(d*(e*sec(c+d*x))^(7/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^2),x,5,2/13*e*sin(c+d*x)/(a^2*d*(e*sec(c+d*x))^(7/2))+14/65*sin(c+d*x)/(a^2*d*e*(e*sec(c+d*x))^(3/2))+42/65*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*e^2*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/13*I*e^2/(d*(e*sec(c+d*x))^(9/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(7/2)*(a+I*a*tan(c+d*x))^2),x,6,2/15*e*sin(c+d*x)/(a^2*d*(e*sec(c+d*x))^(9/2))+6/35*sin(c+d*x)/(a^2*d*e*(e*sec(c+d*x))^(5/2))+2/7*sin(c+d*x)/(a^2*d*e^3*sqrt(e*sec(c+d*x)))+2/7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^2*d*e^4)+4/15*I*e^2/(d*(e*sec(c+d*x))^(11/2)*(a^2+I*a^2*tan(c+d*x)))],
[(e*sec(c+d*x))^(15/2)/(a+I*a*tan(c+d*x))^3,x,6,-22/21*I*e^4*(e*sec(c+d*x))^(7/2)/(a^3*d)+22/15*e^5*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/(a^3*d)-22/5*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+22/5*e^7*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^3*d)-4/3*I*e^2*(e*sec(c+d*x))^(11/2)/(a*d*(a+I*a*tan(c+d*x))^2)],
[(e*sec(c+d*x))^(13/2)/(a+I*a*tan(c+d*x))^3,x,5,-18/5*I*e^4*(e*sec(c+d*x))^(5/2)/(a^3*d)+6*e^5*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/(a^3*d)+6*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^3*d)-4*I*e^2*(e*sec(c+d*x))^(9/2)/(a*d*(a+I*a*tan(c+d*x))^2)],
[(e*sec(c+d*x))^(11/2)/(a+I*a*tan(c+d*x))^3,x,5,14/3*I*e^4*(e*sec(c+d*x))^(3/2)/(a^3*d)+14*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-14*e^5*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^3*d)+4*I*e^2*(e*sec(c+d*x))^(7/2)/(a*d*(a+I*a*tan(c+d*x))^2)],
[(e*sec(c+d*x))^(9/2)/(a+I*a*tan(c+d*x))^3,x,4,10/3*I*e^4*sqrt(e*sec(c+d*x))/(a^3*d)-10/3*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^3*d)+4/3*I*e^2*(e*sec(c+d*x))^(5/2)/(a*d*(a+I*a*tan(c+d*x))^2)],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^3,x,4,-6/5*I*e^4/(a^3*d*sqrt(e*sec(c+d*x)))-6/5*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/5*I*e^2*(e*sec(c+d*x))^(3/2)/(a*d*(a+I*a*tan(c+d*x))^2)],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^3,x,4,-2/21*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^3*d)+4/7*I*e^2*sqrt(e*sec(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^2)-2/21*I*e^2*sqrt(e*sec(c+d*x))/(d*(a^3+I*a^3*tan(c+d*x)))],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^3,x,4,2/15*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/9*I*e^2/(a*d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^2)+2/45*I*e^2/(d*sqrt(e*sec(c+d*x))*(a^3+I*a^3*tan(c+d*x)))],
[(e*sec(c+d*x))^(1/2)/(a+I*a*tan(c+d*x))^3,x,5,10/77*e*sin(c+d*x)/(a^3*d*sqrt(e*sec(c+d*x)))+10/77*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^3*d)+2/11*I*sqrt(e*sec(c+d*x))/(d*(a+I*a*tan(c+d*x))^3)+20/77*I*e^2/(d*(e*sec(c+d*x))^(3/2)*(a^3+I*a^3*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^3),x,5,14/117*e*sin(c+d*x)/(a^3*d*(e*sec(c+d*x))^(3/2))+14/39*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+2/13*I/(d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^3)+28/117*I*e^2/(d*(e*sec(c+d*x))^(5/2)*(a^3+I*a^3*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^3),x,6,6/55*e*sin(c+d*x)/(a^3*d*(e*sec(c+d*x))^(5/2))+2/11*sin(c+d*x)/(a^3*d*e*sqrt(e*sec(c+d*x)))+2/11*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^3*d*e^2)+2/15*I/(d*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^3)+12/55*I*e^2/(d*(e*sec(c+d*x))^(7/2)*(a^3+I*a^3*tan(c+d*x)))],
[(e*sec(c+d*x))^(15/2)/(a+I*a*tan(c+d*x))^4,x,6,-154/15*e^5*(e*sec(c+d*x))^(5/2)*sin(c+d*x)/(a^4*d)+154/5*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))-154/5*e^7*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^4*d)+4*I*e^2*(e*sec(c+d*x))^(11/2)/(a*d*(a+I*a*tan(c+d*x))^3)+44/3*I*e^4*(e*sec(c+d*x))^(7/2)/(d*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(13/2)/(a+I*a*tan(c+d*x))^4,x,5,-10*e^5*(e*sec(c+d*x))^(3/2)*sin(c+d*x)/(a^4*d)-10*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^4*d)+4/3*I*e^2*(e*sec(c+d*x))^(9/2)/(a*d*(a+I*a*tan(c+d*x))^3)+12*I*e^4*(e*sec(c+d*x))^(5/2)/(d*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(11/2)/(a+I*a*tan(c+d*x))^4,x,5,-42/5*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+42/5*e^5*sin(c+d*x)*sqrt(e*sec(c+d*x))/(a^4*d)+4/5*I*e^2*(e*sec(c+d*x))^(7/2)/(a*d*(a+I*a*tan(c+d*x))^3)-28/5*I*e^4*(e*sec(c+d*x))^(3/2)/(d*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(9/2)/(a+I*a*tan(c+d*x))^4,x,4,10/21*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^4*d)+4/7*I*e^2*(e*sec(c+d*x))^(5/2)/(a*d*(a+I*a*tan(c+d*x))^3)-20/21*I*e^4*sqrt(e*sec(c+d*x))/(d*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^4,x,4,-2/15*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/9*I*e^2*(e*sec(c+d*x))^(3/2)/(a*d*(a+I*a*tan(c+d*x))^3)-4/15*I*e^4/(d*sqrt(e*sec(c+d*x))*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^4,x,5,-2/77*e^3*sin(c+d*x)/(a^4*d*sqrt(e*sec(c+d*x)))-2/77*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^4*d)+4/11*I*e^2*sqrt(e*sec(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^3)-4/77*I*e^4/(d*(e*sec(c+d*x))^(3/2)*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^4,x,5,2/117*e^3*sin(c+d*x)/(a^4*d*(e*sec(c+d*x))^(3/2))+2/39*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*d*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x)))+4/13*I*e^2/(a*d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^3)+4/117*I*e^4/(d*(e*sec(c+d*x))^(5/2)*(a^4+I*a^4*tan(c+d*x)))],
[(e*sec(c+d*x))^(1/2)/(a+I*a*tan(c+d*x))^4,x,6,2/33*e*sin(c+d*x)/(a^4*d*sqrt(e*sec(c+d*x)))+2/33*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(e*sec(c+d*x))/(a^4*d)+2/15*I*sqrt(e*sec(c+d*x))/(d*(a+I*a*tan(c+d*x))^4)+14/165*I*sqrt(e*sec(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^3)+4/33*I*e^2/(d*(e*sec(c+d*x))^(3/2)*(a^4+I*a^4*tan(c+d*x)))],

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

# n>0
[(d*sec(e+f*x))^(5/3)*(a+I*a*tan(e+f*x)),x,4,6/5*I*2^(5/6)*a*hypergeom([-5/6,5/6],[11/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(5/3)/(f*(1+I*tan(e+f*x))^(5/6))],
[(d*sec(e+f*x))^(1/3)*(a+I*a*tan(e+f*x)),x,4,6*I*2^(1/6)*a*hypergeom([-1/6,1/6],[7/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(1/3)/(f*(1+I*tan(e+f*x))^(1/6))],
[(a+I*a*tan(e+f*x))/(d*sec(e+f*x))^(1/3),x,4,-3*I*2^(5/6)*a*hypergeom([-1/6,1/6],[5/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(1/6)/(f*(d*sec(e+f*x))^(1/3))],
[(a+I*a*tan(e+f*x))/(d*sec(e+f*x))^(5/3),x,4,-3/5*I*2^(1/6)*a*hypergeom([-5/6,5/6],[1/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(5/6)/(f*(d*sec(e+f*x))^(5/3))],
[(d*sec(e+f*x))^(5/3)*(a+I*a*tan(e+f*x))^2,x,4,12/5*I*2^(5/6)*a^2*hypergeom([-11/6,5/6],[11/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(5/3)/(f*(1+I*tan(e+f*x))^(5/6))],
[(d*sec(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^2,x,4,12*I*2^(1/6)*a^2*hypergeom([-7/6,1/6],[7/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(1/3)/(f*(1+I*tan(e+f*x))^(1/6))],
[(a+I*a*tan(e+f*x))^2/(d*sec(e+f*x))^(1/3),x,4,-6*I*2^(5/6)*hypergeom([-5/6,-1/6],[5/6],1/2*(1-I*tan(e+f*x)))*(a^2+I*a^2*tan(e+f*x))/(f*(d*sec(e+f*x))^(1/3)*(1+I*tan(e+f*x))^(5/6))],
[(a+I*a*tan(e+f*x))^2/(d*sec(e+f*x))^(5/3),x,4,-6/5*I*2^(1/6)*hypergeom([-5/6,-1/6],[1/6],1/2*(1-I*tan(e+f*x)))*(a^2+I*a^2*tan(e+f*x))/(f*(d*sec(e+f*x))^(5/3)*(1+I*tan(e+f*x))^(1/6))],

# n<0
[(d*sec(e+f*x))^(5/3)/(a+I*a*tan(e+f*x)),x,4,3/5*I*hypergeom([5/6,7/6],[11/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(5/3)*(1+I*tan(e+f*x))^(1/6)/(2^(1/6)*f*(a+I*a*tan(e+f*x)))],
[(d*sec(e+f*x))^(1/3)/(a+I*a*tan(e+f*x)),x,4,3*I*hypergeom([1/6,11/6],[7/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(1/3)*(1+I*tan(e+f*x))^(5/6)/(2^(5/6)*f*(a+I*a*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))),x,4,-3/2*I*hypergeom([-1/6,13/6],[5/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(1/6)/(2^(1/6)*a*f*(d*sec(e+f*x))^(1/3))],
[1/((d*sec(e+f*x))^(5/3)*(a+I*a*tan(e+f*x))),x,4,-3/10*I*hypergeom([-5/6,17/6],[1/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(5/6)/(2^(5/6)*a*f*(d*sec(e+f*x))^(5/3))],
[(d*sec(e+f*x))^(5/3)/(a+I*a*tan(e+f*x))^2,x,4,3/10*I*hypergeom([5/6,13/6],[11/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(5/3)*(1+I*tan(e+f*x))^(1/6)/(2^(1/6)*f*(a^2+I*a^2*tan(e+f*x)))],
[(d*sec(e+f*x))^(1/3)/(a+I*a*tan(e+f*x))^2,x,4,3/2*I*hypergeom([1/6,17/6],[7/6],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(1/3)*(1+I*tan(e+f*x))^(5/6)/(2^(5/6)*f*(a^2+I*a^2*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^2),x,4,-3/4*I*hypergeom([-1/6,19/6],[5/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(1/6)/(2^(1/6)*a^2*f*(d*sec(e+f*x))^(1/3))],
[1/((d*sec(e+f*x))^(5/3)*(a+I*a*tan(e+f*x))^2),x,4,-3/20*I*hypergeom([-5/6,23/6],[1/6],1/2*(1-I*tan(e+f*x)))*(1+I*tan(e+f*x))^(5/6)/(2^(5/6)*a^2*f*(d*sec(e+f*x))^(5/3))],

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

# n>0
[sec(c+d*x)^8*sqrt(a+I*a*tan(c+d*x)),x,3,-16/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^4*d)+24/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^5*d)-12/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^6*d)+2/15*I*(a+I*a*tan(c+d*x))^(15/2)/(a^7*d)],
[sec(c+d*x)^6*sqrt(a+I*a*tan(c+d*x)),x,3,-8/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^3*d)+8/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^4*d)-2/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^5*d)],
[sec(c+d*x)^4*sqrt(a+I*a*tan(c+d*x)),x,3,-4/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^2*d)+2/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^3*d)],
[sec(c+d*x)^2*sqrt(a+I*a*tan(c+d*x)),x,2,-2/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[cos(c+d*x)^2*sqrt(a+I*a*tan(c+d*x)),x,5,-3/4*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))+3/4*I*a/(d*sqrt(a+I*a*tan(c+d*x)))-1/2*I*a^2/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*sqrt(a+I*a*tan(c+d*x)),x,7,-35/64*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))+35/64*I*a/(d*sqrt(a+I*a*tan(c+d*x)))+35/96*I*a^2/(d*(a+I*a*tan(c+d*x))^(3/2))-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(3/2))-7/16*I*a^3/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^6*sqrt(a+I*a*tan(c+d*x)),x,9,-231/512*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))+231/512*I*a/(d*sqrt(a+I*a*tan(c+d*x)))+231/640*I*a^3/(d*(a+I*a*tan(c+d*x))^(5/2))-1/6*I*a^6/(d*(a-I*a*tan(c+d*x))^3*(a+I*a*tan(c+d*x))^(5/2))-11/48*I*a^5/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(5/2))-33/64*I*a^4/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+77/256*I*a^2/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^7*sqrt(a+I*a*tan(c+d*x)),x,4,2/13*I*a*sec(c+d*x)^7/(d*sqrt(a+I*a*tan(c+d*x)))+256/3003*I*a^4*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(7/2))+64/429*I*a^3*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(5/2))+24/143*I*a^2*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^5*sqrt(a+I*a*tan(c+d*x)),x,3,2/9*I*a*sec(c+d*x)^5/(d*sqrt(a+I*a*tan(c+d*x)))+64/315*I*a^3*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(5/2))+16/63*I*a^2*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^3*sqrt(a+I*a*tan(c+d*x)),x,2,2/5*I*a*sec(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))+8/15*I*a^2*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)*sqrt(a+I*a*tan(c+d*x)),x,1,2*I*a*sec(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))],
[cos(c+d*x)*sqrt(a+I*a*tan(c+d*x)),x,3,I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(a)/(d*sqrt(2))-I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x)),x,5,5/8*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(a)/(d*sqrt(2))+5/12*I*a*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-5/8*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/3*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d],
[cos(c+d*x)^5*sqrt(a+I*a*tan(c+d*x)),x,7,63/128*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(a)/(d*sqrt(2))+21/64*I*a*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+9/40*I*a*cos(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-63/128*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-21/80*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-1/5*I*cos(c+d*x)^5*sqrt(a+I*a*tan(c+d*x))/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^(3/2),x,3,-16/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^4*d)+24/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^5*d)-4/5*I*(a+I*a*tan(c+d*x))^(15/2)/(a^6*d)+2/17*I*(a+I*a*tan(c+d*x))^(17/2)/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^(3/2),x,3,-8/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^3*d)+8/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^4*d)-2/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^(3/2),x,3,-4/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^2*d)+2/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2),x,2,-2/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a*d)],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2),x,4,-1/2*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-1/2*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^(3/2),x,6,-15/32*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+15/32*I*a^2/(d*sqrt(a+I*a*tan(c+d*x)))-1/4*I*a^4/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x))^2)-5/16*I*a^3/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^(3/2),x,8,-105/256*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+105/256*I*a^2/(d*sqrt(a+I*a*tan(c+d*x)))+35/128*I*a^3/(d*(a+I*a*tan(c+d*x))^(3/2))-1/6*I*a^6/(d*(a-I*a*tan(c+d*x))^3*(a+I*a*tan(c+d*x))^(3/2))-3/16*I*a^5/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(3/2))-21/64*I*a^4/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^5*(a+I*a*tan(c+d*x))^(3/2),x,4,8/33*I*a^2*sec(c+d*x)^5/(d*sqrt(a+I*a*tan(c+d*x)))+2/11*I*a*sec(c+d*x)^5*sqrt(a+I*a*tan(c+d*x))/d+256/1155*I*a^4*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(5/2))+64/231*I*a^3*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2),x,3,16/35*I*a^2*sec(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))+2/7*I*a*sec(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d+64/105*I*a^3*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^(3/2),x,2,8/3*I*a^2*sec(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+2/3*I*a*sec(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^(3/2),x,1,-2*I*a*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2),x,4,1/2*I*a^(3/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))-1/2*I*a*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/3*I*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(3/2),x,6,7/16*I*a^(3/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))+7/24*I*a^2*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-7/16*I*a*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-7/30*I*a*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-1/5*I*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(3/2)/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^(5/2),x,3,-16/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^4*d)+8/5*I*(a+I*a*tan(c+d*x))^(15/2)/(a^5*d)-12/17*I*(a+I*a*tan(c+d*x))^(17/2)/(a^6*d)+2/19*I*(a+I*a*tan(c+d*x))^(19/2)/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^(5/2),x,3,-8/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^3*d)+8/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^4*d)-2/15*I*(a+I*a*tan(c+d*x))^(15/2)/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^(5/2),x,3,-4/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^2*d)+2/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2),x,2,-2/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a*d)],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2),x,4,I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-I*a^3*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^(5/2),x,5,-3/16*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-1/4*I*a^4*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x))^2)-3/16*I*a^3*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^(5/2),x,7,-35/128*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+35/128*I*a^3/(d*sqrt(a+I*a*tan(c+d*x)))-1/6*I*a^6/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x))^3)-7/48*I*a^5/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x))^2)-35/192*I*a^4/(d*sqrt(a+I*a*tan(c+d*x))*(a-I*a*tan(c+d*x)))],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2),x,4,64/105*I*a^3*sec(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))+8/21*I*a^2*sec(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d+256/315*I*a^4*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))+2/9*I*a*sec(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^(5/2),x,3,64/15*I*a^3*sec(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+16/15*I*a^2*sec(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d+2/5*I*a*sec(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^(5/2),x,2,-8*I*a^2*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d+2*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2),x,1,-2/3*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(5/2),x,5,1/4*I*a^(5/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))-1/4*I*a^2*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/6*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d-1/5*I*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(5/2)/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(5/2),x,7,9/32*I*a^(5/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))+3/16*I*a^3*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-9/32*I*a^2*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-3/20*I*a^2*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-9/70*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(3/2)/d-1/7*I*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(5/2)/d],
[sec(c+d*x)^8*(a+I*a*tan(c+d*x))^(7/2),x,3,-16/15*I*(a+I*a*tan(c+d*x))^(15/2)/(a^4*d)+24/17*I*(a+I*a*tan(c+d*x))^(17/2)/(a^5*d)-12/19*I*(a+I*a*tan(c+d*x))^(19/2)/(a^6*d)+2/21*I*(a+I*a*tan(c+d*x))^(21/2)/(a^7*d)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^(7/2),x,3,-8/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^3*d)+8/15*I*(a+I*a*tan(c+d*x))^(15/2)/(a^4*d)-2/17*I*(a+I*a*tan(c+d*x))^(17/2)/(a^5*d)],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^(7/2),x,3,-4/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^2*d)+2/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^3*d)],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^(7/2),x,2,-2/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a*d)],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^(7/2),x,5,3*I*a^(7/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-3*I*a^3*sqrt(a+I*a*tan(c+d*x))/d-I*a^3*(a+I*a*tan(c+d*x))^(3/2)/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^(7/2),x,5,1/8*I*a^(7/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-1/2*I*a^5*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x))^2)+1/8*I*a^4*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x)))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^(7/2),x,6,-5/64*I*a^(7/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-1/6*I*a^6*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x))^3)-5/48*I*a^5*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x))^2)-5/64*I*a^4*sqrt(a+I*a*tan(c+d*x))/(d*(a-I*a*tan(c+d*x)))],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^(7/2),x,4,256/35*I*a^4*sec(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+64/35*I*a^3*sec(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d+24/35*I*a^2*sec(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)/d+2/7*I*a*sec(c+d*x)*(a+I*a*tan(c+d*x))^(5/2)/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^(7/2),x,3,-64/3*I*a^3*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d+16/3*I*a^2*cos(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)/d+2/3*I*a*cos(c+d*x)*(a+I*a*tan(c+d*x))^(5/2)/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(7/2),x,2,8/3*I*a^2*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d-2*I*a*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2)/d],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(7/2),x,1,-2/5*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(5/2)/d],
[cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(7/2),x,6,1/8*I*a^(7/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))-1/8*I*a^3*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/12*I*a^2*cos(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d-1/10*I*a*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(5/2)/d-1/7*I*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(7/2)/d],
[cos(c+d*x)^9*(a+I*a*tan(c+d*x))^(7/2),x,8,11/64*I*a^(7/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))+11/96*I*a^4*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-11/64*I*a^3*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-11/120*I*a^3*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-11/140*I*a^2*cos(c+d*x)^5*(a+I*a*tan(c+d*x))^(3/2)/d-11/126*I*a*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(5/2)/d-1/9*I*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^(7/2)/d],
[cos(c+d*x)^11*(a+I*a*tan(c+d*x))^(7/2),x,10,195/1024*I*a^(7/2)*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2))+65/512*I*a^4*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+39/448*I*a^4*cos(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-195/1024*I*a^3*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-13/128*I*a^3*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-13/168*I*a^3*cos(c+d*x)^5*sqrt(a+I*a*tan(c+d*x))/d-65/924*I*a^2*cos(c+d*x)^7*(a+I*a*tan(c+d*x))^(3/2)/d-5/66*I*a*cos(c+d*x)^9*(a+I*a*tan(c+d*x))^(5/2)/d-1/11*I*cos(c+d*x)^11*(a+I*a*tan(c+d*x))^(7/2)/d],

