# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.2.1 (a+b tan)^m (c+d tan)^n.txt"

lst:=[

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

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

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

# m>0
[tan(c+d*x)^5*(a+I*a*tan(c+d*x)),x,6,-I*a*x-a*log(cos(c+d*x))/d+I*a*tan(c+d*x)/d-1/2*a*tan(c+d*x)^2/d-1/3*I*a*tan(c+d*x)^3/d+1/4*a*tan(c+d*x)^4/d+1/5*I*a*tan(c+d*x)^5/d],
[tan(c+d*x)^4*(a+I*a*tan(c+d*x)),x,5,a*x-I*a*log(cos(c+d*x))/d-a*tan(c+d*x)/d-1/2*I*a*tan(c+d*x)^2/d+1/3*a*tan(c+d*x)^3/d+1/4*I*a*tan(c+d*x)^4/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x)),x,4,I*a*x+a*log(cos(c+d*x))/d-I*a*tan(c+d*x)/d+1/2*a*tan(c+d*x)^2/d+1/3*I*a*tan(c+d*x)^3/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x)),x,3,-a*x+I*a*log(cos(c+d*x))/d+a*tan(c+d*x)/d+1/2*I*a*tan(c+d*x)^2/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x)),x,2,-I*a*x-a*log(cos(c+d*x))/d+I*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],
[cot(c+d*x)*(a+I*a*tan(c+d*x)),x,2,I*a*x+a*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x)),x,3,-a*x-a*cot(c+d*x)/d+I*a*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x)),x,4,-I*a*x-I*a*cot(c+d*x)/d-1/2*a*cot(c+d*x)^2/d-a*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x)),x,5,a*x+a*cot(c+d*x)/d-1/2*I*a*cot(c+d*x)^2/d-1/3*a*cot(c+d*x)^3/d-I*a*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x)),x,6,I*a*x+I*a*cot(c+d*x)/d+1/2*a*cot(c+d*x)^2/d-1/3*I*a*cot(c+d*x)^3/d-1/4*a*cot(c+d*x)^4/d+a*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x)),x,7,-a*x-a*cot(c+d*x)/d+1/2*I*a*cot(c+d*x)^2/d+1/3*a*cot(c+d*x)^3/d-1/4*I*a*cot(c+d*x)^4/d-1/5*a*cot(c+d*x)^5/d+I*a*log(sin(c+d*x))/d],
[tan(c+d*x)^4*(a+I*a*tan(c+d*x))^2,x,6,2*a^2*x-2*I*a^2*log(cos(c+d*x))/d-2*a^2*tan(c+d*x)/d-I*a^2*tan(c+d*x)^2/d+2/3*a^2*tan(c+d*x)^3/d+1/2*I*a^2*tan(c+d*x)^4/d-1/5*a^2*tan(c+d*x)^5/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^2,x,5,2*I*a^2*x+2*a^2*log(cos(c+d*x))/d-2*I*a^2*tan(c+d*x)/d+a^2*tan(c+d*x)^2/d+2/3*I*a^2*tan(c+d*x)^3/d-1/4*a^2*tan(c+d*x)^4/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^2,x,3,-2*a^2*x+2*I*a^2*log(cos(c+d*x))/d+a^2*tan(c+d*x)/d-1/3*I*(a+I*a*tan(c+d*x))^3/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^2,x,3,-2*I*a^2*x-2*a^2*log(cos(c+d*x))/d+I*a^2*tan(c+d*x)/d+1/2*(a+I*a*tan(c+d*x))^2/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],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^2,x,3,2*I*a^2*x+a^2*log(cos(c+d*x))/d+a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^2,x,3,-2*a^2*x-a^2*cot(c+d*x)/d+2*I*a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^2,x,4,-2*I*a^2*x-2*I*a^2*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)^2/d-2*a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^2,x,5,2*a^2*x+2*a^2*cot(c+d*x)/d-I*a^2*cot(c+d*x)^2/d-1/3*a^2*cot(c+d*x)^3/d-2*I*a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^2,x,6,2*I*a^2*x+2*I*a^2*cot(c+d*x)/d+a^2*cot(c+d*x)^2/d-2/3*I*a^2*cot(c+d*x)^3/d-1/4*a^2*cot(c+d*x)^4/d+2*a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x))^2,x,7,-2*a^2*x-2*a^2*cot(c+d*x)/d+I*a^2*cot(c+d*x)^2/d+2/3*a^2*cot(c+d*x)^3/d-1/2*I*a^2*cot(c+d*x)^4/d-1/5*a^2*cot(c+d*x)^5/d+2*I*a^2*log(sin(c+d*x))/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^3,x,6,4*I*a^3*x+4*a^3*log(cos(c+d*x))/d-4*I*a^3*tan(c+d*x)/d+2*a^3*tan(c+d*x)^2/d+4/3*I*a^3*tan(c+d*x)^3/d-11/20*a^3*tan(c+d*x)^4/d-1/5*tan(c+d*x)^4*(a^3+I*a^3*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^3,x,4,-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-1/4*I*(a+I*a*tan(c+d*x))^4/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^3,x,4,-4*I*a^3*x-4*a^3*log(cos(c+d*x))/d+2*I*a^3*tan(c+d*x)/d+1/2*a*(a+I*a*tan(c+d*x))^2/d+1/3*(a+I*a*tan(c+d*x))^3/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],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^3,x,5,4*I*a^3*x+3*a^3*log(cos(c+d*x))/d+a^3*log(sin(c+d*x))/d+(-a^3-I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^3,x,5,-4*a^3*x+I*a^3*log(cos(c+d*x))/d+3*I*a^3*log(sin(c+d*x))/d-cot(c+d*x)*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3,x,4,-4*I*a^3*x-2*I*a^3*cot(c+d*x)/d-4*a^3*log(sin(c+d*x))/d-1/2*a*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^2/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^3,x,5,4*a^3*x+2*a^3*cot(c+d*x)/d-4*I*a^3*log(sin(c+d*x))/d-1/2*I*a*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^2/d-1/3*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^3,x,6,4*I*a^3*x+4*I*a^3*cot(c+d*x)/d+2*a^3*cot(c+d*x)^2/d-3/4*I*a^3*cot(c+d*x)^3/d+4*a^3*log(sin(c+d*x))/d-1/4*cot(c+d*x)^4*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x))^3,x,7,-4*a^3*x-4*a^3*cot(c+d*x)/d+2*I*a^3*cot(c+d*x)^2/d+4/3*a^3*cot(c+d*x)^3/d-11/20*I*a^3*cot(c+d*x)^4/d+4*I*a^3*log(sin(c+d*x))/d-1/5*cot(c+d*x)^5*(a^3+I*a^3*tan(c+d*x))/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^4,x,7,8*I*a^4*x+8*a^4*log(cos(c+d*x))/d-8*I*a^4*tan(c+d*x)/d+4*a^4*tan(c+d*x)^2/d+8/3*I*a^4*tan(c+d*x)^3/d-67/60*a^4*tan(c+d*x)^4/d-1/6*tan(c+d*x)^4*(a^2+I*a^2*tan(c+d*x))^2/d-7/15*tan(c+d*x)^4*(a^4+I*a^4*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^4,x,5,-8*a^4*x+8*I*a^4*log(cos(c+d*x))/d+4*a^4*tan(c+d*x)/d-1/3*I*a*(a+I*a*tan(c+d*x))^3/d-1/5*I*(a+I*a*tan(c+d*x))^5/(a*d)-I*(a^2+I*a^2*tan(c+d*x))^2/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^4,x,5,-8*I*a^4*x-8*a^4*log(cos(c+d*x))/d+4*I*a^4*tan(c+d*x)/d+1/3*a*(a+I*a*tan(c+d*x))^3/d+1/4*(a+I*a*tan(c+d*x))^4/d+(a^2+I*a^2*tan(c+d*x))^2/d],
[(a+I*a*tan(c+d*x))^4,x,4,8*a^4*x-8*I*a^4*log(cos(c+d*x))/d-4*a^4*tan(c+d*x)/d+1/3*I*a*(a+I*a*tan(c+d*x))^3/d+I*(a^2+I*a^2*tan(c+d*x))^2/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^4,x,6,8*I*a^4*x+7*a^4*log(cos(c+d*x))/d+a^4*log(sin(c+d*x))/d-1/2*(a^2+I*a^2*tan(c+d*x))^2/d-3*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^4,x,5,-8*a^4*x+4*I*a^4*log(cos(c+d*x))/d+4*I*a^4*log(sin(c+d*x))/d-cot(c+d*x)*(a^2+I*a^2*tan(c+d*x))^2/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^4,x,6,-8*I*a^4*x-a^4*log(cos(c+d*x))/d-7*a^4*log(sin(c+d*x))/d-1/2*cot(c+d*x)^2*(a^2+I*a^2*tan(c+d*x))^2/d-3*I*cot(c+d*x)*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^4,x,5,8*a^4*x+4*a^4*cot(c+d*x)/d-8*I*a^4*log(sin(c+d*x))/d-1/3*a*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-I*cot(c+d*x)^2*(a^2+I*a^2*tan(c+d*x))^2/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^4,x,6,8*I*a^4*x+4*I*a^4*cot(c+d*x)/d+8*a^4*log(sin(c+d*x))/d-1/3*I*a*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-1/4*cot(c+d*x)^4*(a+I*a*tan(c+d*x))^4/d+cot(c+d*x)^2*(a^2+I*a^2*tan(c+d*x))^2/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x))^4,x,7,-8*a^4*x-8*a^4*cot(c+d*x)/d+4*I*a^4*cot(c+d*x)^2/d+23/15*a^4*cot(c+d*x)^3/d+8*I*a^4*log(sin(c+d*x))/d-1/5*cot(c+d*x)^5*(a^2+I*a^2*tan(c+d*x))^2/d-3/5*I*cot(c+d*x)^4*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^7*(a+I*a*tan(c+d*x))^4,x,8,-8*I*a^4*x-8*I*a^4*cot(c+d*x)/d-4*a^4*cot(c+d*x)^2/d+8/3*I*a^4*cot(c+d*x)^3/d+67/60*a^4*cot(c+d*x)^4/d-8*a^4*log(sin(c+d*x))/d-1/6*cot(c+d*x)^6*(a^2+I*a^2*tan(c+d*x))^2/d-7/15*I*cot(c+d*x)^5*(a^4+I*a^4*tan(c+d*x))/d],