# n<0
[sec(c+d*x)^8/sqrt(a+I*a*tan(c+d*x)),x,3,-16/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^4*d)+8/3*I*(a+I*a*tan(c+d*x))^(9/2)/(a^5*d)-12/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^6*d)+2/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^7*d)],
[sec(c+d*x)^6/sqrt(a+I*a*tan(c+d*x)),x,3,-8/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^3*d)+8/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^4*d)-2/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^5*d)],
[sec(c+d*x)^4/sqrt(a+I*a*tan(c+d*x)),x,3,-4/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)+2/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^3*d)],
[sec(c+d*x)^2/sqrt(a+I*a*tan(c+d*x)),x,2,-2*I*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cos(c+d*x)^2/sqrt(a+I*a*tan(c+d*x)),x,6,-5/8*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+5/8*I/(d*sqrt(a+I*a*tan(c+d*x)))+5/12*I*a/(d*(a+I*a*tan(c+d*x))^(3/2))-1/2*I*a^2/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^4/sqrt(a+I*a*tan(c+d*x)),x,8,-63/128*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+63/128*I/(d*sqrt(a+I*a*tan(c+d*x)))+63/160*I*a^2/(d*(a+I*a*tan(c+d*x))^(5/2))-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(5/2))-9/16*I*a^3/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+21/64*I*a/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^6/sqrt(a+I*a*tan(c+d*x)),x,10,-429/1024*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+429/1024*I/(d*sqrt(a+I*a*tan(c+d*x)))+429/896*I*a^3/(d*(a+I*a*tan(c+d*x))^(7/2))-1/6*I*a^6/(d*(a-I*a*tan(c+d*x))^3*(a+I*a*tan(c+d*x))^(7/2))-13/48*I*a^5/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(7/2))-143/192*I*a^4/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(7/2))+429/1280*I*a^2/(d*(a+I*a*tan(c+d*x))^(5/2))+143/512*I*a/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^9/sqrt(a+I*a*tan(c+d*x)),x,4,256/6435*I*a^4*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(9/2))+64/715*I*a^3*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(7/2))+8/65*I*a^2*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(5/2))+2/15*I*a*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^7/sqrt(a+I*a*tan(c+d*x)),x,3,64/693*I*a^3*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(7/2))+16/99*I*a^2*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(5/2))+2/11*I*a*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^5/sqrt(a+I*a*tan(c+d*x)),x,2,8/35*I*a^2*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(5/2))+2/7*I*a*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^3/sqrt(a+I*a*tan(c+d*x)),x,1,2/3*I*a*sec(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)/sqrt(a+I*a*tan(c+d*x)),x,2,I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[cos(c+d*x)/sqrt(a+I*a*tan(c+d*x)),x,4,3/4*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2)*sqrt(a))+1/2*I*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-3/4*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cos(c+d*x)^3/sqrt(a+I*a*tan(c+d*x)),x,6,35/64*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(d*sqrt(2)*sqrt(a))+35/96*I*cos(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))+1/4*I*cos(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-35/64*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)-7/24*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^(3/2),x,3,-16/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^4*d)+24/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^5*d)-4/3*I*(a+I*a*tan(c+d*x))^(9/2)/(a^6*d)+2/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^(3/2),x,3,-8/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^3*d)+8/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^4*d)-2/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^5*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^(3/2),x,3,-4*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+2/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^3*d)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^(3/2),x,2,2*I/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^(3/2),x,7,-7/16*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+7/16*I/(a*d*sqrt(a+I*a*tan(c+d*x)))+7/20*I*a/(d*(a+I*a*tan(c+d*x))^(5/2))-1/2*I*a^2/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+7/24*I/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^(3/2),x,9,-99/256*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+99/256*I/(a*d*sqrt(a+I*a*tan(c+d*x)))+99/224*I*a^2/(d*(a+I*a*tan(c+d*x))^(7/2))-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(7/2))-11/16*I*a^3/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(7/2))+99/320*I*a/(d*(a+I*a*tan(c+d*x))^(5/2))+33/128*I/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^6/(a+I*a*tan(c+d*x))^(3/2),x,11,-715/2048*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+715/2048*I/(a*d*sqrt(a+I*a*tan(c+d*x)))+715/1152*I*a^3/(d*(a+I*a*tan(c+d*x))^(9/2))-1/6*I*a^6/(d*(a-I*a*tan(c+d*x))^3*(a+I*a*tan(c+d*x))^(9/2))-5/16*I*a^5/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(9/2))-65/64*I*a^4/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(9/2))+715/1792*I*a^2/(d*(a+I*a*tan(c+d*x))^(7/2))+143/512*I*a/(d*(a+I*a*tan(c+d*x))^(5/2))+715/3072*I/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^11/(a+I*a*tan(c+d*x))^(3/2),x,4,256/12155*I*a^4*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(11/2))+64/1105*I*a^3*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(9/2))+8/85*I*a^2*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(7/2))+2/17*I*a*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(5/2))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^(3/2),x,3,64/1287*I*a^3*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(9/2))+16/143*I*a^2*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(7/2))+2/13*I*a*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(5/2))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^(3/2),x,2,8/63*I*a^2*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(7/2))+2/9*I*a*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(5/2))],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^(3/2),x,1,2/5*I*a*sec(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^(5/2))],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^(3/2),x,3,2*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(2)/(a^(3/2)*d)-2*I*sec(c+d*x)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^(3/2),x,3,1/2*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^(3/2),x,5,15/32*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(3/2)*d*sqrt(2))+5/16*I*cos(c+d*x)/(a*d*sqrt(a+I*a*tan(c+d*x)))-15/32*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/4*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^(3/2),x,7,105/256*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(3/2)*d*sqrt(2))+35/128*I*cos(c+d*x)/(a*d*sqrt(a+I*a*tan(c+d*x)))+3/16*I*cos(c+d*x)^3/(a*d*sqrt(a+I*a*tan(c+d*x)))-105/256*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)-7/32*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/6*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^(5/2),x,3,-32/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^5*d)+64/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^6*d)-16/3*I*(a+I*a*tan(c+d*x))^(9/2)/(a^7*d)+16/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^8*d)-2/13*I*(a+I*a*tan(c+d*x))^(13/2)/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^(5/2),x,3,-16/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^4*d)+24/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^5*d)-12/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^6*d)+2/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^(5/2),x,3,-8*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+8/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^4*d)-2/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^5*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^(5/2),x,3,4*I/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+2*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d)],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^(5/2),x,2,2/3*I/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^(5/2),x,8,-9/32*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+9/32*I/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+9/28*I*a/(d*(a+I*a*tan(c+d*x))^(7/2))-1/2*I*a^2/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(7/2))+9/40*I/(d*(a+I*a*tan(c+d*x))^(5/2))+3/16*I/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^(5/2),x,10,-143/512*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+143/512*I/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+143/288*I*a^2/(d*(a+I*a*tan(c+d*x))^(9/2))-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(9/2))-13/16*I*a^3/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(9/2))+143/448*I*a/(d*(a+I*a*tan(c+d*x))^(7/2))+143/640*I/(d*(a+I*a*tan(c+d*x))^(5/2))+143/768*I/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^13/(a+I*a*tan(c+d*x))^(5/2),x,4,256/20995*I*a^4*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(13/2))+64/1615*I*a^3*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(11/2))+24/323*I*a^2*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(9/2))+2/19*I*a*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(7/2))],
[sec(c+d*x)^11/(a+I*a*tan(c+d*x))^(5/2),x,3,64/2145*I*a^3*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(11/2))+16/195*I*a^2*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(9/2))+2/15*I*a*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(7/2))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^(5/2),x,2,8/99*I*a^2*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(9/2))+2/11*I*a*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(7/2))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^(5/2),x,1,2/7*I*a*sec(c+d*x)^7/(d*(a+I*a*tan(c+d*x))^(7/2))],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^(5/2),x,4,4*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(2)/(a^(5/2)*d)-4*I*sec(c+d*x)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-2/3*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^(5/2),x,4,-I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(5/2)*d*sqrt(2))+I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^(5/2),x,4,3/16*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^(5/2))+3/16*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^(5/2),x,6,35/128*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(5/2)*d*sqrt(2))+35/192*I*cos(c+d*x)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-35/128*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/6*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^(5/2))+7/48*I*cos(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^(5/2),x,8,1155/4096*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(5/2)*d*sqrt(2))+385/2048*I*cos(c+d*x)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+33/256*I*cos(c+d*x)^3/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1155/4096*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)-77/512*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/8*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(5/2))+11/96*I*cos(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^10/(a+I*a*tan(c+d*x))^(7/2),x,3,-32/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^5*d)+64/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^6*d)-48/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^7*d)+16/9*I*(a+I*a*tan(c+d*x))^(9/2)/(a^8*d)-2/11*I*(a+I*a*tan(c+d*x))^(11/2)/(a^9*d)],
[sec(c+d*x)^8/(a+I*a*tan(c+d*x))^(7/2),x,3,-16*I*sqrt(a+I*a*tan(c+d*x))/(a^4*d)+8*I*(a+I*a*tan(c+d*x))^(3/2)/(a^5*d)-12/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a^6*d)+2/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a^7*d)],
[sec(c+d*x)^6/(a+I*a*tan(c+d*x))^(7/2),x,3,8*I/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+8*I*sqrt(a+I*a*tan(c+d*x))/(a^4*d)-2/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a^5*d)],
[sec(c+d*x)^4/(a+I*a*tan(c+d*x))^(7/2),x,3,(-2*I)/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+4/3*I/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^2/(a+I*a*tan(c+d*x))^(7/2),x,2,2/5*I/(a*d*(a+I*a*tan(c+d*x))^(5/2))],
[cos(c+d*x)^2/(a+I*a*tan(c+d*x))^(7/2),x,9,-11/64*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(7/2)*d*sqrt(2))+11/64*I/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+11/36*I*a/(d*(a+I*a*tan(c+d*x))^(9/2))-1/2*I*a^2/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(9/2))+11/56*I/(d*(a+I*a*tan(c+d*x))^(7/2))+11/80*I/(a*d*(a+I*a*tan(c+d*x))^(5/2))+11/96*I/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^4/(a+I*a*tan(c+d*x))^(7/2),x,11,-195/1024*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(7/2)*d*sqrt(2))+195/1024*I/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+195/352*I*a^2/(d*(a+I*a*tan(c+d*x))^(11/2))-1/4*I*a^4/(d*(a-I*a*tan(c+d*x))^2*(a+I*a*tan(c+d*x))^(11/2))-15/16*I*a^3/(d*(a-I*a*tan(c+d*x))*(a+I*a*tan(c+d*x))^(11/2))+65/192*I*a/(d*(a+I*a*tan(c+d*x))^(9/2))+195/896*I/(d*(a+I*a*tan(c+d*x))^(7/2))+39/256*I/(a*d*(a+I*a*tan(c+d*x))^(5/2))+65/512*I/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^13/(a+I*a*tan(c+d*x))^(7/2),x,3,64/3315*I*a^3*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(13/2))+16/255*I*a^2*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(11/2))+2/17*I*a*sec(c+d*x)^13/(d*(a+I*a*tan(c+d*x))^(9/2))],
[sec(c+d*x)^11/(a+I*a*tan(c+d*x))^(7/2),x,2,8/143*I*a^2*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(11/2))+2/13*I*a*sec(c+d*x)^11/(d*(a+I*a*tan(c+d*x))^(9/2))],
[sec(c+d*x)^9/(a+I*a*tan(c+d*x))^(7/2),x,1,2/9*I*a*sec(c+d*x)^9/(d*(a+I*a*tan(c+d*x))^(9/2))],
[sec(c+d*x)^7/(a+I*a*tan(c+d*x))^(7/2),x,5,8*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(2)/(a^(7/2)*d)-8*I*sec(c+d*x)/(a^3*d*sqrt(a+I*a*tan(c+d*x)))-2/5*I*sec(c+d*x)^5/(a*d*(a+I*a*tan(c+d*x))^(5/2))-4/3*I*sec(c+d*x)^3/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^5/(a+I*a*tan(c+d*x))^(7/2),x,5,-3*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))*sqrt(2)/(a^(7/2)*d)-2*I*sec(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(5/2))+6*I*sec(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)^3/(a+I*a*tan(c+d*x))^(7/2),x,5,-1/8*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/2*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(5/2))-1/8*I*sec(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[sec(c+d*x)/(a+I*a*tan(c+d*x))^(7/2),x,5,5/64*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/6*I*sec(c+d*x)/(d*(a+I*a*tan(c+d*x))^(7/2))+5/48*I*sec(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(5/2))+5/64*I*sec(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)/(a+I*a*tan(c+d*x))^(7/2),x,7,315/2048*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(7/2)*d*sqrt(2))+105/1024*I*cos(c+d*x)/(a^3*d*sqrt(a+I*a*tan(c+d*x)))-315/2048*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^4*d)+1/8*I*cos(c+d*x)/(d*(a+I*a*tan(c+d*x))^(7/2))+3/32*I*cos(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(5/2))+21/256*I*cos(c+d*x)/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],
[cos(c+d*x)^3/(a+I*a*tan(c+d*x))^(7/2),x,9,3003/16384*I*arctanh(sec(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+I*a*tan(c+d*x))))/(a^(7/2)*d*sqrt(2))+1001/8192*I*cos(c+d*x)/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+429/5120*I*cos(c+d*x)^3/(a^3*d*sqrt(a+I*a*tan(c+d*x)))-3003/16384*I*cos(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^4*d)-1001/10240*I*cos(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/(a^4*d)+1/10*I*cos(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(7/2))+13/160*I*cos(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(5/2))+143/1920*I*cos(c+d*x)^3/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],

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

# n>0
[(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)),x,12,I*a*(e*sec(c+d*x))^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-I*a^(3/2)*e^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*a^(3/2)*e^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/2*I*a^(3/2)*e^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/2*I*a^(3/2)*e^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)),x,10,-I*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(e)/(d*sqrt(2))+I*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(e)/(d*sqrt(2))+I*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)*sqrt(a)*sqrt(e)/d-I*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)*sqrt(a)*sqrt(e)/d],
[sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x)),x,1,-2*I*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(e*sec(c+d*x)))],
[sqrt(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(3/2),x,2,4/3*I*a*sqrt(e*sec(c+d*x))/(d*e^2*sqrt(a+I*a*tan(c+d*x)))-2/3*I*sqrt(a+I*a*tan(c+d*x))/(d*(e*sec(c+d*x))^(3/2))],
[sqrt(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(5/2),x,3,8/15*I*a/(d*e^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-2/5*I*sqrt(a+I*a*tan(c+d*x))/(d*(e*sec(c+d*x))^(5/2))-16/15*I*sqrt(a+I*a*tan(c+d*x))/(d*e^2*sqrt(e*sec(c+d*x)))],
[sqrt(a+I*a*tan(c+d*x))/(e*sec(c+d*x))^(7/2),x,4,12/35*I*a/(d*e^2*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))+32/35*I*a*sqrt(e*sec(c+d*x))/(d*e^4*sqrt(a+I*a*tan(c+d*x)))-2/7*I*sqrt(a+I*a*tan(c+d*x))/(d*(e*sec(c+d*x))^(7/2))-16/35*I*sqrt(a+I*a*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^(3/2),x,13,7/8*I*a^(3/2)*e^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*sqrt(2))-7/8*I*a^(3/2)*e^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*sqrt(2))-7/16*I*a^(3/2)*e^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*sqrt(2))+7/16*I*a^(3/2)*e^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*sqrt(2))+7/12*I*a^2*(e*sec(c+d*x))^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))+1/3*I*a*(e*sec(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x))/d-7/8*I*a*e^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(3/2),x,13,5/4*I*a^2*(e*sec(c+d*x))^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-5/4*I*a^(5/2)*e^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+5/4*I*a^(5/2)*e^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+5/8*I*a^(5/2)*e^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-5/8*I*a^(5/2)*e^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/2*I*a*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(3/2),x,11,3*I*a^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(e)/(d*sqrt(2))-3*I*a^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(e)/(d*sqrt(2))-3/2*I*a^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(e)/(d*sqrt(2))+3/2*I*a^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(e)/(d*sqrt(2))+I*a*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^(3/2)/sqrt(e*sec(c+d*x)),x,12,-I*a^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*a^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*a^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)/(d*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*a^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)/(d*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-4*I*a*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/(e*sec(c+d*x))^(3/2),x,1,-2/3*I*(a+I*a*tan(c+d*x))^(3/2)/(d*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)/(e*sec(c+d*x))^(5/2),x,2,-4/5*I*a*sqrt(a+I*a*tan(c+d*x))/(d*e^2*sqrt(e*sec(c+d*x)))-2/5*I*(a+I*a*tan(c+d*x))^(3/2)/(d*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^(3/2)/(e*sec(c+d*x))^(7/2),x,3,16/21*I*a^2*sqrt(e*sec(c+d*x))/(d*e^4*sqrt(a+I*a*tan(c+d*x)))-8/21*I*a*sqrt(a+I*a*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(3/2))-2/7*I*(a+I*a*tan(c+d*x))^(3/2)/(d*(e*sec(c+d*x))^(7/2))],
[(a+I*a*tan(c+d*x))^(3/2)/(e*sec(c+d*x))^(9/2),x,4,16/45*I*a^2/(d*e^4*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-4/15*I*a*sqrt(a+I*a*tan(c+d*x))/(d*e^2*(e*sec(c+d*x))^(5/2))-32/45*I*a*sqrt(a+I*a*tan(c+d*x))/(d*e^4*sqrt(e*sec(c+d*x)))-2/9*I*(a+I*a*tan(c+d*x))^(3/2)/(d*(e*sec(c+d*x))^(9/2))],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(5/2),x,14,15/8*I*a^3*(e*sec(c+d*x))^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-15/8*I*a^(7/2)*e^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+15/8*I*a^(7/2)*e^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+15/16*I*a^(7/2)*e^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-15/16*I*a^(7/2)*e^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+3/4*I*a^2*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))/d+1/3*I*a*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(5/2),x,12,21/4*I*a^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(e)/(d*sqrt(2))-21/4*I*a^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(e)/(d*sqrt(2))-21/8*I*a^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(e)/(d*sqrt(2))+21/8*I*a^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(e)/(d*sqrt(2))+7/4*I*a^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*I*a*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)/d],
[(a+I*a*tan(c+d*x))^(5/2)/sqrt(e*sec(c+d*x)),x,13,5*I*a^(7/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-5*I*a^(7/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-5/2*I*a^(7/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+5/2*I*a^(7/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-10*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(e*sec(c+d*x)))+I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^(5/2)/(e*sec(c+d*x))^(3/2),x,11,I*a^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*e^(3/2)*sqrt(2))-I*a^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*e^(3/2)*sqrt(2))-I*a^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)/(d*e^(3/2))+I*a^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)/(d*e^(3/2))-4/3*I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^(5/2)/(e*sec(c+d*x))^(5/2),x,1,-2/5*I*(a+I*a*tan(c+d*x))^(5/2)/(d*(e*sec(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))^(5/2)/(e*sec(c+d*x))^(7/2),x,2,-4/21*I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*e^2*(e*sec(c+d*x))^(3/2))-2/7*I*(a+I*a*tan(c+d*x))^(5/2)/(d*(e*sec(c+d*x))^(7/2))],
[(a+I*a*tan(c+d*x))^(5/2)/(e*sec(c+d*x))^(9/2),x,3,-16/45*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*e^4*sqrt(e*sec(c+d*x)))-8/45*I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*e^2*(e*sec(c+d*x))^(5/2))-2/9*I*(a+I*a*tan(c+d*x))^(5/2)/(d*(e*sec(c+d*x))^(9/2))],
[(a+I*a*tan(c+d*x))^(5/2)/(e*sec(c+d*x))^(11/2),x,4,32/77*I*a^3*sqrt(e*sec(c+d*x))/(d*e^6*sqrt(a+I*a*tan(c+d*x)))-16/77*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*e^4*(e*sec(c+d*x))^(3/2))-12/77*I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*e^2*(e*sec(c+d*x))^(7/2))-2/11*I*(a+I*a*tan(c+d*x))^(5/2)/(d*(e*sec(c+d*x))^(11/2))],