# m<0
[tan(c+d*x)^6/(a+I*a*tan(c+d*x)),x,6,5/2*x/a+3*I*log(cos(c+d*x))/(a*d)-5/2*tan(c+d*x)/(a*d)+3/2*I*tan(c+d*x)^2/(a*d)+5/6*tan(c+d*x)^3/(a*d)-3/4*I*tan(c+d*x)^4/(a*d)-1/2*tan(c+d*x)^5/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x)),x,5,-5/2*I*x/a+2*log(cos(c+d*x))/(a*d)+5/2*I*tan(c+d*x)/(a*d)+tan(c+d*x)^2/(a*d)-5/6*I*tan(c+d*x)^3/(a*d)-1/2*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x)),x,4,-3/2*x/a-2*I*log(cos(c+d*x))/(a*d)+3/2*tan(c+d*x)/(a*d)-I*tan(c+d*x)^2/(a*d)-1/2*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x)),x,3,3/2*I*x/a-log(cos(c+d*x))/(a*d)-3/2*I*tan(c+d*x)/(a*d)-1/2*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x)),x,3,1/2*x/a+I*log(cos(c+d*x))/(a*d)+(-1/2*I)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)/(a+I*a*tan(c+d*x)),x,2,-1/2*I*x/a+(-1/2)/(d*(a+I*a*tan(c+d*x)))],
[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)))],
[cot(c+d*x)/(a+I*a*tan(c+d*x)),x,4,-1/2*I*x/a+log(sin(c+d*x))/(a*d)+1/2/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x)),x,4,-3/2*x/a-3/2*cot(c+d*x)/(a*d)-I*log(sin(c+d*x))/(a*d)+1/2*cot(c+d*x)/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^3/(a+I*a*tan(c+d*x)),x,5,3/2*I*x/a+3/2*I*cot(c+d*x)/(a*d)-cot(c+d*x)^2/(a*d)-2*log(sin(c+d*x))/(a*d)+1/2*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^4/(a+I*a*tan(c+d*x)),x,6,5/2*x/a+5/2*cot(c+d*x)/(a*d)+I*cot(c+d*x)^2/(a*d)-5/6*cot(c+d*x)^3/(a*d)+2*I*log(sin(c+d*x))/(a*d)+1/2*cot(c+d*x)^3/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^6/(a+I*a*tan(c+d*x))^2,x,6,-25/4*x/a^2-6*I*log(cos(c+d*x))/(a^2*d)+25/4*tan(c+d*x)/(a^2*d)-3*I*tan(c+d*x)^2/(a^2*d)-25/12*tan(c+d*x)^3/(a^2*d)+3/2*I*tan(c+d*x)^4/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x))^2,x,5,15/4*I*x/a^2-4*log(cos(c+d*x))/(a^2*d)-15/4*I*tan(c+d*x)/(a^2*d)-2*tan(c+d*x)^2/(a^2*d)+5/4*I*tan(c+d*x)^3/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^2,x,4,9/4*x/a^2+2*I*log(cos(c+d*x))/(a^2*d)-9/4*tan(c+d*x)/(a^2*d)+I*tan(c+d*x)^2/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^2,x,6,-3/4*I*x/a^2+log(cos(c+d*x))/(a^2*d)+(-3/4)/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x))^2,x,3,-1/4*x/a^2+3/4*I/(a^2*d*(1+I*tan(c+d*x)))+(-1/4*I)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^2,x,3,-1/4*I*x/a^2+(-1/4)/(d*(a+I*a*tan(c+d*x))^2)+1/4/(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)))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^2,x,4,-3/4*I*x/a^2+log(sin(c+d*x))/(a^2*d)+3/4/(a^2*d*(1+I*tan(c+d*x)))+1/4/(d*(a+I*a*tan(c+d*x))^2)],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^2,x,5,-9/4*x/a^2-9/4*cot(c+d*x)/(a^2*d)-2*I*log(sin(c+d*x))/(a^2*d)+cot(c+d*x)/(a^2*d*(1+I*tan(c+d*x)))+1/4*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^2)],
[cot(c+d*x)^3/(a+I*a*tan(c+d*x))^2,x,6,15/4*I*x/a^2+15/4*I*cot(c+d*x)/(a^2*d)-2*cot(c+d*x)^2/(a^2*d)-4*log(sin(c+d*x))/(a^2*d)+5/4*cot(c+d*x)^2/(a^2*d*(1+I*tan(c+d*x)))+1/4*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^6/(a+I*a*tan(c+d*x))^3,x,6,55/8*x/a^3+7*I*log(cos(c+d*x))/(a^3*d)-55/8*tan(c+d*x)/(a^3*d)+7/2*I*tan(c+d*x)^2/(a^3*d)-1/6*tan(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^3)+13/24*I*tan(c+d*x)^4/(a*d*(a+I*a*tan(c+d*x))^2)+55/24*tan(c+d*x)^3/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x))^3,x,5,-25/8*I*x/a^3+3*log(cos(c+d*x))/(a^3*d)+25/8*I*tan(c+d*x)/(a^3*d)-1/6*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^3)+11/24*I*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^2)+3/2*tan(c+d*x)^2/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^3,x,7,-7/8*x/a^3-I*log(cos(c+d*x))/(a^3*d)-1/6*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^3)+3/8*I*tan(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^2)+7/8*I/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^3,x,4,1/8*I*x/a^3+3/8/(a^3*d*(1+I*tan(c+d*x)))+1/6*I*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^3)+(-1/8)/(a*d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^2/(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)+3/8*I/(a*d*(a+I*a*tan(c+d*x))^2)+(-1/8*I)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^3,x,4,-1/8*I*x/a^3+(-1/6)/(d*(a+I*a*tan(c+d*x))^3)+1/8/(a*d*(a+I*a*tan(c+d*x))^2)+1/8/(d*(a^3+I*a^3*tan(c+d*x)))],
[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)))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^3,x,5,-7/8*I*x/a^3+log(sin(c+d*x))/(a^3*d)+1/6/(d*(a+I*a*tan(c+d*x))^3)+3/8/(a*d*(a+I*a*tan(c+d*x))^2)+7/8/(d*(a^3+I*a^3*tan(c+d*x)))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^3,x,6,-25/8*x/a^3-25/8*cot(c+d*x)/(a^3*d)-3*I*log(sin(c+d*x))/(a^3*d)+1/6*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^3)+11/24*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^2)+3/2*cot(c+d*x)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^6/(a+I*a*tan(c+d*x))^4,x,6,-65/16*x/a^4-4*I*log(cos(c+d*x))/(a^4*d)+65/16*tan(c+d*x)/(a^4*d)-2*I*tan(c+d*x)^2/(a^4*d*(1+I*tan(c+d*x)))+31/48*tan(c+d*x)^3/(a^4*d*(1+I*tan(c+d*x))^2)-1/8*tan(c+d*x)^5/(d*(a+I*a*tan(c+d*x))^4)+7/24*I*tan(c+d*x)^4/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x))^4,x,8,15/16*I*x/a^4-log(cos(c+d*x))/(a^4*d)+15/16/(a^4*d*(1+I*tan(c+d*x)))+7/16*tan(c+d*x)^2/(a^4*d*(1+I*tan(c+d*x))^2)-1/8*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^4)+1/4*I*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^4,x,5,1/16*x/a^4+(-3/16*I)/(a^4*d*(1+I*tan(c+d*x)))+1/8*I*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^4)+1/12*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)+1/16*I/(d*(a^2+I*a^2*tan(c+d*x))^2)],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^4,x,5,1/16*I*x/a^4+3/16/(a^4*d*(1+I*tan(c+d*x)))+1/8*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^4)+1/12*I*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)+(-1/16)/(d*(a^2+I*a^2*tan(c+d*x))^2)],
[tan(c+d*x)^2/(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)+1/4*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)))],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^4,x,5,-1/16*I*x/a^4+(-1/8)/(d*(a+I*a*tan(c+d*x))^4)+1/12/(a*d*(a+I*a*tan(c+d*x))^3)+1/16/(d*(a^2+I*a^2*tan(c+d*x))^2)+1/16/(d*(a^4+I*a^4*tan(c+d*x)))],
[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)))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^4,x,6,-15/16*I*x/a^4+log(sin(c+d*x))/(a^4*d)+7/16/(a^4*d*(1+I*tan(c+d*x))^2)+15/16/(a^4*d*(1+I*tan(c+d*x)))+1/8/(d*(a+I*a*tan(c+d*x))^4)+1/4/(a*d*(a+I*a*tan(c+d*x))^3)],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^4,x,7,-65/16*x/a^4-65/16*cot(c+d*x)/(a^4*d)-4*I*log(sin(c+d*x))/(a^4*d)+31/48*cot(c+d*x)/(a^4*d*(1+I*tan(c+d*x))^2)+2*cot(c+d*x)/(a^4*d*(1+I*tan(c+d*x)))+1/8*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^4)+7/24*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^3)],