# n<0
[(e*sec(c+d*x))^(5/2)/sqrt(a+I*a*tan(c+d*x)),x,11,I*e^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*sqrt(2)*sqrt(a))-I*e^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*sqrt(2)*sqrt(a))-1/2*I*e^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*sqrt(2)*sqrt(a))+1/2*I*e^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*sqrt(2)*sqrt(a))-I*e^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[(e*sec(c+d*x))^(3/2)/sqrt(a+I*a*tan(c+d*x)),x,11,I*e^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*e^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*e^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)*sqrt(a)/(d*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*e^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)*sqrt(a)/(d*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[sqrt(e*sec(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,1,2*I*sqrt(e*sec(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))),x,2,2/3*I/(d*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-4/3*I*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(e*sec(c+d*x)))],
[1/((e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))),x,3,2/5*I/(d*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))+16/15*I*sqrt(e*sec(c+d*x))/(d*e^2*sqrt(a+I*a*tan(c+d*x)))-8/15*I*sqrt(a+I*a*tan(c+d*x))/(a*d*(e*sec(c+d*x))^(3/2))],
[1/((e*sec(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x))),x,4,2/7*I/(d*(e*sec(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x)))+16/35*I/(d*e^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-12/35*I*sqrt(a+I*a*tan(c+d*x))/(a*d*(e*sec(c+d*x))^(5/2))-32/35*I*sqrt(a+I*a*tan(c+d*x))/(a*d*e^2*sqrt(e*sec(c+d*x)))],
[1/((e*sec(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x))),x,5,2/9*I/(d*(e*sec(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x)))+32/105*I/(d*e^2*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))+256/315*I*sqrt(e*sec(c+d*x))/(d*e^4*sqrt(a+I*a*tan(c+d*x)))-16/63*I*sqrt(a+I*a*tan(c+d*x))/(a*d*(e*sec(c+d*x))^(7/2))-128/315*I*sqrt(a+I*a*tan(c+d*x))/(a*d*e^2*(e*sec(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^(3/2),x,13,-I*e^2*(e*sec(c+d*x))^(3/2)/(a*d*sqrt(a+I*a*tan(c+d*x)))-3*I*e^(7/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+3*I*e^(7/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*sqrt(2)*sqrt(a)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+3/2*I*e^(7/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-3/2*I*e^(7/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*sqrt(2)*sqrt(a)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^(3/2),x,11,I*e^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(a^(3/2)*d*sqrt(2))-I*e^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(a^(3/2)*d*sqrt(2))-I*e^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)/(a^(3/2)*d)+I*e^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(2)/(a^(3/2)*d)+4*I*e^2*sqrt(e*sec(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^(3/2),x,1,2/3*I*(e*sec(c+d*x))^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(e*sec(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,2,4/5*I*sqrt(e*sec(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+2/5*I*sqrt(e*sec(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)),x,3,8/21*I/(a*d*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-16/21*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(e*sec(c+d*x)))+2/7*I/(d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/((e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(3/2)),x,4,4/15*I/(a*d*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))+32/45*I*sqrt(e*sec(c+d*x))/(a*d*e^2*sqrt(a+I*a*tan(c+d*x)))-16/45*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d*(e*sec(c+d*x))^(3/2))+2/9*I/(d*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/((e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^(3/2)),x,5,16/77*I/(a*d*(e*sec(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x)))+128/385*I/(a*d*e^2*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-96/385*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d*(e*sec(c+d*x))^(5/2))-256/385*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d*e^2*sqrt(e*sec(c+d*x)))+2/11*I/(d*(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(9/2)/(a+I*a*tan(c+d*x))^(5/2),x,12,-5*I*e^(9/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+5*I*e^(9/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+5/2*I*e^(9/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(a^(5/2)*d*sqrt(2))-5/2*I*e^(9/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(a^(5/2)*d*sqrt(2))+5*I*e^4*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+4*I*e^2*(e*sec(c+d*x))^(5/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^(5/2),x,12,-I*e^(7/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(a^(3/2)*d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*e^(7/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(a^(3/2)*d*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*e^(7/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)/(a^(3/2)*d*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*e^(7/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(2)/(a^(3/2)*d*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+4/3*I*e^2*(e*sec(c+d*x))^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^(5/2),x,1,2/5*I*(e*sec(c+d*x))^(5/2)/(d*(a+I*a*tan(c+d*x))^(5/2))],
[(e*sec(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^(5/2),x,2,2/7*I*(e*sec(c+d*x))^(3/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+4/21*I*(e*sec(c+d*x))^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(e*sec(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,3,16/45*I*sqrt(e*sec(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+2/9*I*sqrt(e*sec(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+8/45*I*sqrt(e*sec(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)),x,4,16/77*I/(a^2*d*sqrt(e*sec(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-32/77*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d*sqrt(e*sec(c+d*x)))+2/11*I/(d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+12/77*I/(a*d*sqrt(e*sec(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/((e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(5/2)),x,5,32/195*I/(a^2*d*(e*sec(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))+256/585*I*sqrt(e*sec(c+d*x))/(a^2*d*e^2*sqrt(a+I*a*tan(c+d*x)))-128/585*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d*(e*sec(c+d*x))^(3/2))+2/13*I/(d*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(5/2))+16/117*I/(a*d*(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form (d Sec[e+f x])^(m/3) (a+a I Tan[e+f x])^(n/2)

# n>0

# n<0
[(e*sec(c+d*x))^(7/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/7*I*2^(2/3)*a*hypergeom([1/3,7/6],[13/6],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(7/3)*(1+I*tan(c+d*x))^(1/3)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(5/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/5*I*2^(1/3)*a*hypergeom([2/3,5/6],[11/6],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(5/3)*(1+I*tan(c+d*x))^(2/3)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[(e*sec(c+d*x))^(2/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/2*I*hypergeom([1/3,7/6],[4/3],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(2/3)*(1+I*tan(c+d*x))^(1/6)/(2^(1/6)*d*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^(1/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3*I*hypergeom([1/6,4/3],[7/6],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(1/3)*(1+I*tan(c+d*x))^(1/3)/(2^(1/3)*d*sqrt(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(1/3)*sqrt(a+I*a*tan(c+d*x))),x,4,-3*I*hypergeom([-1/6,5/3],[5/6],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(2/3)/(2^(2/3)*d*(e*sec(c+d*x))^(1/3)*sqrt(a+I*a*tan(c+d*x)))],
[1/((e*sec(c+d*x))^(4/3)*sqrt(a+I*a*tan(c+d*x))),x,4,-3/8*I*hypergeom([-2/3,13/6],[1/3],1/2*(1-I*tan(c+d*x)))*sqrt(a+I*a*tan(c+d*x))*(1+I*tan(c+d*x))^(1/6)/(2^(1/6)*a*d*(e*sec(c+d*x))^(4/3))],

# Integrands of the form (d Sec[e+f x])^(m/3) (a+a I Tan[e+f x])^(n/3)
[(d*sec(e+f*x))^(2/3)/(a+I*a*tan(e+f*x))^(7/3),x,9,1/4*I*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(7/3))-5/72*x*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(5/3)*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-5/72*I*log(cos(e+f*x))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(5/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-5/24*I*log(2^(1/3)*a^(1/3)-(a-I*a*tan(e+f*x))^(1/3))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(5/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))+5/12*I*arctan((a^(1/3)+2^(2/3)*(a-I*a*tan(e+f*x))^(1/3))/(a^(1/3)*sqrt(3)))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(5/3)*f*sqrt(3)*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))+5/24*I*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(1/3)*(a^2+I*a^2*tan(e+f*x)))],
[(d*sec(e+f*x))^(2/3)/(a+I*a*tan(e+f*x))^(4/3),x,8,1/2*I*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(4/3))-1/6*x*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(2/3)*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-1/6*I*log(cos(e+f*x))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(2/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-1/2*I*log(2^(1/3)*a^(1/3)-(a-I*a*tan(e+f*x))^(1/3))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(2/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))+I*arctan((a^(1/3)+2^(2/3)*(a-I*a*tan(e+f*x))^(1/3))/(a^(1/3)*sqrt(3)))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*a^(2/3)*f*sqrt(3)*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))],
[(d*sec(e+f*x))^(2/3)/(a+I*a*tan(e+f*x))^(1/3),x,6,-1/2*a^(1/3)*x*(d*sec(e+f*x))^(2/3)/(2^(2/3)*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-1/2*I*a^(1/3)*log(cos(e+f*x))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))-3/2*I*a^(1/3)*log(2^(1/3)*a^(1/3)-(a-I*a*tan(e+f*x))^(1/3))*(d*sec(e+f*x))^(2/3)/(2^(2/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))+I*a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a-I*a*tan(e+f*x))^(1/3))/(a^(1/3)*sqrt(3)))*(d*sec(e+f*x))^(2/3)*sqrt(3)/(2^(2/3)*f*(a-I*a*tan(e+f*x))^(1/3)*(a+I*a*tan(e+f*x))^(1/3))],
[(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(2/3),x,1,3*I*a*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(1/3))],
[(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(5/3),x,2,9/2*I*a^2*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(1/3))+3/4*I*a*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(2/3)/f],
[(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(8/3),x,3,54/7*I*a^3*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(1/3))+9/7*I*a^2*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(2/3)/f+3/7*I*a*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(5/3)/f],
[(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(11/3),x,4,486/35*I*a^4*(d*sec(e+f*x))^(2/3)/(f*(a+I*a*tan(e+f*x))^(1/3))+81/35*I*a^3*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(2/3)/f+27/35*I*a^2*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(5/3)/f+3/10*I*a*(d*sec(e+f*x))^(2/3)*(a+I*a*tan(e+f*x))^(8/3)/f],

# Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^n with m symbolic
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^5,x,4,I*2^(5+1/2*m)*a^5*hypergeom([-4-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^3,x,4,I*2^(3+1/2*m)*a^3*hypergeom([-2-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^2,x,4,I*2^(2+1/2*m)*a^2*hypergeom([-1-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x)),x,4,I*2^(1+1/2*m)*a*hypergeom([-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x)),x,4,I*2^(-1+1/2*m)*hypergeom([2-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(a*d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x))^2,x,4,I*2^(-2+1/2*m)*hypergeom([3-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(a^2*d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x))^3,x,4,I*2^(-3+1/2*m)*hypergeom([4-1/2*m,1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m/(a^3*d*m*(1+I*tan(c+d*x))^(1/2*m))],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^(7/2),x,4,I*2^(1/2*(7+m))*a^3*hypergeom([1/2*(-5-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*sqrt(a+I*a*tan(c+d*x))*(1+I*tan(c+d*x))^(1/2*(-1-m))/(d*m)],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^(5/2),x,4,I*2^(1/2*(5+m))*a^2*hypergeom([1/2*(-3-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*sqrt(a+I*a*tan(c+d*x))*(1+I*tan(c+d*x))^(1/2*(-1-m))/(d*m)],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^(3/2),x,4,I*2^(1/2*(3+m))*a*hypergeom([1/2*(-1-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*sqrt(a+I*a*tan(c+d*x))*(1+I*tan(c+d*x))^(1/2*(-1-m))/(d*m)],
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^(1/2),x,4,I*2^(1/2*(1+m))*a*hypergeom([1/2*(1-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*(1+I*tan(c+d*x))^(1/2*(1-m))/(d*m*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x))^(1/2),x,4,I*2^(1/2*(-1+m))*hypergeom([1/2*(3-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*(1+I*tan(c+d*x))^(1/2*(1-m))/(d*m*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x))^(3/2),x,4,I*2^(1/2*(-3+m))*hypergeom([1/2*(5-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*(1+I*tan(c+d*x))^(1/2*(1-m))/(a*d*m*sqrt(a+I*a*tan(c+d*x)))],
[(e*sec(c+d*x))^m/(a+I*a*tan(c+d*x))^(5/2),x,4,I*2^(1/2*(-5+m))*hypergeom([1/2*(7-m),1/2*m],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*(1+I*tan(c+d*x))^(1/2*(1-m))/(a^2*d*m*sqrt(a+I*a*tan(c+d*x)))],

# Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^n with n symbolic
[(e*sec(c+d*x))^m*(a+I*a*tan(c+d*x))^n,x,4,I*2^(1/2*m+n)*hypergeom([1/2*m,1-1/2*m-n],[1/2*(2+m)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^m*(1+I*tan(c+d*x))^(-1/2*m-n)*(a+I*a*tan(c+d*x))^n/(d*m)],
[sec(c+d*x)^6*(a+I*a*tan(c+d*x))^n,x,3,-4*I*(a+I*a*tan(c+d*x))^(3+n)/(a^3*d*(3+n))+4*I*(a+I*a*tan(c+d*x))^(4+n)/(a^4*d*(4+n))-I*(a+I*a*tan(c+d*x))^(5+n)/(a^5*d*(5+n))],
[sec(c+d*x)^4*(a+I*a*tan(c+d*x))^n,x,3,-2*I*(a+I*a*tan(c+d*x))^(2+n)/(a^2*d*(2+n))+I*(a+I*a*tan(c+d*x))^(3+n)/(a^3*d*(3+n))],
[sec(c+d*x)^2*(a+I*a*tan(c+d*x))^n,x,2,-I*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(1+n))],
[cos(c+d*x)^2*(a+I*a*tan(c+d*x))^n,x,2,1/4*I*a*hypergeom([2,-1+n],[n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^(-1+n)/(d*(1-n))],
[cos(c+d*x)^4*(a+I*a*tan(c+d*x))^n,x,2,1/8*I*a^2*hypergeom([3,-2+n],[-1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^(-2+n)/(d*(2-n))],
[cos(c+d*x)^6*(a+I*a*tan(c+d*x))^n,x,2,1/16*I*a^3*hypergeom([4,-3+n],[-2+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^(-3+n)/(d*(3-n))],
[sec(c+d*x)^5*(a+I*a*tan(c+d*x))^n,x,4,1/5*I*2^(5/2+n)*a^2*hypergeom([5/2,-3/2-n],[7/2],1/2*(1-I*tan(c+d*x)))*sec(c+d*x)^5*(1+I*tan(c+d*x))^(-1/2-n)*(a+I*a*tan(c+d*x))^(-2+n)/d],
[sec(c+d*x)^3*(a+I*a*tan(c+d*x))^n,x,4,1/3*I*2^(3/2+n)*a*hypergeom([3/2,-1/2-n],[5/2],1/2*(1-I*tan(c+d*x)))*sec(c+d*x)^3*(1+I*tan(c+d*x))^(-1/2-n)*(a+I*a*tan(c+d*x))^(-1+n)/d],
[sec(c+d*x)*(a+I*a*tan(c+d*x))^n,x,4,I*2^(1/2+n)*a*hypergeom([1/2,1/2-n],[3/2],1/2*(1-I*tan(c+d*x)))*sec(c+d*x)*(1+I*tan(c+d*x))^(1/2-n)*(a+I*a*tan(c+d*x))^(-1+n)/d],
[cos(c+d*x)*(a+I*a*tan(c+d*x))^n,x,4,-I*2^(-1/2+n)*cos(c+d*x)*hypergeom([-1/2,3/2-n],[1/2],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2-n)*(a+I*a*tan(c+d*x))^n/d],
[cos(c+d*x)^3*(a+I*a*tan(c+d*x))^n,x,4,-1/3*I*2^(-3/2+n)*cos(c+d*x)^3*hypergeom([-3/2,5/2-n],[-1/2],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d)],
[cos(c+d*x)^5*(a+I*a*tan(c+d*x))^n,x,4,-1/5*I*2^(-5/2+n)*cos(c+d*x)^5*hypergeom([-5/2,7/2-n],[-3/2],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2-n)*(a+I*a*tan(c+d*x))^(2+n)/(a^2*d)],
[(e*sec(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^n,x,4,1/5*I*2^(9/4+n)*a*hypergeom([5/4,-1/4-n],[9/4],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(5/2)*(1+I*tan(c+d*x))^(-1/4-n)*(a+I*a*tan(c+d*x))^(-1+n)/d],
[(e*sec(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^n,x,4,1/3*I*2^(7/4+n)*a*hypergeom([3/4,1/4-n],[7/4],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(3/2)*(1+I*tan(c+d*x))^(1/4-n)*(a+I*a*tan(c+d*x))^(-1+n)/d],
[(e*sec(c+d*x))^(1/2)*(a+I*a*tan(c+d*x))^n,x,4,I*2^(5/4+n)*a*hypergeom([1/4,3/4-n],[5/4],1/2*(1-I*tan(c+d*x)))*sqrt(e*sec(c+d*x))*(1+I*tan(c+d*x))^(3/4-n)*(a+I*a*tan(c+d*x))^(-1+n)/d],
[(a+I*a*tan(c+d*x))^n/(e*sec(c+d*x))^(1/2),x,4,-I*2^(3/4+n)*hypergeom([-1/4,5/4-n],[3/4],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/4-n)*(a+I*a*tan(c+d*x))^n/(d*sqrt(e*sec(c+d*x)))],
[(a+I*a*tan(c+d*x))^n/(e*sec(c+d*x))^(3/2),x,4,-1/3*I*2^(1/4+n)*hypergeom([-3/4,7/4-n],[1/4],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(3/4-n)*(a+I*a*tan(c+d*x))^n/(d*(e*sec(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))^n/(e*sec(c+d*x))^(5/2),x,4,-1/5*I*2^(-1/4+n)*hypergeom([-5/4,9/4-n],[-1/4],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/4-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(e*sec(c+d*x))^(5/2))],
[(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^n,x,5,I*(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^n/(d*(4-n))+4*I*(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(8-6*n+n^2))-12*I*(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^(2+n)/(a^2*d*(2-n)*(4-n)*n)+24*I*(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^(3+n)/(a^3*d*(4-n)*n*(4-n^2))-24*I*(e*sec(c+d*x))^(-4-n)*(a+I*a*tan(c+d*x))^(4+n)/(a^4*d*n*(64-20*n^2+n^4))],
[(e*sec(c+d*x))^(-3-n)*(a+I*a*tan(c+d*x))^n,x,4,I*(e*sec(c+d*x))^(-3-n)*(a+I*a*tan(c+d*x))^n/(d*(3-n))+3*I*(e*sec(c+d*x))^(-3-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(3-4*n+n^2))-6*I*(e*sec(c+d*x))^(-3-n)*(a+I*a*tan(c+d*x))^(2+n)/(a^2*d*(3-n)*(1-n^2))+6*I*(e*sec(c+d*x))^(-3-n)*(a+I*a*tan(c+d*x))^(3+n)/(a^3*d*(9-10*n^2+n^4))],
[(e*sec(c+d*x))^(-2-n)*(a+I*a*tan(c+d*x))^n,x,3,I*(e*sec(c+d*x))^(-2-n)*(a+I*a*tan(c+d*x))^n/(d*(2-n))-2*I*(e*sec(c+d*x))^(-2-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(2-n)*n)+2*I*(e*sec(c+d*x))^(-2-n)*(a+I*a*tan(c+d*x))^(2+n)/(a^2*d*n*(4-n^2))],
[(e*sec(c+d*x))^(-1-n)*(a+I*a*tan(c+d*x))^n,x,2,I*(e*sec(c+d*x))^(-1-n)*(a+I*a*tan(c+d*x))^n/(d*(1-n))-I*(e*sec(c+d*x))^(-1-n)*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(1-n^2))],
[(a+I*a*tan(c+d*x))^n/(e*sec(c+d*x))^n,x,1,-I*(a+I*a*tan(c+d*x))^n/(d*n*(e*sec(c+d*x))^n)],
[(e*sec(c+d*x))^(1-n)*(a+I*a*tan(c+d*x))^n,x,4,I*2^(1/2*(1+n))*hypergeom([1/2*(1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(1-n)*(1+I*tan(c+d*x))^(1/2*(-1-n))*(a+I*a*tan(c+d*x))^n/(d*(1-n))],
[(e*sec(c+d*x))^(2-n)*(a+I*a*tan(c+d*x))^n,x,4,I*2^(1+1/2*n)*a*hypergeom([1/2*(2-n),-1/2*n],[1/2*(4-n)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(2-n)*(a+I*a*tan(c+d*x))^(-1+n)/(d*(2-n)*(1+I*tan(c+d*x))^(1/2*n))],
[(e*sec(c+d*x))^(3-n)*(a+I*a*tan(c+d*x))^n,x,4,I*2^(1/2*(3+n))*a*hypergeom([1/2*(-1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-I*tan(c+d*x)))*(e*sec(c+d*x))^(3-n)*(1+I*tan(c+d*x))^(1/2*(-1-n))*(a+I*a*tan(c+d*x))^(-1+n)/(d*(3-n))],
[(e*sec(c+d*x))^(6-2*n)*(a+I*a*tan(c+d*x))^n,x,3,8*I*a^3*(e*sec(c+d*x))^(6-2*n)*(a+I*a*tan(c+d*x))^(-3+n)/(d*(60-47*n+12*n^2-n^3))+4*I*a^2*(e*sec(c+d*x))^(6-2*n)*(a+I*a*tan(c+d*x))^(-2+n)/(d*(20-9*n+n^2))+I*a*(e*sec(c+d*x))^(6-2*n)*(a+I*a*tan(c+d*x))^(-1+n)/(d*(5-n))],
[(e*sec(c+d*x))^(5-2*n)*(a+I*a*tan(c+d*x))^n,x,5,-1/5*I*2^(5/2-n)*hypergeom([5/2,1/2*(-3+2*n)],[7/2],1/2*(1+I*tan(c+d*x)))*(e*sec(c+d*x))^(5-2*n)*(1-I*tan(c+d*x))^(-5/2+n)*(a+I*a*tan(c+d*x))^n/d],
[(e*sec(c+d*x))^(4-2*n)*(a+I*a*tan(c+d*x))^n,x,2,2*I*a^2*(e*sec(c+d*x))^(4-2*n)*(a+I*a*tan(c+d*x))^(-2+n)/(d*(6-5*n+n^2))+I*a*(e*sec(c+d*x))^(4-2*n)*(a+I*a*tan(c+d*x))^(-1+n)/(d*(3-n))],
[(e*sec(c+d*x))^(3-2*n)*(a+I*a*tan(c+d*x))^n,x,5,-1/3*I*2^(3/2-n)*hypergeom([3/2,1/2*(-1+2*n)],[5/2],1/2*(1+I*tan(c+d*x)))*(e*sec(c+d*x))^(3-2*n)*(1-I*tan(c+d*x))^(-3/2+n)*(a+I*a*tan(c+d*x))^n/d],
[(e*sec(c+d*x))^(2-2*n)*(a+I*a*tan(c+d*x))^n,x,1,I*a*(e*sec(c+d*x))^(2-2*n)*(a+I*a*tan(c+d*x))^(-1+n)/(d*(1-n))],
[(e*sec(c+d*x))^(1-2*n)*(a+I*a*tan(c+d*x))^n,x,5,-I*2^(1/2-n)*hypergeom([1/2,1/2*(1+2*n)],[3/2],1/2*(1+I*tan(c+d*x)))*(e*sec(c+d*x))^(1-2*n)*(1-I*tan(c+d*x))^(-1/2+n)*(a+I*a*tan(c+d*x))^n/d],
[(a+I*a*tan(c+d*x))^n/(e*sec(c+d*x))^(2*n),x,3,-1/2*I*hypergeom([1,-n],[1-n],1/2*(1-I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n*(e*sec(c+d*x))^(2*n))],
[(e*sec(c+d*x))^(-1-2*n)*(a+I*a*tan(c+d*x))^n,x,5,I*2^(-1/2-n)*hypergeom([-1/2,1/2*(3+2*n)],[1/2],1/2*(1+I*tan(c+d*x)))*(e*sec(c+d*x))^(-1-2*n)*(1-I*tan(c+d*x))^(1/2+n)*(a+I*a*tan(c+d*x))^n/d],
[(e*sec(c+d*x))^(-2-2*n)*(a+I*a*tan(c+d*x))^n,x,4,-1/4*I*hypergeom([2,-1-n],[-n],1/2*(1-I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(1+n)*(e*sec(c+d*x))^(2*(1+n)))],
[(e*sec(c+d*x))^(-3-2*n)*(a+I*a*tan(c+d*x))^n,x,5,1/3*I*2^(-3/2-n)*hypergeom([-3/2,1/2*(5+2*n)],[-1/2],1/2*(1+I*tan(c+d*x)))*(e*sec(c+d*x))^(-3-2*n)*(1-I*tan(c+d*x))^(3/2+n)*(a+I*a*tan(c+d*x))^n/d],
[(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(-2-n),x,4,1/8*I*hypergeom([3,n],[1+n],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(2*n)/(a^2*f*n*(a+I*a*tan(e+f*x))^n)],
[(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(-1-n),x,4,1/4*I*hypergeom([2,n],[1+n],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(2*n)/(a*f*n*(a+I*a*tan(e+f*x))^n)],
[(d*sec(e+f*x))^(2*n)/(a+I*a*tan(e+f*x))^n,x,3,1/2*I*hypergeom([1,n],[1+n],1/2*(1-I*tan(e+f*x)))*(d*sec(e+f*x))^(2*n)/(f*n*(a+I*a*tan(e+f*x))^n)],
[(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(1-n),x,1,I*a*(d*sec(e+f*x))^(2*n)/(f*n*(a+I*a*tan(e+f*x))^n)],
[(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(2-n),x,2,I*a*(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(1-n)/(f*(1+n))+2*I*a^2*(d*sec(e+f*x))^(2*n)/(f*n*(1+n)*(a+I*a*tan(e+f*x))^n)],
[(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(3-n),x,3,4*I*a^2*(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(1-n)/(f*(2+3*n+n^2))+I*a*(d*sec(e+f*x))^(2*n)*(a+I*a*tan(e+f*x))^(2-n)/(f*(2+n))+8*I*a^3*(d*sec(e+f*x))^(2*n)/(f*n*(2+3*n+n^2)*(a+I*a*tan(e+f*x))^n)],

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

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

# n>0
[sec(c+d*x)^6*(a+b*tan(c+d*x)),x,3,1/6*b*sec(c+d*x)^6/d+a*tan(c+d*x)/d+2/3*a*tan(c+d*x)^3/d+1/5*a*tan(c+d*x)^5/d],
[sec(c+d*x)^5*(a+b*tan(c+d*x)),x,4,3/8*a*arctanh(sin(c+d*x))/d+1/5*b*sec(c+d*x)^5/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d],
[sec(c+d*x)^4*(a+b*tan(c+d*x)),x,3,1/4*b*sec(c+d*x)^4/d+a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(a+b*tan(c+d*x)),x,3,1/2*a*arctanh(sin(c+d*x))/d+1/3*b*sec(c+d*x)^3/d+1/2*a*sec(c+d*x)*tan(c+d*x)/d],
[sec(c+d*x)^2*(a+b*tan(c+d*x)),x,3,1/2*b*sec(c+d*x)^2/d+a*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*tan(c+d*x)),x,2,a*arctanh(sin(c+d*x))/d+b*sec(c+d*x)/d],
[cos(c+d*x)*(a+b*tan(c+d*x)),x,2,-b*cos(c+d*x)/d+a*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*tan(c+d*x)),x,3,1/2*a*x-1/2*b*cos(c+d*x)^2/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*tan(c+d*x)),x,3,-1/3*b*cos(c+d*x)^3/d+a*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^4*(a+b*tan(c+d*x)),x,4,3/8*a*x-1/4*b*cos(c+d*x)^4/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[sec(c+d*x)^8*(a+b*tan(c+d*x))^2,x,4,1/4*a*b*sec(c+d*x)^8/d+a^2*tan(c+d*x)/d+1/3*(3*a^2+b^2)*tan(c+d*x)^3/d+3/5*(a^2+b^2)*tan(c+d*x)^5/d+1/7*(a^2+3*b^2)*tan(c+d*x)^7/d+1/9*b^2*tan(c+d*x)^9/d],
[sec(c+d*x)^6*(a+b*tan(c+d*x))^2,x,4,1/3*a*b*sec(c+d*x)^6/d+a^2*tan(c+d*x)/d+1/3*(2*a^2+b^2)*tan(c+d*x)^3/d+1/5*(a^2+2*b^2)*tan(c+d*x)^5/d+1/7*b^2*tan(c+d*x)^7/d],
[sec(c+d*x)^4*(a+b*tan(c+d*x))^2,x,3,1/3*(a^2+b^2)*(a+b*tan(c+d*x))^3/(b^3*d)-1/2*a*(a+b*tan(c+d*x))^4/(b^3*d)+1/5*(a+b*tan(c+d*x))^5/(b^3*d)],
[sec(c+d*x)^2*(a+b*tan(c+d*x))^2,x,2,1/3*(a+b*tan(c+d*x))^3/(b*d)],
[cos(c+d*x)^2*(a+b*tan(c+d*x))^2,x,3,1/2*(a^2+b^2)*x-1/2*cos(c+d*x)^2*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))/d],
[cos(c+d*x)^4*(a+b*tan(c+d*x))^2,x,4,1/8*(3*a^2+b^2)*x-1/4*cos(c+d*x)^4*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))/d-1/8*cos(c+d*x)^2*(2*a*b-(3*a^2+b^2)*tan(c+d*x))/d],
[sec(c+d*x)^7*(a+b*tan(c+d*x))^2,x,6,5/128*(8*a^2-b^2)*arctanh(sin(c+d*x))/d+9/56*a*b*sec(c+d*x)^7/d+5/128*(8*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/d+5/192*(8*a^2-b^2)*sec(c+d*x)^3*tan(c+d*x)/d+1/48*(8*a^2-b^2)*sec(c+d*x)^5*tan(c+d*x)/d+1/8*b*sec(c+d*x)^7*(a+b*tan(c+d*x))/d],
[sec(c+d*x)^5*(a+b*tan(c+d*x))^2,x,5,1/16*(6*a^2-b^2)*arctanh(sin(c+d*x))/d+7/30*a*b*sec(c+d*x)^5/d+1/16*(6*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/d+1/24*(6*a^2-b^2)*sec(c+d*x)^3*tan(c+d*x)/d+1/6*b*sec(c+d*x)^5*(a+b*tan(c+d*x))/d],
[sec(c+d*x)^3*(a+b*tan(c+d*x))^2,x,4,1/8*(4*a^2-b^2)*arctanh(sin(c+d*x))/d+5/12*a*b*sec(c+d*x)^3/d+1/8*(4*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*(a+b*tan(c+d*x))/d],
[sec(c+d*x)*(a+b*tan(c+d*x))^2,x,3,1/2*(2*a^2-b^2)*arctanh(sin(c+d*x))/d+3/2*a*b*sec(c+d*x)/d+1/2*b*sec(c+d*x)*(a+b*tan(c+d*x))/d],
[cos(c+d*x)*(a+b*tan(c+d*x))^2,x,1,b^2*arctanh(sin(c+d*x))/d-2*a*b*cos(c+d*x)/d+(a^2-b^2)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*tan(c+d*x))^2,x,4,-1/6*a*b*cos(c+d*x)^3/d+1/2*(2*a^2+b^2)*sin(c+d*x)/d-1/6*(2*a^2+b^2)*sin(c+d*x)^3/d-1/2*b*cos(c+d*x)^3*(a+b*tan(c+d*x))/d],
[cos(c+d*x)^5*(a+b*tan(c+d*x))^2,x,4,-3/20*a*b*cos(c+d*x)^5/d+1/4*(4*a^2+b^2)*sin(c+d*x)/d-1/6*(4*a^2+b^2)*sin(c+d*x)^3/d+1/20*(4*a^2+b^2)*sin(c+d*x)^5/d-1/4*b*cos(c+d*x)^5*(a+b*tan(c+d*x))/d],
[cos(c+d*x)^7*(a+b*tan(c+d*x))^2,x,4,-5/42*a*b*cos(c+d*x)^7/d+1/6*(6*a^2+b^2)*sin(c+d*x)/d-1/6*(6*a^2+b^2)*sin(c+d*x)^3/d+1/10*(6*a^2+b^2)*sin(c+d*x)^5/d-1/42*(6*a^2+b^2)*sin(c+d*x)^7/d-1/6*b*cos(c+d*x)^7*(a+b*tan(c+d*x))/d],
[sec(c+d*x)^8*(a+b*tan(c+d*x))^3,x,4,3/8*a^2*b*sec(c+d*x)^8/d+a^3*tan(c+d*x)/d+a*(a^2+b^2)*tan(c+d*x)^3/d+1/4*b^3*tan(c+d*x)^4/d+3/5*a*(a^2+3*b^2)*tan(c+d*x)^5/d+1/2*b^3*tan(c+d*x)^6/d+1/7*a*(a^2+9*b^2)*tan(c+d*x)^7/d+3/8*b^3*tan(c+d*x)^8/d+1/3*a*b^2*tan(c+d*x)^9/d+1/10*b^3*tan(c+d*x)^10/d],
[sec(c+d*x)^6*(a+b*tan(c+d*x))^3,x,3,1/4*(a^2+b^2)^2*(a+b*tan(c+d*x))^4/(b^5*d)-4/5*a*(a^2+b^2)*(a+b*tan(c+d*x))^5/(b^5*d)+1/3*(3*a^2+b^2)*(a+b*tan(c+d*x))^6/(b^5*d)-4/7*a*(a+b*tan(c+d*x))^7/(b^5*d)+1/8*(a+b*tan(c+d*x))^8/(b^5*d)],
[sec(c+d*x)^4*(a+b*tan(c+d*x))^3,x,3,1/4*(a^2+b^2)*(a+b*tan(c+d*x))^4/(b^3*d)-2/5*a*(a+b*tan(c+d*x))^5/(b^3*d)+1/6*(a+b*tan(c+d*x))^6/(b^3*d)],
[sec(c+d*x)^2*(a+b*tan(c+d*x))^3,x,2,1/4*(a+b*tan(c+d*x))^4/(b*d)],
[cos(c+d*x)^2*(a+b*tan(c+d*x))^3,x,6,1/2*a*(a^2+3*b^2)*x-b^3*log(cos(c+d*x))/d-1/2*a*b^2*tan(c+d*x)/d-1/2*cos(c+d*x)^2*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d],
[cos(c+d*x)^4*(a+b*tan(c+d*x))^3,x,4,3/8*a*(a^2+b^2)*x-3/8*a*cos(c+d*x)^2*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))/d+1/4*cos(c+d*x)^3*sin(c+d*x)*(a+b*tan(c+d*x))^3/d],
[sec(c+d*x)^5*(a+b*tan(c+d*x))^3,x,6,3/16*a*(2*a^2-b^2)*arctanh(sin(c+d*x))/d+3/16*a*(2*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/d+1/8*a*(2*a^2-b^2)*sec(c+d*x)^3*tan(c+d*x)/d+1/7*b*sec(c+d*x)^5*(a+b*tan(c+d*x))^2/d+1/70*b*sec(c+d*x)^5*(4*(8*a^2-b^2)+15*a*b*tan(c+d*x))/d,3/16*a*(2*a^2-b^2)*arcsinh(tan(c+d*x))*sec(c+d*x)/(d*sqrt(sec(c+d*x)^2))+3/16*a*(2*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/d+1/8*a*(2*a^2-b^2)*sec(c+d*x)^3*tan(c+d*x)/d+1/7*b*sec(c+d*x)^5*(a+b*tan(c+d*x))^2/d+1/70*b*sec(c+d*x)^5*(4*(8*a^2-b^2)+15*a*b*tan(c+d*x))/d],
[sec(c+d*x)^3*(a+b*tan(c+d*x))^3,x,5,1/8*a*(4*a^2-3*b^2)*arctanh(sin(c+d*x))/d+1/8*a*(4*a^2-3*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/5*b*sec(c+d*x)^3*(a+b*tan(c+d*x))^2/d+1/60*b*sec(c+d*x)^3*(8*(6*a^2-b^2)+21*a*b*tan(c+d*x))/d,1/8*a*(4*a^2-3*b^2)*arcsinh(tan(c+d*x))*sec(c+d*x)/(d*sqrt(sec(c+d*x)^2))+1/8*a*(4*a^2-3*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/5*b*sec(c+d*x)^3*(a+b*tan(c+d*x))^2/d+1/60*b*sec(c+d*x)^3*(8*(6*a^2-b^2)+21*a*b*tan(c+d*x))/d],
[sec(c+d*x)*(a+b*tan(c+d*x))^3,x,4,1/2*a*(2*a^2-3*b^2)*arctanh(sin(c+d*x))/d+1/3*b*sec(c+d*x)*(a+b*tan(c+d*x))^2/d+1/6*b*sec(c+d*x)*(4*(4*a^2-b^2)+5*a*b*tan(c+d*x))/d,1/2*a*(2*a^2-3*b^2)*arcsinh(tan(c+d*x))*sec(c+d*x)/(d*sqrt(sec(c+d*x)^2))+1/3*b*sec(c+d*x)*(a+b*tan(c+d*x))^2/d+1/6*b*sec(c+d*x)*(4*(4*a^2-b^2)+5*a*b*tan(c+d*x))/d],
[cos(c+d*x)*(a+b*tan(c+d*x))^3,x,4,3*a*b^2*arctanh(sin(c+d*x))/d-cos(c+d*x)*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d-b*sec(c+d*x)*(2*(a^2-b^2)+a*b*tan(c+d*x))/d,3*a*b^2*arcsinh(tan(c+d*x))*cos(c+d*x)*sqrt(sec(c+d*x)^2)/d-cos(c+d*x)*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d-b*sec(c+d*x)*(2*(a^2-b^2)+a*b*tan(c+d*x))/d],
[cos(c+d*x)^3*(a+b*tan(c+d*x))^3,x,3,-2/3*(a^2+b^2)*cos(c+d*x)*(b-a*tan(c+d*x))/d-1/3*cos(c+d*x)^3*(b-a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d],
[cos(c+d*x)^5*(a+b*tan(c+d*x))^3,x,4,-2/15*(4*a^2+b^2)*cos(c+d*x)*(b-a*tan(c+d*x))/d-1/15*cos(c+d*x)^3*(b-4*a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d+1/5*cos(c+d*x)^4*sin(c+d*x)*(a+b*tan(c+d*x))^3/d],
[cos(c+d*x)^7*(a+b*tan(c+d*x))^3,x,5,8/35*a*(2*a^2+b^2)*sin(c+d*x)/d-3/35*cos(c+d*x)^5*(b-2*a*tan(c+d*x))*(a+b*tan(c+d*x))^2/d+1/7*cos(c+d*x)^6*sin(c+d*x)*(a+b*tan(c+d*x))^3/d-2/35*cos(c+d*x)^3*(b*(6*a^2+b^2)-a*(4*a^2-b^2)*tan(c+d*x))/d],