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

# m>0
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^4,x,6,-I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d+8/35*I*sqrt(a+I*a*tan(c+d*x))/d-2/35*I*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d+2/7*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/d+62/105*I*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3,x,5,arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-8/5*sqrt(a+I*a*tan(c+d*x))/d+2/5*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d-2/15*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2,x,3,I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-2/3*I*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x),x,3,-arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d+2*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x)),x,2,-I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d],
[cot(c+d*x)*sqrt(a+I*a*tan(c+d*x)),x,6,-2*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d+arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d],
[cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x)),x,8,-I*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d+I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x)),x,8,7/4*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d-arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-1/4*I*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/2*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2),x,7,2*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-64/35*a*sqrt(a+I*a*tan(c+d*x))/d+16/35*a*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d+2/7*I*a^2*tan(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-2/7*a^2*tan(c+d*x)^4/(d*sqrt(a+I*a*tan(c+d*x)))-76/105*(a+I*a*tan(c+d*x))^(3/2)/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2),x,4,2*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-2*I*a*sqrt(a+I*a*tan(c+d*x))/d-2/5*I*(a+I*a*tan(c+d*x))^(5/2)/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(3/2),x,4,-2*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*a*sqrt(a+I*a*tan(c+d*x))/d+2/3*(a+I*a*tan(c+d*x))^(3/2)/d],
[(a+I*a*tan(c+d*x))^(3/2),x,3,-2*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*I*a*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(3/2),x,6,-2*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+2*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2),x,8,-3*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+2*I*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-I*a^2/(d*sqrt(a+I*a*tan(c+d*x)))-a^2*cot(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2),x,9,11/4*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-2*a^(3/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-1/2*I*a^2*cot(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-1/2*a^2*cot(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-5/4*I*a*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2),x,7,4*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-368/105*a^2*sqrt(a+I*a*tan(c+d*x))/d+92/105*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d+38/63*I*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/d-2/9*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^4/d-472/315*a*(a+I*a*tan(c+d*x))^(3/2)/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2),x,5,4*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-4*I*a^2*sqrt(a+I*a*tan(c+d*x))/d-2/3*I*a*(a+I*a*tan(c+d*x))^(3/2)/d-2/7*I*(a+I*a*tan(c+d*x))^(7/2)/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(5/2),x,5,-4*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+4*a^2*sqrt(a+I*a*tan(c+d*x))/d+2/3*a*(a+I*a*tan(c+d*x))^(3/2)/d+2/5*(a+I*a*tan(c+d*x))^(5/2)/d],
[(a+I*a*tan(c+d*x))^(5/2),x,4,-4*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+4*I*a^2*sqrt(a+I*a*tan(c+d*x))/d+2/3*I*a*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(5/2),x,7,-2*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+4*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-2*a^2*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2),x,7,-5*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+4*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-a^2*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2),x,8,23/4*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-4*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-9/4*I*a^2*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/2*a^2*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^(5/2),x,9,45/8*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-4*I*a^(5/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+19/8*a^2*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-13/12*I*a^2*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d-1/3*a^2*cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^(7/2),x,5,-8*I*a^(7/2)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+8*I*a^3*sqrt(a+I*a*tan(c+d*x))/d+4/3*I*a^2*(a+I*a*tan(c+d*x))^(3/2)/d+2/5*I*a*(a+I*a*tan(c+d*x))^(5/2)/d],

# m<0
[tan(c+d*x)^5/sqrt(a+I*a*tan(c+d*x)),x,7,-arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))-188/35*sqrt(a+I*a*tan(c+d*x))/(a*d)+47/35*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/(a*d)-9/7*I*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/(a*d)-tan(c+d*x)^4/(d*sqrt(a+I*a*tan(c+d*x)))+223/105*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)],
[tan(c+d*x)^4/sqrt(a+I*a*tan(c+d*x)),x,6,-I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+28/5*I*sqrt(a+I*a*tan(c+d*x))/(a*d)-7/5*I*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/(a*d)-tan(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-23/15*I*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)],
[tan(c+d*x)^3/sqrt(a+I*a*tan(c+d*x)),x,5,arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+4*sqrt(a+I*a*tan(c+d*x))/(a*d)-tan(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))-5/3*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)],
[tan(c+d*x)^2/sqrt(a+I*a*tan(c+d*x)),x,4,I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-I)/(d*sqrt(a+I*a*tan(c+d*x)))-2*I*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[tan(c+d*x)/sqrt(a+I*a*tan(c+d*x)),x,3,-arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-1)/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/sqrt(a+I*a*tan(c+d*x)),x,3,-I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+I/(d*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)/sqrt(a+I*a*tan(c+d*x)),x,7,-2*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+1/(d*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^2/sqrt(a+I*a*tan(c+d*x)),x,8,I*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+cot(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-2*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cot(c+d*x)^3/sqrt(a+I*a*tan(c+d*x)),x,9,11/4*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))-arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+cot(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))+7/4*I*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)-3/2*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x))^(3/2),x,7,-1/2*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+78/5*sqrt(a+I*a*tan(c+d*x))/(a^2*d)-39/10*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/(a^2*d)+19/6*I*tan(c+d*x)^3/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/3*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^(3/2))-151/30*(a+I*a*tan(c+d*x))^(3/2)/(a^3*d)],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^(3/2),x,6,-1/2*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))-10*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+5/2*I*tan(c+d*x)^2/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/3*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))+7/2*I*(a+I*a*tan(c+d*x))^(3/2)/(a^3*d)],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^(3/2),x,5,1/2*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-11/6)/(a*d*sqrt(a+I*a*tan(c+d*x)))-7/3*sqrt(a+I*a*tan(c+d*x))/(a^2*d)-1/3*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x))^(3/2),x,4,1/2*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+3/2*I/(a*d*sqrt(a+I*a*tan(c+d*x)))+(-1/3*I)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^(3/2),x,4,-1/2*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/2/(a*d*sqrt(a+I*a*tan(c+d*x)))+(-1/3)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(a+I*a*tan(c+d*x))^(3/2),x,4,-1/2*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/2*I/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*I/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^(3/2),x,8,-2*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/2*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+3/2/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^(3/2),x,9,3*I*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/2*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+13/6*cot(c+d*x)/(a*d*sqrt(a+I*a*tan(c+d*x)))-7/2*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^3/(a+I*a*tan(c+d*x))^(3/2),x,10,23/4*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-1/2*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+17/6*cot(c+d*x)^2/(a*d*sqrt(a+I*a*tan(c+d*x)))+21/4*I*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)-11/3*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^5/(a+I*a*tan(c+d*x))^(5/2),x,7,-1/4*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))-89/5*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+89/20*tan(c+d*x)^2/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1/5*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^(5/2))+7/10*I*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(3/2))+361/60*(a+I*a*tan(c+d*x))^(3/2)/(a^4*d)],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^(5/2),x,6,-1/4*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+151/60*I/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+83/30*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d)-1/5*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(5/2))+17/30*I*tan(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^(5/2),x,5,1/4*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+31/20/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1/5*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(5/2))+(-13/30)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x))^(5/2),x,5,1/4*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+(-1/4*I)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+(-1/5*I)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/2*I/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^(5/2),x,5,-1/4*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/4/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+(-1/5)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(a+I*a*tan(c+d*x))^(5/2),x,5,-1/4*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/4*I/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*I/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*I/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^(5/2),x,9,-2*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+1/4*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+7/4/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5/(d*(a+I*a*tan(c+d*x))^(5/2))+1/2/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^(5/2),x,10,5*I*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+1/4*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+41/12*cot(c+d*x)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-21/4*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(5/2))+19/30*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(a+I*a*tan(c+d*x))^(7/2),x,6,-1/8*I*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(7/2)*d*sqrt(2))+1/8*I/(a^3*d*sqrt(a+I*a*tan(c+d*x)))+1/7*I/(d*(a+I*a*tan(c+d*x))^(7/2))+1/10*I/(a*d*(a+I*a*tan(c+d*x))^(5/2))+1/12*I/(a^2*d*(a+I*a*tan(c+d*x))^(3/2))],

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

# m>0
[(d*tan(e+f*x))^(5/2)*(a+I*a*tan(e+f*x)),x,5,-2*(-1)^(3/4)*a*d^(5/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f-2*I*a*d^2*sqrt(d*tan(e+f*x))/f+2/3*a*d*(d*tan(e+f*x))^(3/2)/f+2/5*I*a*(d*tan(e+f*x))^(5/2)/f],
[(d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x)),x,4,2*(-1)^(1/4)*a*d^(3/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f+2*a*d*sqrt(d*tan(e+f*x))/f+2/3*I*a*(d*tan(e+f*x))^(3/2)/f],
[sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x)),x,3,2*(-1)^(3/4)*a*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/f+2*I*a*sqrt(d*tan(e+f*x))/f],
[(a+I*a*tan(e+f*x))/sqrt(d*tan(e+f*x)),x,2,-2*(-1)^(1/4)*a*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(d))],
[(a+I*a*tan(e+f*x))/(d*tan(e+f*x))^(3/2),x,3,-2*(-1)^(3/4)*a*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(3/2)*f)-2*a/(d*f*sqrt(d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))/(d*tan(e+f*x))^(5/2),x,4,2*(-1)^(1/4)*a*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(5/2)*f)-2*I*a/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*a/(d*f*(d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))/(d*tan(e+f*x))^(7/2),x,5,2*(-1)^(3/4)*a*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(7/2)*f)+2*a/(d^3*f*sqrt(d*tan(e+f*x)))-2/5*a/(d*f*(d*tan(e+f*x))^(5/2))-2/3*I*a/(d^2*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(5/2)*(a-I*a*tan(e+f*x)),x,5,2*(-1)^(3/4)*a*d^(5/2)*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f+2*I*a*d^2*sqrt(d*tan(e+f*x))/f+2/3*a*d*(d*tan(e+f*x))^(3/2)/f-2/5*I*a*(d*tan(e+f*x))^(5/2)/f],
[(d*tan(e+f*x))^(3/2)*(a-I*a*tan(e+f*x)),x,4,2*(-1)^(1/4)*a*d^(3/2)*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f+2*a*d*sqrt(d*tan(e+f*x))/f-2/3*I*a*(d*tan(e+f*x))^(3/2)/f],
[sqrt(d*tan(e+f*x))*(a-I*a*tan(e+f*x)),x,3,-2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/f-2*I*a*sqrt(d*tan(e+f*x))/f],
[(a-I*a*tan(e+f*x))/sqrt(d*tan(e+f*x)),x,2,-2*(-1)^(1/4)*a*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(d))],
[(a-I*a*tan(e+f*x))/(d*tan(e+f*x))^(3/2),x,3,2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(3/2)*f)-2*a/(d*f*sqrt(d*tan(e+f*x)))],
[(a-I*a*tan(e+f*x))/(d*tan(e+f*x))^(5/2),x,4,2*(-1)^(1/4)*a*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(5/2)*f)+2*I*a/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*a/(d*f*(d*tan(e+f*x))^(3/2))],
[(a-I*a*tan(e+f*x))/(d*tan(e+f*x))^(7/2),x,5,-2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(7/2)*f)+2*a/(d^3*f*sqrt(d*tan(e+f*x)))-2/5*a/(d*f*(d*tan(e+f*x))^(5/2))+2/3*I*a/(d^2*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(5/2)*(a+I*a*tan(e+f*x))^2,x,6,-4*(-1)^(3/4)*a^2*d^(5/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f-4*I*a^2*d^2*sqrt(d*tan(e+f*x))/f+4/3*a^2*d*(d*tan(e+f*x))^(3/2)/f+4/5*I*a^2*(d*tan(e+f*x))^(5/2)/f-2/7*a^2*(d*tan(e+f*x))^(7/2)/(d*f)],
[(d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))^2,x,5,4*(-1)^(1/4)*a^2*d^(3/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f+4*a^2*d*sqrt(d*tan(e+f*x))/f+4/3*I*a^2*(d*tan(e+f*x))^(3/2)/f-2/5*a^2*(d*tan(e+f*x))^(5/2)/(d*f)],
[sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2,x,4,4*(-1)^(3/4)*a^2*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/f+4*I*a^2*sqrt(d*tan(e+f*x))/f-2/3*a^2*(d*tan(e+f*x))^(3/2)/(d*f)],
[(a+I*a*tan(e+f*x))^2/sqrt(d*tan(e+f*x)),x,3,-4*(-1)^(1/4)*a^2*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(d))-2*a^2*sqrt(d*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^2/(d*tan(e+f*x))^(3/2),x,3,-4*(-1)^(3/4)*a^2*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(3/2)*f)-2*a^2/(d*f*sqrt(d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(d*tan(e+f*x))^(5/2),x,4,4*(-1)^(1/4)*a^2*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(5/2)*f)-4*I*a^2/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*a^2/(d*f*(d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2/(d*tan(e+f*x))^(7/2),x,5,4*(-1)^(3/4)*a^2*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(7/2)*f)+4*a^2/(d^3*f*sqrt(d*tan(e+f*x)))-2/5*a^2/(d*f*(d*tan(e+f*x))^(5/2))-4/3*I*a^2/(d^2*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(5/2)*(a+I*a*tan(e+f*x))^3,x,7,-8*(-1)^(3/4)*a^3*d^(5/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f-8*I*a^3*d^2*sqrt(d*tan(e+f*x))/f+8/3*a^3*d*(d*tan(e+f*x))^(3/2)/f+8/5*I*a^3*(d*tan(e+f*x))^(5/2)/f-40/63*a^3*(d*tan(e+f*x))^(7/2)/(d*f)-2/9*(d*tan(e+f*x))^(7/2)*(a^3+I*a^3*tan(e+f*x))/(d*f)],
[(d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))^3,x,6,8*(-1)^(1/4)*a^3*d^(3/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/f+8*a^3*d*sqrt(d*tan(e+f*x))/f+8/3*I*a^3*(d*tan(e+f*x))^(3/2)/f-32/35*a^3*(d*tan(e+f*x))^(5/2)/(d*f)-2/7*(d*tan(e+f*x))^(5/2)*(a^3+I*a^3*tan(e+f*x))/(d*f)],
[sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3,x,5,8*(-1)^(3/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/f+8*I*a^3*sqrt(d*tan(e+f*x))/f-8/5*a^3*(d*tan(e+f*x))^(3/2)/(d*f)-2/5*(d*tan(e+f*x))^(3/2)*(a^3+I*a^3*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^3/sqrt(d*tan(e+f*x)),x,4,-8*(-1)^(1/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(d))-16/3*a^3*sqrt(d*tan(e+f*x))/(d*f)-2/3*sqrt(d*tan(e+f*x))*(a^3+I*a^3*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^3/(d*tan(e+f*x))^(3/2),x,4,-8*(-1)^(3/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(3/2)*f)-2*(a^3+I*a^3*tan(e+f*x))/(d*f*sqrt(d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(d*tan(e+f*x))^(5/2),x,4,8*(-1)^(1/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(5/2)*f)-16/3*I*a^3/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*(a^3+I*a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^3/(d*tan(e+f*x))^(7/2),x,5,8*(-1)^(3/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(7/2)*f)+8*a^3/(d^3*f*sqrt(d*tan(e+f*x)))-8/5*I*a^3/(d^2*f*(d*tan(e+f*x))^(3/2))-2/5*(a^3+I*a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^3/(d*tan(e+f*x))^(9/2),x,6,-8*(-1)^(1/4)*a^3*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(d^(9/2)*f)+8*I*a^3/(d^4*f*sqrt(d*tan(e+f*x)))-32/35*I*a^3/(d^2*f*(d*tan(e+f*x))^(5/2))+8/3*a^3/(d^3*f*(d*tan(e+f*x))^(3/2))-2/7*(a^3+I*a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(7/2))],