# n<0
[sec(c+d*x)^6/(a+b*tan(c+d*x)),x,3,(a^2+b^2)^2*log(a+b*tan(c+d*x))/(b^5*d)-a*(a^2+2*b^2)*tan(c+d*x)/(b^4*d)+1/2*(a^2+2*b^2)*tan(c+d*x)^2/(b^3*d)-1/3*a*tan(c+d*x)^3/(b^2*d)+1/4*tan(c+d*x)^4/(b*d)],
[sec(c+d*x)^4/(a+b*tan(c+d*x)),x,3,(a^2+b^2)*log(a+b*tan(c+d*x))/(b^3*d)-a*tan(c+d*x)/(b^2*d)+1/2*tan(c+d*x)^2/(b*d)],
[sec(c+d*x)^2/(a+b*tan(c+d*x)),x,2,log(a+b*tan(c+d*x))/(b*d)],
[cos(c+d*x)^2/(a+b*tan(c+d*x)),x,7,1/2*a*(a^2+3*b^2)*x/(a^2+b^2)^2+b^3*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+1/2*cos(c+d*x)^2*(b+a*tan(c+d*x))/((a^2+b^2)*d)],
[cos(c+d*x)^4/(a+b*tan(c+d*x)),x,8,1/8*a*(3*a^4+10*a^2*b^2+15*b^4)*x/(a^2+b^2)^3+b^5*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/4*cos(c+d*x)^4*(b+a*tan(c+d*x))/((a^2+b^2)*d)+1/8*cos(c+d*x)^2*(4*b^3+a*(3*a^2+7*b^2)*tan(c+d*x))/((a^2+b^2)^2*d)],
[sec(c+d*x)^5/(a+b*tan(c+d*x)),x,9,-1/2*a*(2*a^2+3*b^2)*arctanh(sin(c+d*x))/(b^4*d)-(a^2+b^2)^(3/2)*arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))/(b^4*d)+(a^2+b^2)*sec(c+d*x)/(b^3*d)+1/3*sec(c+d*x)^3/(b*d)-1/2*a*sec(c+d*x)*tan(c+d*x)/(b^2*d),-1/2*a*arctanh(sin(c+d*x))/(b^2*d)-a*(a^2+b^2)*arctanh(sin(c+d*x))/(b^4*d)-(a^2+b^2)^(3/2)*arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))/(b^4*d)+(a^2+b^2)*sec(c+d*x)/(b^3*d)+1/3*sec(c+d*x)^3/(b*d)-1/2*a*sec(c+d*x)*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^3/(a+b*tan(c+d*x)),x,5,-a*arctanh(sin(c+d*x))/(b^2*d)+sec(c+d*x)/(b*d)-arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))*sqrt(a^2+b^2)/(b^2*d)],
[sec(c+d*x)/(a+b*tan(c+d*x)),x,2,-arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))/(d*sqrt(a^2+b^2))],
[cos(c+d*x)/(a+b*tan(c+d*x)),x,5,-b^2*arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))/((a^2+b^2)^(3/2)*d)+b*cos(c+d*x)/((a^2+b^2)*d)+a*sin(c+d*x)/((a^2+b^2)*d)],
[cos(c+d*x)^3/(a+b*tan(c+d*x)),x,9,-b^4*arctanh(cos(c+d*x)*(b-a*tan(c+d*x))/sqrt(a^2+b^2))/((a^2+b^2)^(5/2)*d)+b^3*cos(c+d*x)/((a^2+b^2)^2*d)+1/3*b*cos(c+d*x)^3/((a^2+b^2)*d)+a*b^2*sin(c+d*x)/((a^2+b^2)^2*d)+a*sin(c+d*x)/((a^2+b^2)*d)-1/3*a*sin(c+d*x)^3/((a^2+b^2)*d)],
[sec(c+d*x)^8/(a+b*tan(c+d*x))^2,x,3,-6*a*(a^2+b^2)^2*log(a+b*tan(c+d*x))/(b^7*d)+(5*a^4+9*a^2*b^2+3*b^4)*tan(c+d*x)/(b^6*d)-a*(2*a^2+3*b^2)*tan(c+d*x)^2/(b^5*d)+(a^2+b^2)*tan(c+d*x)^3/(b^4*d)-1/2*a*tan(c+d*x)^4/(b^3*d)+1/5*tan(c+d*x)^5/(b^2*d)-(a^2+b^2)^3/(b^7*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^6/(a+b*tan(c+d*x))^2,x,3,-4*a*(a^2+b^2)*log(a+b*tan(c+d*x))/(b^5*d)+(3*a^2+2*b^2)*tan(c+d*x)/(b^4*d)-a*tan(c+d*x)^2/(b^3*d)+1/3*tan(c+d*x)^3/(b^2*d)-(a^2+b^2)^2/(b^5*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^4/(a+b*tan(c+d*x))^2,x,3,-2*a*log(a+b*tan(c+d*x))/(b^3*d)+tan(c+d*x)/(b^2*d)+(-a^2-b^2)/(b^3*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^2/(a+b*tan(c+d*x))^2,x,2,(-1)/(b*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)^2/(a+b*tan(c+d*x))^2,x,7,1/2*(a^4+6*a^2*b^2-3*b^4)*x/(a^2+b^2)^3+4*a*b^3*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*b*(a^2-3*b^2)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))+1/2*cos(c+d*x)^2*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)^4/(a+b*tan(c+d*x))^2,x,8,3/8*(a^6+5*a^4*b^2+15*a^2*b^4-5*b^6)*x/(a^2+b^2)^4+6*a*b^5*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+3/8*b*(a^2-b^2)*(a^2+5*b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))+1/4*cos(c+d*x)^4*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))-1/8*cos(c+d*x)^2*(b*(a^2-5*b^2)-3*a*(a^2+3*b^2)*tan(c+d*x))/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^7/(a+b*tan(c+d*x))^2,x,8,5/8*(8*a^4+12*a^2*b^2+3*b^4)*arcsinh(tan(c+d*x))*sec(c+d*x)/(b^6*d*sqrt(sec(c+d*x)^2))+5*a*(a^2+b^2)^(3/2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/(b^6*d*sqrt(sec(c+d*x)^2))-5/12*sec(c+d*x)^3*(4*a-3*b*tan(c+d*x))/(b^3*d)-sec(c+d*x)^5/(b*d*(a+b*tan(c+d*x)))-5/8*sec(c+d*x)*(8*a*(a^2+b^2)-b*(4*a^2+3*b^2)*tan(c+d*x))/(b^5*d)],
[sec(c+d*x)^5/(a+b*tan(c+d*x))^2,x,7,3/2*(2*a^2+b^2)*arcsinh(tan(c+d*x))*sec(c+d*x)/(b^4*d*sqrt(sec(c+d*x)^2))+3*a*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)*sqrt(a^2+b^2)/(b^4*d*sqrt(sec(c+d*x)^2))-3/2*sec(c+d*x)*(2*a-b*tan(c+d*x))/(b^3*d)-sec(c+d*x)^3/(b*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^3/(a+b*tan(c+d*x))^2,x,6,arctanh(sin(c+d*x))/(b^2*d)+a*arctanh((b*cos(c+d*x)-a*sin(c+d*x))/sqrt(a^2+b^2))/(b^2*d*sqrt(a^2+b^2))-sec(c+d*x)/(b*d*(a+b*tan(c+d*x))),arcsinh(tan(c+d*x))*sec(c+d*x)/(b^2*d*sqrt(sec(c+d*x)^2))+a*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/(b^2*d*sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2))-sec(c+d*x)/(b*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)/(a+b*tan(c+d*x))^2,x,4,-a*arctanh((b*cos(c+d*x)-a*sin(c+d*x))/sqrt(a^2+b^2))/((a^2+b^2)^(3/2)*d)-b*sec(c+d*x)/((a^2+b^2)*d*(a+b*tan(c+d*x))),-a*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/((a^2+b^2)^(3/2)*d*sqrt(sec(c+d*x)^2))-b*sec(c+d*x)/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)/(a+b*tan(c+d*x))^2,x,5,-3*a*b^2*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*cos(c+d*x)*sqrt(sec(c+d*x)^2)/((a^2+b^2)^(5/2)*d)+b*(a^2-2*b^2)*sec(c+d*x)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))+cos(c+d*x)*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)^3/(a+b*tan(c+d*x))^2,x,6,-5*a*b^4*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*cos(c+d*x)*sqrt(sec(c+d*x)^2)/((a^2+b^2)^(7/2)*d)+1/3*b*(2*a^4+9*a^2*b^2-8*b^4)*sec(c+d*x)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))+1/3*cos(c+d*x)^3*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))-1/3*cos(c+d*x)*(b*(a^2-4*b^2)-a*(2*a^2+7*b^2)*tan(c+d*x))/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^8/(a+b*tan(c+d*x))^3,x,3,3*(a^2+b^2)*(5*a^2+b^2)*log(a+b*tan(c+d*x))/(b^7*d)-a*(10*a^2+9*b^2)*tan(c+d*x)/(b^6*d)+3/2*(2*a^2+b^2)*tan(c+d*x)^2/(b^5*d)-a*tan(c+d*x)^3/(b^4*d)+1/4*tan(c+d*x)^4/(b^3*d)-1/2*(a^2+b^2)^3/(b^7*d*(a+b*tan(c+d*x))^2)+6*a*(a^2+b^2)^2/(b^7*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^6/(a+b*tan(c+d*x))^3,x,3,2*(3*a^2+b^2)*log(a+b*tan(c+d*x))/(b^5*d)-3*a*tan(c+d*x)/(b^4*d)+1/2*tan(c+d*x)^2/(b^3*d)-1/2*(a^2+b^2)^2/(b^5*d*(a+b*tan(c+d*x))^2)+4*a*(a^2+b^2)/(b^5*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^4/(a+b*tan(c+d*x))^3,x,3,log(a+b*tan(c+d*x))/(b^3*d)+1/2*(-a^2-b^2)/(b^3*d*(a+b*tan(c+d*x))^2)+2*a/(b^3*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^2/(a+b*tan(c+d*x))^3,x,2,(-1/2)/(b*d*(a+b*tan(c+d*x))^2)],
[cos(c+d*x)^2/(a+b*tan(c+d*x))^3,x,7,1/2*a*(a^4+10*a^2*b^2-15*b^4)*x/(a^2+b^2)^4+2*b^3*(5*a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+1/2*b*(a^2-2*b^2)/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+1/2*cos(c+d*x)^2*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/2*a*b*(a^2-11*b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)^4/(a+b*tan(c+d*x))^3,x,8,3/8*a*(a^6+7*a^4*b^2+35*a^2*b^4-35*b^6)*x/(a^2+b^2)^5+3*b^5*(7*a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^5*d)+3/8*b*(a^4+5*a^2*b^2-4*b^4)/((a^2+b^2)^3*d*(a+b*tan(c+d*x))^2)+1/4*cos(c+d*x)^4*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+3/8*a*b*(a^4+6*a^2*b^2-27*b^4)/((a^2+b^2)^4*d*(a+b*tan(c+d*x)))-1/8*cos(c+d*x)^2*(2*b*(a^2-3*b^2)-a*(3*a^2+11*b^2)*tan(c+d*x))/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)],
[sec(c+d*x)^7/(a+b*tan(c+d*x))^3,x,8,-5/2*a*(4*a^2+3*b^2)*arcsinh(tan(c+d*x))*sec(c+d*x)/(b^6*d*sqrt(sec(c+d*x)^2))-5/2*(4*a^2+b^2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)*sqrt(a^2+b^2)/(b^6*d*sqrt(sec(c+d*x)^2))-1/2*sec(c+d*x)^5/(b*d*(a+b*tan(c+d*x))^2)+5/6*sec(c+d*x)^3*(4*a+b*tan(c+d*x))/(b^3*d*(a+b*tan(c+d*x)))+5/2*sec(c+d*x)*(4*a^2+b^2-2*a*b*tan(c+d*x))/(b^5*d)],
[sec(c+d*x)^5/(a+b*tan(c+d*x))^3,x,7,-3*a*arctanh(sin(c+d*x))/(b^4*d)-3/2*(2*a^2+b^2)*arctanh((b*cos(c+d*x)-a*sin(c+d*x))/sqrt(a^2+b^2))/(b^4*d*sqrt(a^2+b^2))-1/2*sec(c+d*x)^3/(b*d*(a+b*tan(c+d*x))^2)+3/2*sec(c+d*x)*(2*a+b*tan(c+d*x))/(b^3*d*(a+b*tan(c+d*x))),-3*a*arcsinh(tan(c+d*x))*sec(c+d*x)/(b^4*d*sqrt(sec(c+d*x)^2))-3/2*(2*a^2+b^2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/(b^4*d*sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2))-1/2*sec(c+d*x)^3/(b*d*(a+b*tan(c+d*x))^2)+3/2*sec(c+d*x)*(2*a+b*tan(c+d*x))/(b^3*d*(a+b*tan(c+d*x)))],
[sec(c+d*x)^3/(a+b*tan(c+d*x))^3,x,4,-1/2*arctanh((b*cos(c+d*x)-a*sin(c+d*x))/sqrt(a^2+b^2))/((a^2+b^2)^(3/2)*d)-1/2*sec(c+d*x)*(b-a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2),-1/2*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/((a^2+b^2)^(3/2)*d*sqrt(sec(c+d*x)^2))-1/2*sec(c+d*x)*(b-a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)],
[sec(c+d*x)/(a+b*tan(c+d*x))^3,x,5,-1/2*(2*a^2-b^2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*sec(c+d*x)/((a^2+b^2)^(5/2)*d*sqrt(sec(c+d*x)^2))-1/2*b*sec(c+d*x)/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)-3/2*a*b*sec(c+d*x)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)/(a+b*tan(c+d*x))^3,x,6,-3/2*b^2*(4*a^2-b^2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*cos(c+d*x)*sqrt(sec(c+d*x)^2)/((a^2+b^2)^(7/2)*d)+1/2*b*(2*a^2-3*b^2)*sec(c+d*x)/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+cos(c+d*x)*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/2*a*b*(2*a^2-13*b^2)*sec(c+d*x)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cos(c+d*x)^3/(a+b*tan(c+d*x))^3,x,7,-5/2*b^4*(6*a^2-b^2)*arctanh((b-a*tan(c+d*x))/(sqrt(a^2+b^2)*sqrt(sec(c+d*x)^2)))*cos(c+d*x)*sqrt(sec(c+d*x)^2)/((a^2+b^2)^(9/2)*d)+1/6*b*(4*a^4+24*a^2*b^2-15*b^4)*sec(c+d*x)/((a^2+b^2)^3*d*(a+b*tan(c+d*x))^2)+1/3*cos(c+d*x)^3*(b+a*tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/6*a*b*(4*a^4+28*a^2*b^2-81*b^4)*sec(c+d*x)/((a^2+b^2)^4*d*(a+b*tan(c+d*x)))-1/3*cos(c+d*x)*(b*(2*a^2-5*b^2)-a*(2*a^2+9*b^2)*tan(c+d*x))/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)],