# m<0
[(d*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x)),x,13,(5/4-7/4*I)*d^(7/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(-5/4+7/4*I)*d^(7/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(5/8+7/8*I)*d^(7/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))+(-5/8-7/8*I)*d^(7/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))+5/2*d^3*sqrt(d*tan(e+f*x))/(a*f)-7/6*I*d^2*(d*tan(e+f*x))^(3/2)/(a*f)-1/2*d*(d*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x)),x,12,(-3/4-5/4*I)*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(3/4+5/4*I)*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(3/8-5/8*I)*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))+(-3/8+5/8*I)*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))-5/2*I*d^2*sqrt(d*tan(e+f*x))/(a*f)-1/2*d*(d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x)),x,11,(-1/4+3/4*I)*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(1/4-3/4*I)*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2))+(-1/8-3/8*I)*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))+(1/8+3/8*I)*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2))-1/2*d*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x)))],
[sqrt(d*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,3,1/2*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a*f)+1/2*I*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x)))],
[1/(sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))),x,11,(-3/4+1/4*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2)*sqrt(d))+(3/4-1/4*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(2)*sqrt(d))+(-3/8-1/8*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2)*sqrt(d))+(3/8+1/8*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*f*sqrt(2)*sqrt(d))+1/2*sqrt(d*tan(e+f*x))/(d*f*(a+I*a*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))),x,12,(5/4+3/4*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(3/2)*f*sqrt(2))+(-5/4-3/4*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(3/2)*f*sqrt(2))+(-5/8+3/8*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*d^(3/2)*f*sqrt(2))+(5/8-3/8*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*d^(3/2)*f*sqrt(2))+(-5/2)/(a*d*f*sqrt(d*tan(e+f*x)))+1/2/(d*f*sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(5/2)*(a+I*a*tan(e+f*x))),x,13,(7/4-5/4*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(5/2)*f*sqrt(2))+(-7/4+5/4*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(5/2)*f*sqrt(2))+(7/8+5/8*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*d^(5/2)*f*sqrt(2))+(-7/8-5/8*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a*d^(5/2)*f*sqrt(2))+5/2*I/(a*d^2*f*sqrt(d*tan(e+f*x)))+(-7/6)/(a*d*f*(d*tan(e+f*x))^(3/2))+1/2/(d*f*(d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^(9/2)/(a+I*a*tan(e+f*x))^2,x,14,(-49/16-45/16*I)*d^(9/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(49/16+45/16*I)*d^(9/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(49/32-45/32*I)*d^(9/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+(-49/32+45/32*I)*d^(9/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))-45/8*I*d^4*sqrt(d*tan(e+f*x))/(a^2*f)-49/24*d^3*(d*tan(e+f*x))^(3/2)/(a^2*f)+9/8*I*d^2*(d*tan(e+f*x))^(5/2)/(a^2*f*(1+I*tan(e+f*x)))-1/4*d*(d*tan(e+f*x))^(7/2)/(f*(a+I*a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^2,x,13,(-25/16+21/16*I)*d^(7/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(25/16-21/16*I)*d^(7/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(-25/32-21/32*I)*d^(7/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+(25/32+21/32*I)*d^(7/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))-25/8*d^3*sqrt(d*tan(e+f*x))/(a^2*f)+7/8*I*d^2*(d*tan(e+f*x))^(3/2)/(a^2*f*(1+I*tan(e+f*x)))-1/4*d*(d*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^2,x,12,(9/16+5/16*I)*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(-9/16-5/16*I)*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(-9/32+5/32*I)*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+(9/32-5/32*I)*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+5/8*I*d^2*sqrt(d*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))-1/4*d*(d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^2,x,12,(1/16+3/16*I)*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(-1/16-3/16*I)*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+(1/32-3/32*I)*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+(-1/32+3/32*I)*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+3/8*d*sqrt(d*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))-1/4*d*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^2)],
[sqrt(d*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,12,(-1/16+3/16*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^2*f*sqrt(2))+(1/16-3/16*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^2*f*sqrt(2))+(1/32+3/32*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^2*f*sqrt(2))+(-1/32-3/32*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^2*f*sqrt(2))+1/8*I*sqrt(d*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))+1/4*I*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^2)],
[1/(sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2),x,12,(-9/16+5/16*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2)*sqrt(d))+(9/16-5/16*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2)*sqrt(d))+(-9/32-5/32*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2)*sqrt(d))+(9/32+5/32*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2)*sqrt(d))+5/8*sqrt(d*tan(e+f*x))/(a^2*d*f*(1+I*tan(e+f*x)))+1/4*sqrt(d*tan(e+f*x))/(d*f*(a+I*a*tan(e+f*x))^2)],
[1/((d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))^2),x,13,(25/16+21/16*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(3/2)*f*sqrt(2))+(-25/16-21/16*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(3/2)*f*sqrt(2))+(-25/32+21/32*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(3/2)*f*sqrt(2))+(25/32-21/32*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(3/2)*f*sqrt(2))+(-25/8)/(a^2*d*f*sqrt(d*tan(e+f*x)))+7/8/(a^2*d*f*sqrt(d*tan(e+f*x))*(1+I*tan(e+f*x)))+1/4/(d*f*sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2)],
[1/((d*tan(e+f*x))^(5/2)*(a+I*a*tan(e+f*x))^2),x,14,(49/16-45/16*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(5/2)*f*sqrt(2))+(-49/16+45/16*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(5/2)*f*sqrt(2))+(49/32+45/32*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(5/2)*f*sqrt(2))+(-49/32-45/32*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(5/2)*f*sqrt(2))+45/8*I/(a^2*d^2*f*sqrt(d*tan(e+f*x)))+(-49/24)/(a^2*d*f*(d*tan(e+f*x))^(3/2))+9/8/(a^2*d*f*(1+I*tan(e+f*x))*(d*tan(e+f*x))^(3/2))+1/4/(d*f*(d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(9/2)/(a+I*a*tan(e+f*x))^3,x,14,(7/4+15/8*I)*d^(9/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))+(-7/4-15/8*I)*d^(9/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))+(-7/8+15/16*I)*d^(9/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))+(7/8-15/16*I)*d^(9/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))+15/4*I*d^4*sqrt(d*tan(e+f*x))/(a^3*f)-1/6*d*(d*tan(e+f*x))^(7/2)/(f*(a+I*a*tan(e+f*x))^3)+5/12*I*d^2*(d*tan(e+f*x))^(5/2)/(a*f*(a+I*a*tan(e+f*x))^2)+7/6*d^3*(d*tan(e+f*x))^(3/2)/(f*(a^3+I*a^3*tan(e+f*x)))],
[(d*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^3,x,13,(5/16-7/16*I)*d^(7/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))+(-5/16+7/16*I)*d^(7/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))+(5/32+7/32*I)*d^(7/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))+(-5/32-7/32*I)*d^(7/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))-1/6*d*(d*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^3)+1/3*I*d^2*(d*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^2)+5/8*d^3*sqrt(d*tan(e+f*x))/(f*(a^3+I*a^3*tan(e+f*x)))],
[(d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^3,x,16,1/8*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))-1/8*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2))-1/16*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))+1/16*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2))-1/6*d*(d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^3)+1/4*I*d^2*sqrt(d*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^2)-1/4*I*d^2*sqrt(d*tan(e+f*x))/(f*(a^3+I*a^3*tan(e+f*x)))],
[(d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^3,x,7,1/8*(-1)^(1/4)*d^(3/2)*arctan((-1)^(3/4)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f)-1/6*d*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^3)+1/6*d*sqrt(d*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^2)+1/8*d*sqrt(d*tan(e+f*x))/(f*(a^3+I*a^3*tan(e+f*x)))],
[sqrt(d*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,14,1/8*I*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^3*f*sqrt(2))-1/8*I*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^3*f*sqrt(2))+1/16*I*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^3*f*sqrt(2))-1/16*I*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^3*f*sqrt(2))+1/6*I*sqrt(d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^3)+1/12*I*sqrt(d*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^2)],
[1/(sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3),x,13,(-7/16+5/16*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2)*sqrt(d))+(7/16-5/16*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(2)*sqrt(d))+(-7/32-5/32*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2)*sqrt(d))+(7/32+5/32*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*f*sqrt(2)*sqrt(d))+1/6*sqrt(d*tan(e+f*x))/(d*f*(a+I*a*tan(e+f*x))^3)+1/3*sqrt(d*tan(e+f*x))/(a*d*f*(a+I*a*tan(e+f*x))^2)+5/8*sqrt(d*tan(e+f*x))/(d*f*(a^3+I*a^3*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(3/2)*(a+I*a*tan(e+f*x))^3),x,14,(15/8+7/4*I)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*d^(3/2)*f*sqrt(2))+(-15/8-7/4*I)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*d^(3/2)*f*sqrt(2))+(-15/16+7/8*I)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*d^(3/2)*f*sqrt(2))+(15/16-7/8*I)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^3*d^(3/2)*f*sqrt(2))+(-15/4)/(a^3*d*f*sqrt(d*tan(e+f*x)))+1/6/(d*f*sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3)+5/12/(a*d*f*sqrt(d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2)+7/6/(d*f*sqrt(d*tan(e+f*x))*(a^3+I*a^3*tan(e+f*x)))],

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

# m>0
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2),x,9,7/4*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d-1/4*I*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2),x,9,-(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(-1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)),x,7,-2*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(-1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d],
[sqrt(a+I*a*tan(c+d*x))/sqrt(tan(c+d*x)),x,2,(1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d],
[sqrt(a+I*a*tan(c+d*x))/tan(c+d*x)^(3/2),x,3,(1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d-2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[sqrt(a+I*a*tan(c+d*x))/tan(c+d*x)^(5/2),x,5,(-1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d-2/3*I*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[sqrt(a+I*a*tan(c+d*x))/tan(c+d*x)^(7/2),x,6,(-1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+26/15*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/5*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*I*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2),x,11,23/8*(-1)^(3/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-9/8*I*a*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+7/12*a*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d+1/3*I*a^2*tan(c+d*x)^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))-1/3*a^2*tan(c+d*x)^(7/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2),x,10,-11/4*(-1)^(1/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+5/4*a*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*I*a^2*tan(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-1/2*a^2*tan(c+d*x)^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2),x,9,-3*(-1)^(3/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+I*a^2*sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))-a^2*tan(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/sqrt(tan(c+d*x)),x,7,2*(-1)^(1/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(3/2),x,3,(2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*a*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(5/2),x,4,(-2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*I*a*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(7/2),x,7,(-2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+12/5*a*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/5*a^2/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2))-2/5*I*a^2/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-4/5*I*a*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(9/2),x,8,(2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+268/105*I*a*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/7*a^2/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(7/2))-2/7*I*a^2/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2))-16/35*I*a*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+76/105*a*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2),x,11,363/64*(-1)^(3/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-149/64*I*a^2*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+107/96*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d+17/24*I*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)/d-1/4*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(7/2)/d],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2),x,10,-45/8*(-1)^(1/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+19/8*a^2*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+13/12*I*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d-1/3*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2),x,9,-23/4*(-1)^(3/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+9/4*I*a^2*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-1/2*a^2*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[(a+I*a*tan(c+d*x))^(5/2)/sqrt(tan(c+d*x)),x,8,5*(-1)^(1/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-a^2*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^(5/2)/tan(c+d*x)^(3/2),x,8,2*(-1)^(3/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(5/2)/tan(c+d*x)^(5/2),x,4,(-4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-4*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(5/2)/tan(c+d*x)^(7/2),x,5,(-4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+4*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*I*a*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))-2/5*(a+I*a*tan(c+d*x))^(5/2)/(d*tan(c+d*x)^(5/2))],
[(a+I*a*tan(c+d*x))^(5/2)/tan(c+d*x)^(9/2),x,7,(4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+104/21*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/7*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-6/7*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+32/21*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(5/2)/tan(c+d*x)^(11/2),x,8,(4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-1576/315*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/9*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(9/2))-38/63*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+92/105*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+472/315*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],