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

# n>0
[(d*sec(e+f*x))^(7/2)*(a+b*tan(e+f*x)),x,5,2/7*b*(d*sec(e+f*x))^(7/2)/f+2/5*a*d*(d*sec(e+f*x))^(5/2)*sin(e+f*x)/f-6/5*a*d^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))+6/5*a*d^3*sin(e+f*x)*sqrt(d*sec(e+f*x))/f],
[(d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x)),x,4,2/5*b*(d*sec(e+f*x))^(5/2)/f+2/3*a*d*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/f+2/3*a*d^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/f],
[(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x)),x,4,2/3*b*(d*sec(e+f*x))^(3/2)/f-2*a*d^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))+2*a*d*sin(e+f*x)*sqrt(d*sec(e+f*x))/f],
[(d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x)),x,3,2*b*sqrt(d*sec(e+f*x))/f+2*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/f],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(1/2),x,3,-2*b/(f*sqrt(d*sec(e+f*x)))+2*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(3/2),x,4,-2/3*b/(f*(d*sec(e+f*x))^(3/2))+2/3*a*sin(e+f*x)/(d*f*sqrt(d*sec(e+f*x)))+2/3*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/(d^2*f)],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(5/2),x,4,-2/5*b/(f*(d*sec(e+f*x))^(5/2))+2/5*a*sin(e+f*x)/(d*f*(d*sec(e+f*x))^(3/2))+6/5*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(d^2*f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(7/2),x,5,-2/7*b/(f*(d*sec(e+f*x))^(7/2))+2/7*a*sin(e+f*x)/(d*f*(d*sec(e+f*x))^(5/2))+10/21*a*sin(e+f*x)/(d^3*f*sqrt(d*sec(e+f*x)))+10/21*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/(d^4*f)],
[(d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))^2,x,5,18/35*a*b*(d*sec(e+f*x))^(5/2)/f+2/21*(7*a^2-2*b^2)*d*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/f+2/21*(7*a^2-2*b^2)*d^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/f+2/7*b*(d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^2,x,5,14/15*a*b*(d*sec(e+f*x))^(3/2)/f-2/5*(5*a^2-2*b^2)*d^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))+2/5*(5*a^2-2*b^2)*d*sin(e+f*x)*sqrt(d*sec(e+f*x))/f+2/5*b*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x))^2,x,4,10/3*a*b*sqrt(d*sec(e+f*x))/f+2/3*(3*a^2-2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/f+2/3*b*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))/f],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(1/2),x,4,-6*a*b/(f*sqrt(d*sec(e+f*x)))+2*(a^2-2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))+2*b*(a+b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(3/2),x,5,2/3*a*b/(f*(d*sec(e+f*x))^(3/2))+2/3*(a^2+2*b^2)*sin(e+f*x)/(d*f*sqrt(d*sec(e+f*x)))+2/3*(a^2+2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/(d^2*f)-2*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(5/2),x,5,-2/15*a*b/(f*(d*sec(e+f*x))^(5/2))+2/15*(3*a^2+2*b^2)*sin(e+f*x)/(d*f*(d*sec(e+f*x))^(3/2))+2/5*(3*a^2+2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(d^2*f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))-2/3*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(5/2))],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(7/2),x,6,-6/35*a*b/(f*(d*sec(e+f*x))^(7/2))+2/35*(5*a^2+2*b^2)*sin(e+f*x)/(d*f*(d*sec(e+f*x))^(5/2))+2/21*(5*a^2+2*b^2)*sin(e+f*x)/(d^3*f*sqrt(d*sec(e+f*x)))+2/21*(5*a^2+2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x))/(d^4*f)-2/5*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(7/2))],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(9/2),x,6,-10/63*a*b/(f*(d*sec(e+f*x))^(9/2))+2/63*(7*a^2+2*b^2)*sin(e+f*x)/(d*f*(d*sec(e+f*x))^(7/2))+2/45*(7*a^2+2*b^2)*sin(e+f*x)/(d^3*f*(d*sec(e+f*x))^(3/2))+2/15*(7*a^2+2*b^2)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(d^4*f*sqrt(cos(e+f*x))*sqrt(d*sec(e+f*x)))-2/7*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(9/2))],
[(d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))^3,x,5,2/21*a*(7*a^2-6*b^2)*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/(f*(sec(e+f*x)^2)^(1/4))+2/21*a*(7*a^2-6*b^2)*d^2*sqrt(d*sec(e+f*x))*tan(e+f*x)/f+2/9*b*d^2*sec(e+f*x)^2*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2/f+2/315*b*d^2*sec(e+f*x)^2*sqrt(d*sec(e+f*x))*(14*(11*a^2-2*b^2)+65*a*b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^3,x,5,-2/5*a*(5*a^2-6*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/(f*(sec(e+f*x)^2)^(3/4))+2/5*a*(5*a^2-6*b^2)*cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/f+2/7*b*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^2/f+2/105*b*(d*sec(e+f*x))^(3/2)*(10*(9*a^2-2*b^2)+33*a*b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x))^3,x,4,2*a*(a^2-2*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/(f*(sec(e+f*x)^2)^(1/4))+2/5*b*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2/f+2/5*b*sqrt(d*sec(e+f*x))*(2*(7*a^2-2*b^2)+3*a*b*tan(e+f*x))/f],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(1/2),x,5,2*a*(a^2-6*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/(f*sqrt(d*sec(e+f*x)))-2*a*(a^2-6*b^2)*tan(e+f*x)/(f*sqrt(d*sec(e+f*x)))-2*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(f*sqrt(d*sec(e+f*x)))-2/3*b*sec(e+f*x)^2*(2*(3*a^2-2*b^2)+3*a*b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(3/2),x,4,2/3*a*(a^2+6*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/(f*(d*sec(e+f*x))^(3/2))-2/3*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(f*(d*sec(e+f*x))^(3/2))-2/3*b*sec(e+f*x)^2*(2*(a^2-2*b^2)+a*b*tan(e+f*x))/(f*(d*sec(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(5/2),x,5,6/5*a*(a^2+2*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/(d^2*f*sqrt(d*sec(e+f*x)))-6/5*a*(a^2+2*b^2)*tan(e+f*x)/(d^2*f*sqrt(d*sec(e+f*x)))-2/5*cos(e+f*x)^2*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^2*f*sqrt(d*sec(e+f*x)))-2/5*(2*b*(a^2+2*b^2)-a*(3*a^2+5*b^2)*tan(e+f*x))/(d^2*f*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(7/2),x,4,2/21*a*(5*a^2+6*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/(d^2*f*(d*sec(e+f*x))^(3/2))-2/7*cos(e+f*x)^2*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^2*f*(d*sec(e+f*x))^(3/2))-2/21*(2*b*(3*a^2+2*b^2)-a*(5*a^2+3*b^2)*tan(e+f*x))/(d^2*f*(d*sec(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(9/2),x,4,2/15*a*(7*a^2+6*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/(d^4*f*sqrt(d*sec(e+f*x)))-2/9*cos(e+f*x)^4*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^4*f*sqrt(d*sec(e+f*x)))-2/45*cos(e+f*x)^2*(2*b*(5*a^2+2*b^2)-a*(7*a^2+b^2)*tan(e+f*x))/(d^4*f*sqrt(d*sec(e+f*x)))],
[(a+b*tan(e+f*x))^3/(d*sec(e+f*x))^(11/2),x,5,10/77*a*(3*a^2+2*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/(d^4*f*(d*sec(e+f*x))^(3/2))+10/77*a*(3*a^2+2*b^2)*tan(e+f*x)/(d^4*f*(d*sec(e+f*x))^(3/2))-2/11*cos(e+f*x)^4*(b-a*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^4*f*(d*sec(e+f*x))^(3/2))-2/77*cos(e+f*x)^2*(2*b*(7*a^2+2*b^2)-a*(9*a^2-b^2)*tan(e+f*x))/(d^4*f*(d*sec(e+f*x))^(3/2))],

# n<0
[(d*sec(e+f*x))^(7/2)/(a+b*tan(e+f*x)),x,17,2/3*d^2*(d*sec(e+f*x))^(3/2)/(b*f)+(a^2+b^2)^(3/4)*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*f*(sec(e+f*x)^2)^(3/4))-(a^2+b^2)^(3/4)*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*f*(sec(e+f*x)^2)^(3/4))+2*a*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/(b^2*f*(sec(e+f*x)^2)^(3/4))-2*a*d^2*cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/(b^2*f)-a*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(a^2+b^2)*sqrt(-tan(e+f*x)^2)/(b^3*f*(sec(e+f*x)^2)^(3/4))+a*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(a^2+b^2)*sqrt(-tan(e+f*x)^2)/(b^3*f*(sec(e+f*x)^2)^(3/4))],
[(d*sec(e+f*x))^(5/2)/(a+b*tan(e+f*x)),x,17,2*d^2*sqrt(d*sec(e+f*x))/(b*f)-(a^2+b^2)^(1/4)*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*f*(sec(e+f*x)^2)^(1/4))-(a^2+b^2)^(1/4)*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*f*(sec(e+f*x)^2)^(1/4))-2*a*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/(b^2*f*(sec(e+f*x)^2)^(1/4))+a*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*f*(sec(e+f*x)^2)^(1/4))+a*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*f*(sec(e+f*x)^2)^(1/4))],
[(d*sec(e+f*x))^(3/2)/(a+b*tan(e+f*x)),x,13,arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(1/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(1/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-a*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*f*(sec(e+f*x)^2)^(3/4)*sqrt(a^2+b^2))+a*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*f*(sec(e+f*x)^2)^(3/4)*sqrt(a^2+b^2))],
[(d*sec(e+f*x))^(1/2)/(a+b*tan(e+f*x)),x,14,-arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(3/4)*f*(sec(e+f*x)^2)^(1/4))-arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(3/4)*f*(sec(e+f*x)^2)^(1/4))+a*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))+a*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))],
[1/((d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x))),x,17,b^(3/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(5/4)*f*sqrt(d*sec(e+f*x)))-b^(3/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(5/4)*f*sqrt(d*sec(e+f*x)))+2*a*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)*f*sqrt(d*sec(e+f*x)))-a*b*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(3/2)*f*sqrt(d*sec(e+f*x)))+a*b*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(3/2)*f*sqrt(d*sec(e+f*x)))-2*a*tan(e+f*x)/((a^2+b^2)*f*sqrt(d*sec(e+f*x)))+2*(b+a*tan(e+f*x))/((a^2+b^2)*f*sqrt(d*sec(e+f*x)))],
[1/((d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))),x,17,-b^(5/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(7/4)*f*(d*sec(e+f*x))^(3/2))-b^(5/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(7/4)*f*(d*sec(e+f*x))^(3/2))+2/3*a*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)*f*(d*sec(e+f*x))^(3/2))+a*b^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^2*f*(d*sec(e+f*x))^(3/2))+a*b^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^2*f*(d*sec(e+f*x))^(3/2))+2/3*(b+a*tan(e+f*x))/((a^2+b^2)*f*(d*sec(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))),x,18,b^(7/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(9/4)*d^2*f*sqrt(d*sec(e+f*x)))-b^(7/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(9/4)*d^2*f*sqrt(d*sec(e+f*x)))+2/5*a*(3*a^2+8*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^2*d^2*f*sqrt(d*sec(e+f*x)))-a*b^3*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(5/2)*d^2*f*sqrt(d*sec(e+f*x)))+a*b^3*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(5/2)*d^2*f*sqrt(d*sec(e+f*x)))-2/5*a*(3*a^2+8*b^2)*tan(e+f*x)/((a^2+b^2)^2*d^2*f*sqrt(d*sec(e+f*x)))+2/5*cos(e+f*x)^2*(b+a*tan(e+f*x))/((a^2+b^2)*d^2*f*sqrt(d*sec(e+f*x)))+2/5*(5*b^3+a*(3*a^2+8*b^2)*tan(e+f*x))/((a^2+b^2)^2*d^2*f*sqrt(d*sec(e+f*x)))],
[(d*sec(e+f*x))^(7/2)/(a+b*tan(e+f*x))^2,x,17,-3/2*a*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*(a^2+b^2)^(1/4)*f*(sec(e+f*x)^2)^(3/4))+3/2*a*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*(a^2+b^2)^(1/4)*f*(sec(e+f*x)^2)^(3/4))-3*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/(b^2*f*(sec(e+f*x)^2)^(3/4))+3*d^2*cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/(b^2*f)+3/2*a^2*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b^3*f*(sec(e+f*x)^2)^(3/4)*sqrt(a^2+b^2))-3/2*a^2*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b^3*f*(sec(e+f*x)^2)^(3/4)*sqrt(a^2+b^2))-d^2*(d*sec(e+f*x))^(3/2)/(b*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(5/2)/(a+b*tan(e+f*x))^2,x,17,1/2*a*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*(a^2+b^2)^(3/4)*f*(sec(e+f*x)^2)^(1/4))+1/2*a*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*(a^2+b^2)^(3/4)*f*(sec(e+f*x)^2)^(1/4))+d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/(b^2*f*(sec(e+f*x)^2)^(1/4))-1/2*a^2*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*(a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))-1/2*a^2*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*(a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))-d^2*sqrt(d*sec(e+f*x))/(b*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(3/2)/(a+b*tan(e+f*x))^2,x,17,-sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)*f*(sec(e+f*x)^2)^(3/4))+cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/((a^2+b^2)*f)+1/2*a*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(5/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-1/2*a*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(5/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-1/2*a^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*(a^2+b^2)^(3/2)*f*(sec(e+f*x)^2)^(3/4))+1/2*a^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*(a^2+b^2)^(3/2)*f*(sec(e+f*x)^2)^(3/4))-b*(d*sec(e+f*x))^(3/2)/((a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(1/2)/(a+b*tan(e+f*x))^2,x,17,-sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/((a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))-3/2*a*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(7/4)*f*(sec(e+f*x)^2)^(1/4))-3/2*a*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(7/4)*f*(sec(e+f*x)^2)^(1/4))+3/2*a^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^2*f*(sec(e+f*x)^2)^(1/4))+3/2*a^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^2*f*(sec(e+f*x)^2)^(1/4))-b*sqrt(d*sec(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x))^2),x,18,5/2*a*b^(3/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(9/4)*f*sqrt(d*sec(e+f*x)))-5/2*a*b^(3/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(9/4)*f*sqrt(d*sec(e+f*x)))+(2*a^2-3*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^2*f*sqrt(d*sec(e+f*x)))-5/2*a^2*b*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(5/2)*f*sqrt(d*sec(e+f*x)))+5/2*a^2*b*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(5/2)*f*sqrt(d*sec(e+f*x)))-(2*a^2-3*b^2)*tan(e+f*x)/((a^2+b^2)^2*f*sqrt(d*sec(e+f*x)))+b*(2*a^2-3*b^2)*sec(e+f*x)^2/((a^2+b^2)^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))+2*(b+a*tan(e+f*x))/((a^2+b^2)*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^2),x,18,-7/2*a*b^(5/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(11/4)*f*(d*sec(e+f*x))^(3/2))-7/2*a*b^(5/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(11/4)*f*(d*sec(e+f*x))^(3/2))+1/3*(2*a^2-5*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^2*f*(d*sec(e+f*x))^(3/2))+7/2*a^2*b^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^3*f*(d*sec(e+f*x))^(3/2))+7/2*a^2*b^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^3*f*(d*sec(e+f*x))^(3/2))+1/3*b*(2*a^2-5*b^2)*sec(e+f*x)^2/((a^2+b^2)^2*f*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x)))+2/3*(b+a*tan(e+f*x))/((a^2+b^2)*f*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))^2),x,19,9/2*a*b^(7/2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(13/4)*d^2*f*sqrt(d*sec(e+f*x)))-9/2*a*b^(7/2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(13/4)*d^2*f*sqrt(d*sec(e+f*x)))+3/5*(2*a^4+10*a^2*b^2-7*b^4)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^3*d^2*f*sqrt(d*sec(e+f*x)))-9/2*a^2*b^3*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(7/2)*d^2*f*sqrt(d*sec(e+f*x)))+9/2*a^2*b^3*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(7/2)*d^2*f*sqrt(d*sec(e+f*x)))-3/5*(2*a^4+10*a^2*b^2-7*b^4)*tan(e+f*x)/((a^2+b^2)^3*d^2*f*sqrt(d*sec(e+f*x)))+3/5*b*(2*a^4+10*a^2*b^2-7*b^4)*sec(e+f*x)^2/((a^2+b^2)^3*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))+2/5*cos(e+f*x)^2*(b+a*tan(e+f*x))/((a^2+b^2)*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))-2/5*(b*(2*a^2-7*b^2)-3*a*(a^2+4*b^2)*tan(e+f*x))/((a^2+b^2)^2*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(7/2)/(a+b*tan(e+f*x))^3,x,18,3/8*(a^2+2*b^2)*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*(a^2+b^2)^(5/4)*f*(sec(e+f*x)^2)^(3/4))-3/8*(a^2+2*b^2)*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/(b^(5/2)*(a^2+b^2)^(5/4)*f*(sec(e+f*x)^2)^(3/4))+3/4*a*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/(b^2*(a^2+b^2)*f*(sec(e+f*x)^2)^(3/4))-3/4*a*d^2*cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/(b^2*(a^2+b^2)*f)-3/8*a*(a^2+2*b^2)*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b^3*(a^2+b^2)^(3/2)*f*(sec(e+f*x)^2)^(3/4))+3/8*a*(a^2+2*b^2)*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b^3*(a^2+b^2)^(3/2)*f*(sec(e+f*x)^2)^(3/4))-1/2*d^2*(d*sec(e+f*x))^(3/2)/(b*f*(a+b*tan(e+f*x))^2)+3/4*a*d^2*(d*sec(e+f*x))^(3/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(5/2)/(a+b*tan(e+f*x))^3,x,18,1/8*(a^2-2*b^2)*d^2*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*(a^2+b^2)^(7/4)*f*(sec(e+f*x)^2)^(1/4))+1/8*(a^2-2*b^2)*d^2*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(d*sec(e+f*x))/(b^(3/2)*(a^2+b^2)^(7/4)*f*(sec(e+f*x)^2)^(1/4))+1/4*a*d^2*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/(b^2*(a^2+b^2)*f*(sec(e+f*x)^2)^(1/4))-1/8*a*(a^2-2*b^2)*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*(a^2+b^2)^2*f*(sec(e+f*x)^2)^(1/4))-1/8*a*(a^2-2*b^2)*d^2*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/(b^2*(a^2+b^2)^2*f*(sec(e+f*x)^2)^(1/4))-1/2*d^2*sqrt(d*sec(e+f*x))/(b*f*(a+b*tan(e+f*x))^2)+1/4*a*d^2*sqrt(d*sec(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(3/2)/(a+b*tan(e+f*x))^3,x,18,-5/4*a*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^2*f*(sec(e+f*x)^2)^(3/4))+5/4*a*cos(e+f*x)*(d*sec(e+f*x))^(3/2)*sin(e+f*x)/((a^2+b^2)^2*f)+1/8*(3*a^2-2*b^2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(9/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-1/8*(3*a^2-2*b^2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^(9/4)*f*(sec(e+f*x)^2)^(3/4)*sqrt(b))-1/8*a*(3*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*(a^2+b^2)^(5/2)*f*(sec(e+f*x)^2)^(3/4))+1/8*a*(3*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(d*sec(e+f*x))^(3/2)*sqrt(-tan(e+f*x)^2)/(b*(a^2+b^2)^(5/2)*f*(sec(e+f*x)^2)^(3/4))-1/2*b*(d*sec(e+f*x))^(3/2)/((a^2+b^2)*f*(a+b*tan(e+f*x))^2)-5/4*a*b*(d*sec(e+f*x))^(3/2)/((a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[(d*sec(e+f*x))^(1/2)/(a+b*tan(e+f*x))^3,x,18,-7/4*a*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*sqrt(d*sec(e+f*x))/((a^2+b^2)^2*f*(sec(e+f*x)^2)^(1/4))-3/8*(5*a^2-2*b^2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(11/4)*f*(sec(e+f*x)^2)^(1/4))-3/8*(5*a^2-2*b^2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*sqrt(b)*sqrt(d*sec(e+f*x))/((a^2+b^2)^(11/4)*f*(sec(e+f*x)^2)^(1/4))+3/8*a*(5*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^3*f*(sec(e+f*x)^2)^(1/4))+3/8*a*(5*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*sqrt(d*sec(e+f*x))*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^3*f*(sec(e+f*x)^2)^(1/4))-1/2*b*sqrt(d*sec(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^2)-7/4*a*b*sqrt(d*sec(e+f*x))/((a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/2)*(a+b*tan(e+f*x))^3),x,19,5/8*b^(3/2)*(7*a^2-2*b^2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(13/4)*f*sqrt(d*sec(e+f*x)))-5/8*b^(3/2)*(7*a^2-2*b^2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(13/4)*f*sqrt(d*sec(e+f*x)))+1/4*a*(8*a^2-37*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^3*f*sqrt(d*sec(e+f*x)))-5/8*a*b*(7*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(7/2)*f*sqrt(d*sec(e+f*x)))+5/8*a*b*(7*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(7/2)*f*sqrt(d*sec(e+f*x)))-1/4*a*(8*a^2-37*b^2)*tan(e+f*x)/((a^2+b^2)^3*f*sqrt(d*sec(e+f*x)))+1/2*b*(4*a^2-5*b^2)*sec(e+f*x)^2/((a^2+b^2)^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2)+2*(b+a*tan(e+f*x))/((a^2+b^2)*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2)+1/4*a*b*(8*a^2-37*b^2)*sec(e+f*x)^2/((a^2+b^2)^3*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^3),x,19,-7/8*b^(5/2)*(9*a^2-2*b^2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(15/4)*f*(d*sec(e+f*x))^(3/2))-7/8*b^(5/2)*(9*a^2-2*b^2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^(15/4)*f*(d*sec(e+f*x))^(3/2))+1/12*a*(8*a^2-69*b^2)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticF(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(3/4)/((a^2+b^2)^3*f*(d*sec(e+f*x))^(3/2))+7/8*a*b^2*(9*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^4*f*(d*sec(e+f*x))^(3/2))+7/8*a*b^2*(9*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(3/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^4*f*(d*sec(e+f*x))^(3/2))+1/6*b*(4*a^2-7*b^2)*sec(e+f*x)^2/((a^2+b^2)^2*f*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^2)+2/3*(b+a*tan(e+f*x))/((a^2+b^2)*f*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x))^2)+1/12*a*b*(8*a^2-69*b^2)*sec(e+f*x)^2/((a^2+b^2)^3*f*(d*sec(e+f*x))^(3/2)*(a+b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(5/2)*(a+b*tan(e+f*x))^3),x,20,9/8*b^(7/2)*(11*a^2-2*b^2)*arctan((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(17/4)*d^2*f*sqrt(d*sec(e+f*x)))-9/8*b^(7/2)*(11*a^2-2*b^2)*arctanh((sec(e+f*x)^2)^(1/4)*sqrt(b)/(a^2+b^2)^(1/4))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^(17/4)*d^2*f*sqrt(d*sec(e+f*x)))+3/20*a*(8*a^4+64*a^2*b^2-139*b^4)*sqrt(cos(1/2*arctan(tan(e+f*x)))^2)/cos(1/2*arctan(tan(e+f*x)))*EllipticE(sin(1/2*arctan(tan(e+f*x))),sqrt(2))*(sec(e+f*x)^2)^(1/4)/((a^2+b^2)^4*d^2*f*sqrt(d*sec(e+f*x)))-9/8*a*b^3*(11*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),-b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(9/2)*d^2*f*sqrt(d*sec(e+f*x)))+9/8*a*b^3*(11*a^2-2*b^2)*cot(e+f*x)*EllipticPi((sec(e+f*x)^2)^(1/4),b/sqrt(a^2+b^2),I)*(sec(e+f*x)^2)^(1/4)*sqrt(-tan(e+f*x)^2)/((a^2+b^2)^(9/2)*d^2*f*sqrt(d*sec(e+f*x)))-3/20*a*(8*a^4+64*a^2*b^2-139*b^4)*tan(e+f*x)/((a^2+b^2)^4*d^2*f*sqrt(d*sec(e+f*x)))+3/10*b*(4*a^4+28*a^2*b^2-15*b^4)*sec(e+f*x)^2/((a^2+b^2)^3*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2)+2/5*cos(e+f*x)^2*(b+a*tan(e+f*x))/((a^2+b^2)*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2)+3/20*a*b*(8*a^4+64*a^2*b^2-139*b^4)*sec(e+f*x)^2/((a^2+b^2)^4*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x)))-2/5*(b*(4*a^2-9*b^2)-a*(3*a^2+16*b^2)*tan(e+f*x))/((a^2+b^2)^2*d^2*f*sqrt(d*sec(e+f*x))*(a+b*tan(e+f*x))^2)],

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

# n>0
[(d*sec(e+f*x))^(5/3)*(a+b*tan(e+f*x)),x,3,3/5*b*(d*sec(e+f*x))^(5/3)/f+3/2*a*d*hypergeom([-1/3,1/2],[2/3],cos(e+f*x)^2)*(d*sec(e+f*x))^(2/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[(d*sec(e+f*x))^(1/3)*(a+b*tan(e+f*x)),x,3,3*b*(d*sec(e+f*x))^(1/3)/f-3/2*a*d*hypergeom([1/3,1/2],[4/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(2/3)*sqrt(sin(e+f*x)^2))],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(1/3),x,3,-3*b/(f*(d*sec(e+f*x))^(1/3))-3/4*a*d*hypergeom([1/2,2/3],[5/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(4/3)*sqrt(sin(e+f*x)^2))],
[(a+b*tan(e+f*x))/(d*sec(e+f*x))^(5/3),x,3,-3/5*b/(f*(d*sec(e+f*x))^(5/3))-3/8*a*d*hypergeom([1/2,4/3],[7/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(8/3)*sqrt(sin(e+f*x)^2))],
[(d*sec(e+f*x))^(5/3)*(a+b*tan(e+f*x))^2,x,4,33/40*a*b*(d*sec(e+f*x))^(5/3)/f+3/16*(8*a^2-3*b^2)*d*hypergeom([-1/3,1/2],[2/3],cos(e+f*x)^2)*(d*sec(e+f*x))^(2/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))+3/8*b*(d*sec(e+f*x))^(5/3)*(a+b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(1/3)*(a+b*tan(e+f*x))^2,x,4,21/4*a*b*(d*sec(e+f*x))^(1/3)/f-3/8*(4*a^2-3*b^2)*d*hypergeom([1/3,1/2],[4/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(2/3)*sqrt(sin(e+f*x)^2))+3/4*b*(d*sec(e+f*x))^(1/3)*(a+b*tan(e+f*x))/f],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(1/3),x,4,-15/2*a*b/(f*(d*sec(e+f*x))^(1/3))-3/8*(2*a^2-3*b^2)*d*hypergeom([1/2,2/3],[5/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(4/3)*sqrt(sin(e+f*x)^2))+3/2*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(1/3))],
[(a+b*tan(e+f*x))^2/(d*sec(e+f*x))^(5/3),x,4,3/10*a*b/(f*(d*sec(e+f*x))^(5/3))-3/16*(2*a^2+3*b^2)*d*hypergeom([1/2,4/3],[7/3],cos(e+f*x)^2)*sin(e+f*x)/(f*(d*sec(e+f*x))^(8/3)*sqrt(sin(e+f*x)^2))-3/2*b*(a+b*tan(e+f*x))/(f*(d*sec(e+f*x))^(5/3))],

# n<0
[(d*sec(e+f*x))^(5/3)/(a+b*tan(e+f*x)),x,16,-arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(1/6)*f*(sec(e+f*x)^2)^(5/6))+1/4*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(1/6)*f*(sec(e+f*x)^2)^(5/6))-1/4*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(1/6)*f*(sec(e+f*x)^2)^(5/6))-1/2*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(5/3)*sqrt(3)/(b^(2/3)*(a^2+b^2)^(1/6)*f*(sec(e+f*x)^2)^(5/6))+1/2*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(5/3)*sqrt(3)/(b^(2/3)*(a^2+b^2)^(1/6)*f*(sec(e+f*x)^2)^(5/6))+AppellF1(1/2,1,1/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(5/3)*tan(e+f*x)/(a*f*(sec(e+f*x)^2)^(5/6))],
[(d*sec(e+f*x))^(1/3)/(a+b*tan(e+f*x)),x,16,-b^(2/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(5/6)*f*(sec(e+f*x)^2)^(1/6))+1/4*b^(2/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(5/6)*f*(sec(e+f*x)^2)^(1/6))-1/4*b^(2/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(5/6)*f*(sec(e+f*x)^2)^(1/6))+1/2*b^(2/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(1/3)*sqrt(3)/((a^2+b^2)^(5/6)*f*(sec(e+f*x)^2)^(1/6))-1/2*b^(2/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(1/3)*sqrt(3)/((a^2+b^2)^(5/6)*f*(sec(e+f*x)^2)^(1/6))+AppellF1(1/2,1,5/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(1/3)*tan(e+f*x)/(a*f*(sec(e+f*x)^2)^(1/6))],
[1/((d*sec(e+f*x))^(1/3)*(a+b*tan(e+f*x))),x,17,3*b/((a^2+b^2)*f*(d*sec(e+f*x))^(1/3))-b^(4/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(7/6)*f*(d*sec(e+f*x))^(1/3))+1/4*b^(4/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(7/6)*f*(d*sec(e+f*x))^(1/3))-1/4*b^(4/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(7/6)*f*(d*sec(e+f*x))^(1/3))-1/2*b^(4/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(1/6)*sqrt(3)/((a^2+b^2)^(7/6)*f*(d*sec(e+f*x))^(1/3))+1/2*b^(4/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(1/6)*sqrt(3)/((a^2+b^2)^(7/6)*f*(d*sec(e+f*x))^(1/3))+AppellF1(1/2,1,7/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(1/6)*tan(e+f*x)/(a*f*(d*sec(e+f*x))^(1/3))],
[1/((d*sec(e+f*x))^(5/3)*(a+b*tan(e+f*x))),x,17,3/5*b/((a^2+b^2)*f*(d*sec(e+f*x))^(5/3))-b^(8/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(11/6)*f*(d*sec(e+f*x))^(5/3))+1/4*b^(8/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(11/6)*f*(d*sec(e+f*x))^(5/3))-1/4*b^(8/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(11/6)*f*(d*sec(e+f*x))^(5/3))+1/2*b^(8/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(5/6)*sqrt(3)/((a^2+b^2)^(11/6)*f*(d*sec(e+f*x))^(5/3))-1/2*b^(8/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(5/6)*sqrt(3)/((a^2+b^2)^(11/6)*f*(d*sec(e+f*x))^(5/3))+AppellF1(1/2,1,11/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(5/6)*tan(e+f*x)/(a*f*(d*sec(e+f*x))^(5/3))],
[(d*sec(e+f*x))^(5/3)/(a+b*tan(e+f*x))^2,x,18,-1/3*a*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(7/6)*f*(sec(e+f*x)^2)^(5/6))+1/12*a*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(7/6)*f*(sec(e+f*x)^2)^(5/6))-1/12*a*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(7/6)*f*(sec(e+f*x)^2)^(5/6))-1/2*a*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(7/6)*f*(sec(e+f*x)^2)^(5/6)*sqrt(3))+1/2*a*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(5/3)/(b^(2/3)*(a^2+b^2)^(7/6)*f*(sec(e+f*x)^2)^(5/6)*sqrt(3))+AppellF1(1/2,2,1/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(5/3)*tan(e+f*x)/(a^2*f*(sec(e+f*x)^2)^(5/6))+1/3*b^2*AppellF1(3/2,2,1/6,5/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(5/3)*tan(e+f*x)^3/(a^4*f*(sec(e+f*x)^2)^(5/6))-a*b*(d*sec(e+f*x))^(5/3)/((a^2+b^2)*f*(a^2-b^2*tan(e+f*x)^2))],
[(d*sec(e+f*x))^(1/3)/(a+b*tan(e+f*x))^2,x,18,-5/3*a*b^(2/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(11/6)*f*(sec(e+f*x)^2)^(1/6))+5/12*a*b^(2/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(11/6)*f*(sec(e+f*x)^2)^(1/6))-5/12*a*b^(2/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(11/6)*f*(sec(e+f*x)^2)^(1/6))+5/2*a*b^(2/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(11/6)*f*(sec(e+f*x)^2)^(1/6)*sqrt(3))-5/2*a*b^(2/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(d*sec(e+f*x))^(1/3)/((a^2+b^2)^(11/6)*f*(sec(e+f*x)^2)^(1/6)*sqrt(3))+AppellF1(1/2,2,5/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(1/3)*tan(e+f*x)/(a^2*f*(sec(e+f*x)^2)^(1/6))+1/3*b^2*AppellF1(3/2,2,5/6,5/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*sec(e+f*x))^(1/3)*tan(e+f*x)^3/(a^4*f*(sec(e+f*x)^2)^(1/6))-a*b*(d*sec(e+f*x))^(1/3)/((a^2+b^2)*f*(a^2-b^2*tan(e+f*x)^2))],
[1/((d*sec(e+f*x))^(1/3)*(a+b*tan(e+f*x))^2),x,19,7*a*b/((a^2+b^2)^2*f*(d*sec(e+f*x))^(1/3))-7/3*a*b^(4/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(13/6)*f*(d*sec(e+f*x))^(1/3))+7/12*a*b^(4/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(13/6)*f*(d*sec(e+f*x))^(1/3))-7/12*a*b^(4/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(13/6)*f*(d*sec(e+f*x))^(1/3))-7/2*a*b^(4/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(13/6)*f*(d*sec(e+f*x))^(1/3)*sqrt(3))+7/2*a*b^(4/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(13/6)*f*(d*sec(e+f*x))^(1/3)*sqrt(3))+AppellF1(1/2,2,7/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(1/6)*tan(e+f*x)/(a^2*f*(d*sec(e+f*x))^(1/3))+1/3*b^2*AppellF1(3/2,2,7/6,5/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(1/6)*tan(e+f*x)^3/(a^4*f*(d*sec(e+f*x))^(1/3))-a*b/((a^2+b^2)*f*(d*sec(e+f*x))^(1/3)*(a^2-b^2*tan(e+f*x)^2))],
[1/((d*sec(e+f*x))^(5/3)*(a+b*tan(e+f*x))^2),x,19,11/5*a*b/((a^2+b^2)^2*f*(d*sec(e+f*x))^(5/3))-11/3*a*b^(8/3)*arctanh(b^(1/3)*(sec(e+f*x)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(17/6)*f*(d*sec(e+f*x))^(5/3))+11/12*a*b^(8/3)*log((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(17/6)*f*(d*sec(e+f*x))^(5/3))-11/12*a*b^(8/3)*log((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(e+f*x)^2)^(1/6)+b^(2/3)*(sec(e+f*x)^2)^(1/3))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(17/6)*f*(d*sec(e+f*x))^(5/3))+11/2*a*b^(8/3)*arctan(1/sqrt(3)-2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(17/6)*f*(d*sec(e+f*x))^(5/3)*sqrt(3))-11/2*a*b^(8/3)*arctan(1/sqrt(3)+2*b^(1/3)*(sec(e+f*x)^2)^(1/6)/((a^2+b^2)^(1/6)*sqrt(3)))*(sec(e+f*x)^2)^(5/6)/((a^2+b^2)^(17/6)*f*(d*sec(e+f*x))^(5/3)*sqrt(3))+AppellF1(1/2,2,11/6,3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(5/6)*tan(e+f*x)/(a^2*f*(d*sec(e+f*x))^(5/3))+1/3*b^2*AppellF1(3/2,2,11/6,5/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(sec(e+f*x)^2)^(5/6)*tan(e+f*x)^3/(a^4*f*(d*sec(e+f*x))^(5/3))-a*b/((a^2+b^2)*f*(d*sec(e+f*x))^(5/3)*(a^2-b^2*tan(e+f*x)^2))],

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

# Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n with n symbolic
[(d*sec(e+f*x))^m*(a+b*tan(e+f*x))^n,x,3,b*AppellF1(1+n,1-1/2*m,1-1/2*m,2+n,(a+b*tan(e+f*x))/(a+sqrt(-b^2)),(a+b*tan(e+f*x))/(a-sqrt(-b^2)))*(d*sec(e+f*x))^m*(a+b*tan(e+f*x))^(1+n)/((a^2+b^2)*f*(1+n)*(1+(a+b*tan(e+f*x))/(-a+sqrt(-b^2)))^(1/2*m)*(1+(-a-b*tan(e+f*x))/(a+sqrt(-b^2)))^(1/2*m)),AppellF1(1+n,1-1/2*m,1-1/2*m,2+n,(a+b*tan(e+f*x))/(a-sqrt(-b^2)),(a+b*tan(e+f*x))/(a+sqrt(-b^2)))*cos(e+f*x)^2*(d*sec(e+f*x))^m*(a+b*tan(e+f*x))^(1+n)*(1+(-a-b*tan(e+f*x))/(a-sqrt(-b^2)))^(1-1/2*m)*(1+(-a-b*tan(e+f*x))/(a+sqrt(-b^2)))^(1-1/2*m)/(b*f*(1+n))],
[sec(c+d*x)^6*(a+b*tan(c+d*x))^n,x,3,(a^2+b^2)^2*(a+b*tan(c+d*x))^(1+n)/(b^5*d*(1+n))-4*a*(a^2+b^2)*(a+b*tan(c+d*x))^(2+n)/(b^5*d*(2+n))+2*(3*a^2+b^2)*(a+b*tan(c+d*x))^(3+n)/(b^5*d*(3+n))-4*a*(a+b*tan(c+d*x))^(4+n)/(b^5*d*(4+n))+(a+b*tan(c+d*x))^(5+n)/(b^5*d*(5+n))],
[sec(c+d*x)^4*(a+b*tan(c+d*x))^n,x,3,(a^2+b^2)*(a+b*tan(c+d*x))^(1+n)/(b^3*d*(1+n))-2*a*(a+b*tan(c+d*x))^(2+n)/(b^3*d*(2+n))+(a+b*tan(c+d*x))^(3+n)/(b^3*d*(3+n))],
[sec(c+d*x)^2*(a+b*tan(c+d*x))^n,x,2,(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n))],
[cos(c+d*x)^2*(a+b*tan(c+d*x))^n,x,6,-1/4*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*(-a*n+(1+a^2/b^2-n)*sqrt(-b^2))*(a+b*tan(c+d*x))^(1+n)/((1+a^2/b^2)*b*d*(1+n)*(a-sqrt(-b^2)))+1/4*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*(a*n+(1+a^2/b^2-n)*sqrt(-b^2))*(a+b*tan(c+d*x))^(1+n)/((a^2+b^2)*d*(1+n)*(a+sqrt(-b^2)))+1/2*cos(c+d*x)^2*(b+a*tan(c+d*x))*(a+b*tan(c+d*x))^(1+n)/((a^2+b^2)*d)],
[cos(c+d*x)^4*(a+b*tan(c+d*x))^n,x,7,1/16*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*(a*(5+3*a^2/b^2-2*n)*n/b^2-(3*a^4+a^2*b^2*(6-2*n-n^2)+b^4*(3-4*n+n^2))*sqrt(-b^2)/b^6)*(a+b*tan(c+d*x))^(1+n)/((1+a^2/b^2)^2*d*(1+n)*(a-sqrt(-b^2)))+1/16*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*(a*(5+3*a^2/b^2-2*n)*n/b^2+(3*a^4+a^2*b^2*(6-2*n-n^2)+b^4*(3-4*n+n^2))*sqrt(-b^2)/b^6)*(a+b*tan(c+d*x))^(1+n)/((1+a^2/b^2)^2*d*(1+n)*(a+sqrt(-b^2)))+1/4*cos(c+d*x)^4*(b+a*tan(c+d*x))*(a+b*tan(c+d*x))^(1+n)/((a^2+b^2)*d)+1/8*b*cos(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)*(b^2*(3-n)+a^2*(1+n)+a*b*(5+3*a^2/b^2-2*n)*tan(c+d*x))/((a^2+b^2)^2*d)],
[sec(c+d*x)^3*(a+b*tan(c+d*x))^n,x,3,AppellF1(1+n,-1/2,-1/2,2+n,(a+b*tan(c+d*x))/(a-sqrt(-b^2)),(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*sec(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n)*sqrt(1+(-a-b*tan(c+d*x))/(a-sqrt(-b^2)))*sqrt(1+(-a-b*tan(c+d*x))/(a+sqrt(-b^2))))],
[sec(c+d*x)*(a+b*tan(c+d*x))^n,x,3,AppellF1(1+n,1/2,1/2,2+n,(a+b*tan(c+d*x))/(a-sqrt(-b^2)),(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*cos(c+d*x)*sqrt(1+(-a-b*tan(c+d*x))/(a-sqrt(-b^2)))*sqrt(1+(-a-b*tan(c+d*x))/(a+sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n))],
[cos(c+d*x)*(a+b*tan(c+d*x))^n,x,3,AppellF1(1+n,3/2,3/2,2+n,(a+b*tan(c+d*x))/(a-sqrt(-b^2)),(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*cos(c+d*x)^3*(a+b*tan(c+d*x))^(1+n)*(1+(-a-b*tan(c+d*x))/(a-sqrt(-b^2)))^(3/2)*(1+(-a-b*tan(c+d*x))/(a+sqrt(-b^2)))^(3/2)/(b*d*(1+n))],
[cos(c+d*x)^3*(a+b*tan(c+d*x))^n,x,3,AppellF1(1+n,5/2,5/2,2+n,(a+b*tan(c+d*x))/(a-sqrt(-b^2)),(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*cos(c+d*x)^5*(a+b*tan(c+d*x))^(1+n)*(1+(-a-b*tan(c+d*x))/(a-sqrt(-b^2)))^(5/2)*(1+(-a-b*tan(c+d*x))/(a+sqrt(-b^2)))^(5/2)/(b*d*(1+n))],