# m<0
[tan(c+d*x)^(7/2)/sqrt(a+I*a*tan(c+d*x)),x,10,11/4*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+7/4*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)-3/2*I*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/(a*d)-tan(c+d*x)^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(5/2)/sqrt(a+I*a*tan(c+d*x)),x,9,-(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))-2*I*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)-tan(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(3/2)/sqrt(a+I*a*tan(c+d*x)),x,8,-2*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(-1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))-sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(1/2)/sqrt(a+I*a*tan(c+d*x)),x,3,(-1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+I*sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/2)),x,4,(1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)),x,5,(1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+1/(d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-3*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)),x,6,(-1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+7/3*I*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))+1/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-5/3*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(7/2)),x,7,(-1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+61/15*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))+1/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2))-7/5*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+23/15*I*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(7/2)/(a+I*a*tan(c+d*x))^(3/2),x,10,-3*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+(1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)-7/2*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+13/6*I*tan(c+d*x)^(3/2)/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/3*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(5/2)/(a+I*a*tan(c+d*x))^(3/2),x,9,2*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+(1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+3/2*I*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/3*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)/(a+I*a*tan(c+d*x))^(3/2),x,4,(-1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/2*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*I*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(1/2)/(a+I*a*tan(c+d*x))^(3/2),x,4,(-1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/2*I*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^(3/2)),x,5,(1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+7/6*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)),x,6,(1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+11/6/(a*d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-25/6*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))+1/3/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)),x,7,(-1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+13/2*I*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))+5/2/(a*d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-7/2*sqrt(a+I*a*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))+1/3/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(9/2)/(a+I*a*tan(c+d*x))^(5/2),x,11,5*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+(-1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+21/4*I*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+41/12*tan(c+d*x)^(3/2)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1/5*tan(c+d*x)^(7/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+19/30*I*tan(c+d*x)^(5/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(7/2)/(a+I*a*tan(c+d*x))^(5/2),x,10,2*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+(1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+7/4*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1/5*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/2*I*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(5/2)/(a+I*a*tan(c+d*x))^(5/2),x,5,(1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)-1/4*I*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*I*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)/(a+I*a*tan(c+d*x))^(5/2),x,5,(-1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+1/4*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*I*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(1/2)/(a+I*a*tan(c+d*x))^(5/2),x,6,(-1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)-1/20*I*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*I*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+1/10*I*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^(5/2)),x,6,(1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+67/60*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+13/30*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)),x,7,(1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+151/60/(a^2*d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-317/60*sqrt(a+I*a*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))+1/5/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+17/30/(a*d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)),x,8,(-1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+707/60*I*sqrt(a+I*a*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))+89/20/(a^2*d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-361/60*sqrt(a+I*a*tan(c+d*x))/(a^3*d*tan(c+d*x)^(3/2))+1/5/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2))+7/10/(a*d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],

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

# m>0

# m<0
[tan(c+d*x)^(10/3)/(a+I*a*tan(c+d*x)),x,25,7/12*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)-7/12*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)-7/6*arctan(tan(c+d*x)^(1/3))/(a*d)-5/6*I*log(1+tan(c+d*x)^(2/3))/(a*d)+5/12*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)-5/2*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))+7/8*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-7/8*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+7/2*tan(c+d*x)^(1/3)/(a*d)-5/4*I*tan(c+d*x)^(4/3)/(a*d)-1/2*tan(c+d*x)^(7/3)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(8/3)/(a+I*a*tan(c+d*x)),x,24,-5/12*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)+5/12*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)+5/6*arctan(tan(c+d*x)^(1/3))/(a*d)+2/3*I*log(1+tan(c+d*x)^(2/3))/(a*d)-1/3*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)-2*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))+5/8*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-5/8*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-2*I*tan(c+d*x)^(2/3)/(a*d)-1/2*tan(c+d*x)^(5/3)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(4/3)/(a+I*a*tan(c+d*x)),x,23,-1/12*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)+1/12*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)+1/6*arctan(tan(c+d*x)^(1/3))/(a*d)+1/3*I*log(1+tan(c+d*x)^(2/3))/(a*d)-1/6*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)+I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))-1/8*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+1/8*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-1/2*tan(c+d*x)^(1/3)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(2/3)/(a+I*a*tan(c+d*x)),x,23,-1/12*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)+1/12*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)+1/6*arctan(tan(c+d*x)^(1/3))/(a*d)-1/6*I*log(1+tan(c+d*x)^(2/3))/(a*d)+1/12*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)+1/2*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))+1/8*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-1/8*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+1/2*I*tan(c+d*x)^(2/3)/(d*(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(1/3)*(a+I*a*tan(c+d*x))),x,23,1/12*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)-1/12*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)-1/6*I*arctan(tan(c+d*x)^(1/3))/(a*d)+1/3*log(1+tan(c+d*x)^(2/3))/(a*d)-1/6*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)-arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))-1/8*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+1/8*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+1/2*tan(c+d*x)^(2/3)/(d*(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(5/3)*(a+I*a*tan(c+d*x))),x,24,5/12*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)-5/12*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)-5/6*I*arctan(tan(c+d*x)^(1/3))/(a*d)+2/3*log(1+tan(c+d*x)^(2/3))/(a*d)-1/3*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)+2*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))+5/8*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-5/8*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+(-2)/(a*d*tan(c+d*x)^(2/3))+1/2/(d*tan(c+d*x)^(2/3)*(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(7/3)*(a+I*a*tan(c+d*x))),x,25,-7/12*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a*d)+7/12*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a*d)+7/6*I*arctan(tan(c+d*x)^(1/3))/(a*d)-5/6*log(1+tan(c+d*x)^(2/3))/(a*d)+5/12*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a*d)+5/2*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a*d*sqrt(3))+7/8*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))-7/8*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a*d*sqrt(3))+(-5/4)/(a*d*tan(c+d*x)^(4/3))+7/2*I/(a*d*tan(c+d*x)^(1/3))+1/2/(d*tan(c+d*x)^(4/3)*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(14/3)/(a+I*a*tan(c+d*x))^2,x,26,-121/72*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)+121/72*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)+121/36*arctan(tan(c+d*x)^(1/3))/(a^2*d)+14/9*I*log(1+tan(c+d*x)^(2/3))/(a^2*d)-7/9*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)-14/3*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))+121/48*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-121/48*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-14/3*I*tan(c+d*x)^(2/3)/(a^2*d)-121/60*tan(c+d*x)^(5/3)/(a^2*d)+7/6*I*tan(c+d*x)^(8/3)/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^(11/3)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(10/3)/(a+I*a*tan(c+d*x))^2,x,25,-49/72*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)+49/72*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)+49/36*arctan(tan(c+d*x)^(1/3))/(a^2*d)+5/9*I*log(1+tan(c+d*x)^(2/3))/(a^2*d)-5/18*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+5/3*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))-49/48*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+49/48*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-49/12*tan(c+d*x)^(1/3)/(a^2*d)+5/6*I*tan(c+d*x)^(4/3)/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^(7/3)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(8/3)/(a+I*a*tan(c+d*x))^2,x,24,25/72*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)-25/72*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)-25/36*arctan(tan(c+d*x)^(1/3))/(a^2*d)-2/9*I*log(1+tan(c+d*x)^(2/3))/(a^2*d)+1/9*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+2/3*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))-25/48*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+25/48*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+2/3*I*tan(c+d*x)^(2/3)/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^(5/3)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(4/3)/(a+I*a*tan(c+d*x))^2,x,24,1/72*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)-1/72*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)-1/36*arctan(tan(c+d*x)^(1/3))/(a^2*d)+1/9*I*log(1+tan(c+d*x)^(2/3))/(a^2*d)-1/18*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+1/3*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))+1/48*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-1/48*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+1/3*tan(c+d*x)^(1/3)/(a^2*d*(1+I*tan(c+d*x)))-1/4*tan(c+d*x)^(1/3)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(2/3)/(a+I*a*tan(c+d*x))^2,x,24,-1/72*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)+1/72*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)+1/36*arctan(tan(c+d*x)^(1/3))/(a^2*d)-1/9*I*log(1+tan(c+d*x)^(2/3))/(a^2*d)+1/18*I*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+1/3*I*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))+1/48*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-1/48*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+1/3*I*tan(c+d*x)^(2/3)/(a^2*d*(1+I*tan(c+d*x)))+1/4*tan(c+d*x)^(5/3)/(d*(a+I*a*tan(c+d*x))^2)],
[1/(tan(c+d*x)^(1/3)*(a+I*a*tan(c+d*x))^2),x,24,7/72*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)-7/72*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)-7/36*I*arctan(tan(c+d*x)^(1/3))/(a^2*d)+2/9*log(1+tan(c+d*x)^(2/3))/(a^2*d)-1/9*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)-2/3*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))-7/48*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+7/48*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+7/12*tan(c+d*x)^(2/3)/(a^2*d*(1+I*tan(c+d*x)))+1/4*tan(c+d*x)^(2/3)/(d*(a+I*a*tan(c+d*x))^2)],
[1/(tan(c+d*x)^(5/3)*(a+I*a*tan(c+d*x))^2),x,25,55/72*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)-55/72*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)-55/36*I*arctan(tan(c+d*x)^(1/3))/(a^2*d)+8/9*log(1+tan(c+d*x)^(2/3))/(a^2*d)-4/9*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+8/3*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))+55/48*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-55/48*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+(-8/3)/(a^2*d*tan(c+d*x)^(2/3))+11/12/(a^2*d*(1+I*tan(c+d*x))*tan(c+d*x)^(2/3))+1/4/(d*tan(c+d*x)^(2/3)*(a+I*a*tan(c+d*x))^2)],
[1/(tan(c+d*x)^(7/3)*(a+I*a*tan(c+d*x))^2),x,26,-91/72*I*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/(a^2*d)+91/72*I*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/(a^2*d)+91/36*I*arctan(tan(c+d*x)^(1/3))/(a^2*d)-25/18*log(1+tan(c+d*x)^(2/3))/(a^2*d)+25/36*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/(a^2*d)+25/6*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))/(a^2*d*sqrt(3))+91/48*I*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))-91/48*I*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))/(a^2*d*sqrt(3))+(-25/12)/(a^2*d*tan(c+d*x)^(4/3))+13/12/(a^2*d*(1+I*tan(c+d*x))*tan(c+d*x)^(4/3))+91/12*I/(a^2*d*tan(c+d*x)^(1/3))+1/4/(d*tan(c+d*x)^(4/3)*(a+I*a*tan(c+d*x))^2)],

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

# m>0
[tan(c+d*x)^(4/3)*(a+I*a*tan(c+d*x))^(1/2),x,4,3/7*a*AppellF1(7/3,1/2,1,10/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(7/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(2/3)*(a+I*a*tan(c+d*x))^(1/2),x,4,3/5*a*AppellF1(5/3,1/2,1,8/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(5/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(1/3)*(a+I*a*tan(c+d*x))^(1/2),x,4,3/4*a*AppellF1(4/3,1/2,1,7/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(4/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(1/2)/tan(c+d*x)^(1/3),x,4,3/2*a*AppellF1(2/3,1/2,1,5/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(2/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(1/2)/tan(c+d*x)^(2/3),x,4,3*a*AppellF1(1/3,1/2,1,4/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(1/2)/tan(c+d*x)^(4/3),x,4,-3*a*AppellF1(-1/3,1/2,1,2/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/3))],
[tan(c+d*x)^(4/3)*(a+I*a*tan(c+d*x))^(3/2),x,4,3/7*a*AppellF1(7/3,-1/2,1,10/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(7/3)/(d*sqrt(1+I*tan(c+d*x)))],
[tan(c+d*x)^(2/3)*(a+I*a*tan(c+d*x))^(3/2),x,4,3/5*a*AppellF1(5/3,-1/2,1,8/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/3)/(d*sqrt(1+I*tan(c+d*x)))],
[tan(c+d*x)^(1/3)*(a+I*a*tan(c+d*x))^(3/2),x,4,3/4*a*AppellF1(4/3,-1/2,1,7/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(4/3)/(d*sqrt(1+I*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(1/3),x,4,3/2*a*AppellF1(2/3,-1/2,1,5/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(2/3)/(d*sqrt(1+I*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(2/3),x,4,3*a*AppellF1(1/3,-1/2,1,4/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/3)/(d*sqrt(1+I*tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)/tan(c+d*x)^(4/3),x,4,-3*a*AppellF1(-1/3,-1/2,1,2/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1/3))],

# m<0
[tan(c+d*x)^(4/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/7*AppellF1(7/3,3/2,1,10/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(7/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(2/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/5*AppellF1(5/3,3/2,1,8/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(5/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(1/3)/sqrt(a+I*a*tan(c+d*x)),x,4,3/4*AppellF1(4/3,3/2,1,7/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(4/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/3)),x,4,3/2*AppellF1(2/3,3/2,1,5/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(2/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(2/3)),x,4,3*AppellF1(1/3,3/2,1,4/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1/3)/(d*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(4/3)),x,4,-3*AppellF1(-1/3,3/2,1,2/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/3))],
[tan(c+d*x)^(4/3)/(a+I*a*tan(c+d*x))^(3/2),x,4,3/7*AppellF1(7/3,5/2,1,10/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(7/3)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(2/3)/(a+I*a*tan(c+d*x))^(3/2),x,4,3/5*AppellF1(5/3,5/2,1,8/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(5/3)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(1/3)/(a+I*a*tan(c+d*x))^(3/2),x,4,3/4*AppellF1(4/3,5/2,1,7/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(4/3)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(1/3)*(a+I*a*tan(c+d*x))^(3/2)),x,4,3/2*AppellF1(2/3,5/2,1,5/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(2/3)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(2/3)*(a+I*a*tan(c+d*x))^(3/2)),x,4,3*AppellF1(1/3,5/2,1,4/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1/3)/(a*d*sqrt(a+I*a*tan(c+d*x)))],
[1/(tan(c+d*x)^(4/3)*(a+I*a*tan(c+d*x))^(3/2)),x,4,-3*AppellF1(-1/3,5/2,1,2/3,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1/3))],

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

# m>0
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^(1/3),x,8,-1/2*I*a^(1/3)*x/2^(2/3)-1/2*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)-3/2*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)-18/7*(a+I*a*tan(c+d*x))^(1/3)/d+3/7*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(1/3)/d-3/28*(a+I*a*tan(c+d*x))^(4/3)/(a*d)],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(1/3),x,6,1/2*a^(1/3)*x/2^(2/3)-1/2*I*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)-3/2*I*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+I*a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)-3/4*I*(a+I*a*tan(c+d*x))^(4/3)/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(1/3),x,6,1/2*I*a^(1/3)*x/2^(2/3)+1/2*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)+3/2*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)+3*(a+I*a*tan(c+d*x))^(1/3)/d],
[(a+I*a*tan(c+d*x))^(1/3),x,5,-1/2*a^(1/3)*x/2^(2/3)+1/2*I*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)+3/2*I*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-I*a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(1/3),x,11,-1/2*I*a^(1/3)*x/2^(2/3)-1/2*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)-1/2*a^(1/3)*log(tan(c+d*x))/d+3/2*a^(1/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3/2*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-a^(1/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(1/3),x,12,1/2*a^(1/3)*x/2^(2/3)-1/2*I*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)-1/6*I*a^(1/3)*log(tan(c+d*x))/d+1/2*I*a^(1/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3/2*I*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-I*a^(1/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(d*sqrt(3))+I*a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)-cot(c+d*x)*(a+I*a*tan(c+d*x))^(1/3)/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(1/3),x,13,1/2*I*a^(1/3)*x/2^(2/3)+1/2*a^(1/3)*log(cos(c+d*x))/(2^(2/3)*d)+4/9*a^(1/3)*log(tan(c+d*x))/d-4/3*a^(1/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d+3/2*a^(1/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+8/3*a^(1/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(d*sqrt(3))-a^(1/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)-1/6*I*cot(c+d*x)*(a+I*a*tan(c+d*x))^(1/3)/d-1/2*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(1/3)/d],
[(a+I*a*tan(c+d*x))^(2/3),x,5,-1/2*a^(2/3)*x/2^(1/3)+1/2*I*a^(2/3)*log(cos(c+d*x))/(2^(1/3)*d)+3/2*I*a^(2/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+I*a^(2/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^(4/3),x,9,-I*a^(4/3)*x/2^(2/3)-a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)-3*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d-3*a*(a+I*a*tan(c+d*x))^(1/3)/d-9/20*(a+I*a*tan(c+d*x))^(4/3)/d+3/10*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(4/3)/d-6/35*(a+I*a*tan(c+d*x))^(7/3)/(a*d)],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(4/3),x,7,a^(4/3)*x/2^(2/3)-I*a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)-3*I*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+I*2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d-3*I*a*(a+I*a*tan(c+d*x))^(1/3)/d-3/7*I*(a+I*a*tan(c+d*x))^(7/3)/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(4/3),x,7,I*a^(4/3)*x/2^(2/3)+a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)+3*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+3*a*(a+I*a*tan(c+d*x))^(1/3)/d+3/4*(a+I*a*tan(c+d*x))^(4/3)/d],
[(a+I*a*tan(c+d*x))^(4/3),x,6,-a^(4/3)*x/2^(2/3)+I*a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)+3*I*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-I*2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+3*I*a*(a+I*a*tan(c+d*x))^(1/3)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(4/3),x,13,-I*a^(4/3)*x/2^(2/3)-a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)-1/2*a^(4/3)*log(tan(c+d*x))/d+3/2*a^(4/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-a^(4/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(4/3),x,13,a^(4/3)*x/2^(2/3)-I*a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)-2/3*I*a^(4/3)*log(tan(c+d*x))/d+2*I*a^(4/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3*I*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-4*I*a^(4/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(d*sqrt(3))+I*2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+I*a*(a+I*a*tan(c+d*x))^(1/3)/d-cot(c+d*x)*(a+I*a*tan(c+d*x))^(4/3)/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(4/3),x,13,I*a^(4/3)*x/2^(2/3)+a^(4/3)*log(cos(c+d*x))/(2^(2/3)*d)+11/18*a^(4/3)*log(tan(c+d*x))/d-11/6*a^(4/3)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d+3*a^(4/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)+11/3*a^(4/3)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(d*sqrt(3))-2^(1/3)*a^(4/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d-2/3*I*a*cot(c+d*x)*(a+I*a*tan(c+d*x))^(1/3)/d-1/2*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(4/3)/d],
[(a+I*a*tan(c+d*x))^(5/3),x,6,-a^(5/3)*x/2^(1/3)+I*a^(5/3)*log(cos(c+d*x))/(2^(1/3)*d)+3*I*a^(5/3)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+I*2^(2/3)*a^(5/3)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d+3/2*I*a*(a+I*a*tan(c+d*x))^(2/3)/d],

# m<0
[tan(c+d*x)^m/(a+I*a*tan(c+d*x))^(1/3),x,3,AppellF1(1+m,4/3,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*(1+I*tan(c+d*x))^(1/3)*tan(c+d*x)^(1+m)/(d*(1+m)*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)^(1/2)/(a+I*a*tan(c+d*x))^(1/3),x,4,2/3*AppellF1(3/2,4/3,1,5/2,-I*tan(c+d*x),I*tan(c+d*x))*(1+I*tan(c+d*x))^(1/3)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^(1/3),x,9,-1/4*x/(2^(1/3)*a^(1/3))+1/4*I*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)+3/4*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)+1/2*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)-15/8*I*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(1/3))+3/8*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(1/3))+45/8*I*(a+I*a*tan(c+d*x))^(2/3)/(a*d)-39/20*I*(a+I*a*tan(c+d*x))^(5/3)/(a^2*d)],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^(1/3),x,8,-1/4*I*x/(2^(1/3)*a^(1/3))-1/4*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)-3/4*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)-1/2*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+21/10/(d*(a+I*a*tan(c+d*x))^(1/3))+3/5*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(1/3))+3/10*(a+I*a*tan(c+d*x))^(2/3)/(a*d)],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x))^(1/3),x,7,1/4*x/(2^(1/3)*a^(1/3))-1/4*I*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)-3/4*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)-1/2*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+(-3/2*I)/(d*(a+I*a*tan(c+d*x))^(1/3))-3/2*I*(a+I*a*tan(c+d*x))^(2/3)/(a*d)],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^(1/3),x,6,1/4*I*x/(2^(1/3)*a^(1/3))+1/4*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)+3/4*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)+1/2*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+(-3/2)/(d*(a+I*a*tan(c+d*x))^(1/3))],
[1/(a+I*a*tan(c+d*x))^(1/3),x,6,-1/4*x/(2^(1/3)*a^(1/3))+1/4*I*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)+3/4*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)+1/2*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+3/2*I/(d*(a+I*a*tan(c+d*x))^(1/3))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^(1/3),x,13,-1/4*I*x/(2^(1/3)*a^(1/3))-1/4*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)-1/2*log(tan(c+d*x))/(a^(1/3)*d)+3/2*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*d)-3/4*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)+arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(1/3)*d)-1/2*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+3/2/(d*(a+I*a*tan(c+d*x))^(1/3))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^(1/3),x,13,1/4*x/(2^(1/3)*a^(1/3))-1/4*I*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)+1/6*I*log(tan(c+d*x))/(a^(1/3)*d)-1/2*I*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*d)-3/4*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)-I*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(a^(1/3)*d*sqrt(3))-1/2*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+(-5/2*I)/(d*(a+I*a*tan(c+d*x))^(1/3))-cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(1/3))],
[1/(a+I*a*tan(c+d*x))^(2/3),x,6,-1/4*x/(2^(2/3)*a^(2/3))+1/4*I*log(cos(c+d*x))/(2^(2/3)*a^(2/3)*d)+3/4*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*a^(2/3)*d)-1/2*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*a^(2/3)*d)+3/4*I/(d*(a+I*a*tan(c+d*x))^(2/3))],
[tan(c+d*x)^m/(a+I*a*tan(c+d*x))^(4/3),x,3,AppellF1(1+m,7/3,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*(1+I*tan(c+d*x))^(1/3)*tan(c+d*x)^(1+m)/(a*d*(1+m)*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)^(1/2)/(a+I*a*tan(c+d*x))^(4/3),x,4,2/3*AppellF1(3/2,7/3,1,5/2,-I*tan(c+d*x),I*tan(c+d*x))*(1+I*tan(c+d*x))^(1/3)*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)^4/(a+I*a*tan(c+d*x))^(4/3),x,9,-1/8*x/(2^(1/3)*a^(4/3))+1/8*I*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)+3/8*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)+1/4*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)-39/40*I*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(4/3))+3/5*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(4/3))+(-51/10*I)/(a*d*(a+I*a*tan(c+d*x))^(1/3))-87/40*I*(a+I*a*tan(c+d*x))^(2/3)/(a^2*d)],
[tan(c+d*x)^3/(a+I*a*tan(c+d*x))^(4/3),x,8,-1/8*I*x/(2^(1/3)*a^(4/3))-1/8*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)-3/8*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)-1/4*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+15/8/(d*(a+I*a*tan(c+d*x))^(4/3))+3/2*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(4/3))+(-27/4)/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)^2/(a+I*a*tan(c+d*x))^(4/3),x,7,1/8*x/(2^(1/3)*a^(4/3))-1/8*I*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)-3/8*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)-1/4*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+(-3/8*I)/(d*(a+I*a*tan(c+d*x))^(4/3))+9/4*I/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[tan(c+d*x)/(a+I*a*tan(c+d*x))^(4/3),x,7,1/8*I*x/(2^(1/3)*a^(4/3))+1/8*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)+3/8*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)+1/4*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+(-3/8)/(d*(a+I*a*tan(c+d*x))^(4/3))+3/4/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[1/(a+I*a*tan(c+d*x))^(4/3),x,7,-1/8*x/(2^(1/3)*a^(4/3))+1/8*I*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)+3/8*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)+1/4*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+3/8*I/(d*(a+I*a*tan(c+d*x))^(4/3))+3/4*I/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[cot(c+d*x)/(a+I*a*tan(c+d*x))^(4/3),x,15,-1/8*I*x/(2^(1/3)*a^(4/3))-1/8*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)-1/2*log(tan(c+d*x))/(a^(4/3)*d)+3/2*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(a^(4/3)*d)-3/8*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)+arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(4/3)*d)-1/4*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+3/8/(d*(a+I*a*tan(c+d*x))^(4/3))+9/4/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[cot(c+d*x)^2/(a+I*a*tan(c+d*x))^(4/3),x,14,1/8*x/(2^(1/3)*a^(4/3))-1/8*I*log(cos(c+d*x))/(2^(1/3)*a^(4/3)*d)+2/3*I*log(tan(c+d*x))/(a^(4/3)*d)-2*I*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(a^(4/3)*d)-3/8*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(4/3)*d)-4*I*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(a^(4/3)*d*sqrt(3))-1/4*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(4/3)*d)+(-11/8*I)/(d*(a+I*a*tan(c+d*x))^(4/3))-cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(4/3))+(-19/4*I)/(a*d*(a+I*a*tan(c+d*x))^(1/3))],
[1/(a+I*a*tan(c+d*x))^(5/3),x,7,-1/8*x/(2^(2/3)*a^(5/3))+1/8*I*log(cos(c+d*x))/(2^(2/3)*a^(5/3)*d)+3/8*I*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*a^(5/3)*d)-1/4*I*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*a^(5/3)*d)+3/10*I/(d*(a+I*a*tan(c+d*x))^(5/3))+3/8*I/(a*d*(a+I*a*tan(c+d*x))^(2/3))],

# Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n with n synbolic
[(e*tan(c+d*x))^m*(a+I*a*tan(c+d*x)),x,2,a*hypergeom([1,1+m],[2+m],I*tan(c+d*x))*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(e*tan(c+d*x))^m*(a-I*a*tan(c+d*x)),x,2,a*hypergeom([1,1+m],[2+m],-I*tan(c+d*x))*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^4,x,6,-2*a^4*(16+11*n+2*n^2)*(d*tan(e+f*x))^(1+n)/(d*f*(3+n)*(2+3*n+n^2))+8*a^4*hypergeom([1,1+n],[2+n],I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))-(d*tan(e+f*x))^(1+n)*(a^2+I*a^2*tan(e+f*x))^2/(d*f*(3+n))-2*(4+n)*(d*tan(e+f*x))^(1+n)*(a^4+I*a^4*tan(e+f*x))/(d*f*(2+n)*(3+n))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^3,x,5,-a^3*(5+2*n)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*(2+n))+4*a^3*hypergeom([1,1+n],[2+n],I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))-(d*tan(e+f*x))^(1+n)*(a^3+I*a^3*tan(e+f*x))/(d*f*(2+n))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^2,x,4,-a^2*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+2*a^2*hypergeom([1,1+n],[2+n],I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x)),x,2,a*hypergeom([1,1+n],[2+n],I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x)),x,6,1/2*(1-n)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(a*d*f*(1+n))+1/2*I*n*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(a*d^2*f*(2+n))+1/2*(d*tan(e+f*x))^(1+n)/(d*f*(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^2,x,7,1/4*(1-n)^2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(a^2*d*f*(1+n))+1/4*(2-n)*(d*tan(e+f*x))^(1+n)/(a^2*d*f*(1+I*tan(e+f*x)))+1/4*I*(2-n)*n*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(a^2*d^2*f*(2+n))+1/4*(d*tan(e+f*x))^(1+n)/(d*f*(a+I