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

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

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

# n>0
[(e*cos(c+d*x))^(7/2)*(a+I*a*tan(c+d*x)),x,6,-2/7*I*a*(e*cos(c+d*x))^(7/2)/d+10/21*a*(e*cos(c+d*x))^(7/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(d*cos(c+d*x)^(7/2))+2/7*a*(e*cos(c+d*x))^(7/2)*tan(c+d*x)/d+10/21*a*(e*cos(c+d*x))^(7/2)*sec(c+d*x)^2*tan(c+d*x)/d],
[(e*cos(c+d*x))^(5/2)*(a+I*a*tan(c+d*x)),x,5,-2/5*I*a*(e*cos(c+d*x))^(5/2)/d+6/5*a*(e*cos(c+d*x))^(5/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*cos(c+d*x)^(5/2))+2/5*a*(e*cos(c+d*x))^(5/2)*tan(c+d*x)/d],
[(e*cos(c+d*x))^(3/2)*(a+I*a*tan(c+d*x)),x,5,-2/3*I*a*(e*cos(c+d*x))^(3/2)/d+2/3*a*(e*cos(c+d*x))^(3/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(d*cos(c+d*x)^(3/2))+2/3*a*(e*cos(c+d*x))^(3/2)*tan(c+d*x)/d],
[sqrt(e*cos(c+d*x))*(a+I*a*tan(c+d*x)),x,4,-2*I*a*sqrt(e*cos(c+d*x))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+I*a*tan(c+d*x))/sqrt(e*cos(c+d*x)),x,4,2*I*a/(d*sqrt(e*cos(c+d*x)))+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))],
[(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(3/2),x,5,2/3*I*a/(d*(e*cos(c+d*x))^(3/2))-2*a*cos(c+d*x)^(3/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*(e*cos(c+d*x))^(3/2))+2*a*sin(c+d*x)/(d*e*sqrt(e*cos(c+d*x))),2/3*I*a/(d*(e*cos(c+d*x))^(3/2))-2*a*cos(c+d*x)^(3/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*(e*cos(c+d*x))^(3/2))+2*a*cos(c+d*x)*sin(c+d*x)/(d*(e*cos(c+d*x))^(3/2))],
[(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(5/2),x,5,2/5*I*a/(d*(e*cos(c+d*x))^(5/2))+2/3*a*cos(c+d*x)^(5/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(d*(e*cos(c+d*x))^(5/2))+2/3*a*cos(c+d*x)*sin(c+d*x)/(d*(e*cos(c+d*x))^(5/2))],
[(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(7/2),x,6,2/7*I*a/(d*(e*cos(c+d*x))^(7/2))-6/5*a*cos(c+d*x)^(7/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*(e*cos(c+d*x))^(7/2))+2/5*a*cos(c+d*x)*sin(c+d*x)/(d*(e*cos(c+d*x))^(7/2))+6/5*a*cos(c+d*x)^3*sin(c+d*x)/(d*(e*cos(c+d*x))^(7/2))],

# n<0
[(e*cos(c+d*x))^(7/2)/(a+I*a*tan(c+d*x))^2,x,7,2/7*(e*cos(c+d*x))^(7/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*cos(c+d*x)^(7/2))+2/15*cos(c+d*x)*(e*cos(c+d*x))^(7/2)*sin(c+d*x)/(a^2*d)+6/35*(e*cos(c+d*x))^(7/2)*tan(c+d*x)/(a^2*d)+2/7*(e*cos(c+d*x))^(7/2)*sec(c+d*x)^2*tan(c+d*x)/(a^2*d)+4/15*I*cos(c+d*x)^2*(e*cos(c+d*x))^(7/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*cos(c+d*x))^(5/2)/(a+I*a*tan(c+d*x))^2,x,6,42/65*(e*cos(c+d*x))^(5/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*cos(c+d*x)^(5/2))+2/13*cos(c+d*x)*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/(a^2*d)+14/65*(e*cos(c+d*x))^(5/2)*tan(c+d*x)/(a^2*d)+4/13*I*cos(c+d*x)^2*(e*cos(c+d*x))^(5/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+I*a*tan(c+d*x))^2,x,6,10/33*(e*cos(c+d*x))^(3/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*cos(c+d*x)^(3/2))+2/11*cos(c+d*x)*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(a^2*d)+10/33*(e*cos(c+d*x))^(3/2)*tan(c+d*x)/(a^2*d)+4/11*I*cos(c+d*x)^2*(e*cos(c+d*x))^(3/2)/(d*(a^2+I*a^2*tan(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+I*a*tan(c+d*x))^2,x,5,2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*sqrt(cos(c+d*x)))+2/9*I*sqrt(e*cos(c+d*x))/(d*(a+I*a*tan(c+d*x))^2)+2/9*I*sqrt(e*cos(c+d*x))/(d*(a^2+I*a^2*tan(c+d*x))),2/9*cos(c+d*x)*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*sqrt(cos(c+d*x)))+4/9*I*cos(c+d*x)^2*sqrt(e*cos(c+d*x))/(d*(a^2+I*a^2*tan(c+d*x)))],
[1/(sqrt(e*cos(c+d*x))*(a+I*a*tan(c+d*x))^2),x,5,2/7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(a^2*d*sqrt(e*cos(c+d*x)))+2/7*I/(d*sqrt(e*cos(c+d*x))*(a+I*a*tan(c+d*x))^2)+2/7*I/(d*sqrt(e*cos(c+d*x))*(a^2+I*a^2*tan(c+d*x))),2/7*cos(c+d*x)*sin(c+d*x)/(a^2*d*sqrt(e*cos(c+d*x)))+2/7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(a^2*d*sqrt(e*cos(c+d*x)))+4/7*I*cos(c+d*x)^2/(d*sqrt(e*cos(c+d*x))*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+I*a*tan(c+d*x))^2),x,4,2/5*cos(c+d*x)^(3/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*(e*cos(c+d*x))^(3/2))+4/5*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(3/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+I*a*tan(c+d*x))^2),x,4,-2/3*cos(c+d*x)^(5/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*(e*cos(c+d*x))^(5/2))+4/3*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(5/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+I*a*tan(c+d*x))^2),x,5,6*cos(c+d*x)^(7/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*(e*cos(c+d*x))^(7/2))-6*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(e*cos(c+d*x))^(7/2))+4*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(7/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(9/2)*(a+I*a*tan(c+d*x))^2),x,5,10/3*cos(c+d*x)^(9/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*(e*cos(c+d*x))^(9/2))+10/3*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(e*cos(c+d*x))^(9/2))-4*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(9/2)*(a^2+I*a^2*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(11/2)*(a+I*a*tan(c+d*x))^2),x,6,-14/5*cos(c+d*x)^(11/2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d*(e*cos(c+d*x))^(11/2))+14/15*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(e*cos(c+d*x))^(11/2))+14/5*cos(c+d*x)^5*sin(c+d*x)/(a^2*d*(e*cos(c+d*x))^(11/2))-4/3*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(11/2)*(a^2+I*a^2*tan(c+d*x)))],

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

# n>0
[(e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x)),x,5,12/35*I*a*(e*cos(c+d*x))^(7/2)*sec(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))+32/35*I*a*(e*cos(c+d*x))^(7/2)*sec(c+d*x)^4/(d*sqrt(a+I*a*tan(c+d*x)))-2/7*I*(e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x))/d-16/35*I*(e*cos(c+d*x))^(7/2)*sec(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x)),x,4,8/15*I*a*(e*cos(c+d*x))^(5/2)*sec(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-2/5*I*(e*cos(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x))/d-16/15*I*(e*cos(c+d*x))^(5/2)*sec(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)),x,3,4/3*I*a*e*sec(c+d*x)*sqrt(e*cos(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))-2/3*I*(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))/d,4/3*I*a*(e*cos(c+d*x))^(3/2)*sec(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-2/3*I*(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x)),x,2,-2*I*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x))/sqrt(e*cos(c+d*x)),x,10,-I*log(a*sqrt(e)-sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))+cos(c+d*x)*sqrt(e)*(a+I*a*tan(c+d*x)))*sqrt(a)/(d*sqrt(2)*sqrt(e))+I*log(a*sqrt(e)+sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))+cos(c+d*x)*sqrt(e)*(a+I*a*tan(c+d*x)))*sqrt(a)/(d*sqrt(2)*sqrt(e))+I*arctan(1-sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sqrt(2)*sqrt(a)/(d*sqrt(e))-I*arctan(1+sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sqrt(2)*sqrt(a)/(d*sqrt(e))],
[sqrt(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(3/2),x,13,I*a/(d*(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))-I*a^(3/2)*arctan(1-sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sec(c+d*x)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*a^(3/2)*arctan(1+sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sec(c+d*x)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/2*I*a^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/sqrt(e)+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/2*I*a^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/sqrt(e)+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))),I*a/(d*(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x)))-I*a^(3/2)*e^(3/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*(e*cos(c+d*x))^(3/2)*(e*sec(c+d*x))^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*a^(3/2)*e^(3/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*(e*cos(c+d*x))^(3/2)*(e*sec(c+d*x))^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/2*I*a^(3/2)*e^(3/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*(e*cos(c+d*x))^(3/2)*(e*sec(c+d*x))^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/2*I*a^(3/2)*e^(3/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*(e*cos(c+d*x))^(3/2)*(e*sec(c+d*x))^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[sqrt(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(5/2),x,13,3/4*I*e^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(a)/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2))-3/4*I*e^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sqrt(a)/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2))-3/8*I*e^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(a)/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2))+3/8*I*e^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))*sqrt(a)/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2))+1/2*I*a/(d*(e*cos(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x)))-3/4*I*cos(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(d*(e*cos(c+d*x))^(5/2))],
[sqrt(a+I*a*tan(c+d*x))/(e*cos(c+d*x))^(7/2),x,15,1/3*I*a/(d*(e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x)))+5/8*I*a*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x)))-5/8*I*a^(3/2)*e^(7/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+5/8*I*a^(3/2)*e^(7/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+5/16*I*a^(3/2)*e^(7/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-5/16*I*a^(3/2)*e^(7/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-5/12*I*cos(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(d*(e*cos(c+d*x))^(7/2))],

# n<0
[(e*cos(c+d*x))^(5/2)/sqrt(a+I*a*tan(c+d*x)),x,5,2/7*I*(e*cos(c+d*x))^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))+16/35*I*(e*cos(c+d*x))^(5/2)*sec(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-12/35*I*(e*cos(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x))/(a*d)-32/35*I*(e*cos(c+d*x))^(5/2)*sec(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[(e*cos(c+d*x))^(3/2)/sqrt(a+I*a*tan(c+d*x)),x,4,2/5*I*(e*cos(c+d*x))^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))+16/15*I*(e*cos(c+d*x))^(3/2)*sec(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-8/15*I*(e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[sqrt(e*cos(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,3,2/3*I*sqrt(e*cos(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))-4/3*I*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[1/(sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x))),x,2,2*I/(d*sqrt(e*cos(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*sqrt(a+I*a*tan(c+d*x))),x,11,I*log(a*sqrt(e)-sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))+cos(c+d*x)*sqrt(e)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*log(a*sqrt(e)+sqrt(2)*sqrt(a)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))+cos(c+d*x)*sqrt(e)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*e^(3/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-I*arctan(1-sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sec(c+d*x)*sqrt(2)*sqrt(a)/(d*e^(3/2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+I*arctan(1+sqrt(2)*sqrt(e*cos(c+d*x))*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e)))*sec(c+d*x)*sqrt(2)*sqrt(a)/(d*e^(3/2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*sqrt(a+I*a*tan(c+d*x))),x,12,I*e^(5/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2)*sqrt(a))-I*e^(5/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2)*sqrt(a))-1/2*I*e^(5/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2)*sqrt(a))+1/2*I*e^(5/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a+I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a+I*a*tan(c+d*x)))/(d*(e*cos(c+d*x))^(5/2)*(e*sec(c+d*x))^(5/2)*sqrt(2)*sqrt(a))-I*cos(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a*d*(e*cos(c+d*x))^(5/2))],
[1/((e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x))),x,14,3/4*I*cos(c+d*x)^2/(d*(e*cos(c+d*x))^(7/2)*sqrt(a+I*a*tan(c+d*x)))-3/4*I*e^(7/2)*arctan(1-sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(a)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+3/4*I*e^(7/2)*arctan(1+sqrt(2)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/(sqrt(a)*sqrt(e*sec(c+d*x))))*sec(c+d*x)*sqrt(a)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+3/8*I*e^(7/2)*log(a-sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-3/8*I*e^(7/2)*log(a+sqrt(2)*sqrt(a)*sqrt(e)*sqrt(a-I*a*tan(c+d*x))/sqrt(e*sec(c+d*x))+cos(c+d*x)*(a-I*a*tan(c+d*x)))*sec(c+d*x)*sqrt(a)/(d*(e*cos(c+d*x))^(7/2)*(e*sec(c+d*x))^(7/2)*sqrt(2)*sqrt(a-I*a*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/2*I*cos(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a*d*(e*cos(c+d*x))^(7/2))],

# Integrands of the form (d Cos[e+f x])^m (a+a I Tan[e+f x])^n with m symbolic
[(e*cos(c+d*x))^m*(a+I*a*tan(c+d*x))^n,x,5,-I*2^(-1/2*m+n)*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*(2+m-2*n)],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*(m-2*n))*(a+I*a*tan(c+d*x))^n/(d*m)],
[(e*cos(c+d*x))^m*(a+I*a*tan(c+d*x))^2,x,5,-I*2^(2-1/2*m)*a^2*(e*cos(c+d*x))^m*hypergeom([1/2*(-2+m),-1/2*m],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*m)/(d*m)],
[(e*cos(c+d*x))^m*(a+I*a*tan(c+d*x)),x,5,-I*2^(1-1/2*m)*a*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*m],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*m)/(d*m)],
[(e*cos(c+d*x))^m/(a+I*a*tan(c+d*x)),x,5,-I*2^(-1-1/2*m)*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*(4+m)],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*m)/(a*d*m)],
[(e*cos(c+d*x))^m/(a+I*a*tan(c+d*x))^2,x,5,-I*2^(-2-1/2*m)*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*(6+m)],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*m)/(a^2*d*m)],
[(e*cos(c+d*x))^m*sqrt(a+I*a*tan(c+d*x)),x,5,-I*2^(1/2-1/2*m)*a*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*(1+m)],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*(1+m))/(d*m*sqrt(a+I*a*tan(c+d*x)))],
[(e*cos(c+d*x))^m/sqrt(a+I*a*tan(c+d*x)),x,5,-I*2^(-1/2-1/2*m)*(e*cos(c+d*x))^m*hypergeom([-1/2*m,1/2*(3+m)],[1-1/2*m],1/2*(1-I*tan(c+d*x)))*(1+I*tan(c+d*x))^(1/2*(1+m))/(d*m*sqrt(a+I*a*tan(c+d*x)))],

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

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n with m symbolic
[(d*cos(e+f*x))^m*(a+b*tan(e+f*x))^3,x,5,-a*(3*b^2-a^2*(1-m))*(d*cos(e+f*x))^m*hypergeom([1/2,1/2*(2+m)],[3/2],-tan(e+f*x)^2)*(sec(e+f*x)^2)^(1/2*m)*tan(e+f*x)/(f*(1-m))+b*(d*cos(e+f*x))^m*(a+b*tan(e+f*x))^2/(f*(2-m))+b*(d*cos(e+f*x))^m*(2*(b^2-a^2*(3-m))*(1-m)+a*b*(4-m)*m*tan(e+f*x))/(f*(1-m)*(2-m)*m)],
[(d*cos(e+f*x))^m*(a+b*tan(e+f*x))^2,x,5,-a*b*(2-m)*(d*cos(e+f*x))^m/(f*(1-m)*m)+(b^2-a^2*(1-m))*cos(e+f*x)*(d*cos(e+f*x))^m*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(e+f*x)^2)*sin(e+f*x)/(f*(1-m^2)*sqrt(sin(e+f*x)^2))+b*(d*cos(e+f*x))^m*(a+b*tan(e+f*x))/(f*(1-m))],
[(d*cos(e+f*x))^m*(a+b*tan(e+f*x)),x,4,-b*(d*cos(e+f*x))^m/(f*m)-a*(d*cos(e+f*x))^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(e+f*x)^2)*sin(e+f*x)/(d*f*(1+m)*sqrt(sin(e+f*x)^2)),-b*(d*cos(e+f*x))^m/(f*m)-a*cos(e+f*x)*(d*cos(e+f*x))^m*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(e+f*x)^2)*sin(e+f*x)/(f*(1+m)*sqrt(sin(e+f*x)^2))],
[(d*cos(e+f*x))^m/(a+b*tan(e+f*x)),x,7,b*(d*cos(e+f*x))^m*hypergeom([1,-1/2*m],[1-1/2*m],b^2*sec(e+f*x)^2/(a^2+b^2))/((a^2+b^2)*f*m)+AppellF1(1/2,1,1/2*(2+m),3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*cos(e+f*x))^m*(sec(e+f*x)^2)^(1/2*m)*tan(e+f*x)/(a*f)],
[(d*cos(e+f*x))^m/(a+b*tan(e+f*x))^2,x,8,2*a*b*(d*cos(e+f*x))^m*hypergeom([2,-1/2*m],[1-1/2*m],b^2*sec(e+f*x)^2/(a^2+b^2))/((a^2+b^2)^2*f*m)+AppellF1(1/2,2,1/2*(2+m),3/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*cos(e+f*x))^m*(sec(e+f*x)^2)^(1/2*m)*tan(e+f*x)/(a^2*f)+1/3*b^2*AppellF1(3/2,2,1/2*(2+m),5/2,b^2*tan(e+f*x)^2/a^2,-tan(e+f*x)^2)*(d*cos(e+f*x))^m*(sec(e+f*x)^2)^(1/2*m)*tan(e+f*x)^3/(a^4*f)],

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n with n symbolic
[(d*cos(e+f*x))^m*(a+b*tan(e+f*x))^n,x,4,AppellF1(1+n,1/2*(2+m),1/2*(2+m),2+n,(a+b*tan(e+f*x))/(a-sqrt(-b^2)),(a+b*tan(e+f*x))/(a+sqrt(-b^2)))*cos(e+f*x)^2*(d*cos(e+f*x))^m*(a+b*tan(e+f*x))^(1+n)*(1+(-a-b*tan(e+f*x))/(a-sqrt(-b^2)))^(1/2*(2+m))*(1+(-a-b*tan(e+f*x))/(a+sqrt(-b^2)))^(1/2*(2+m))/(b*f*(1+n))]]:
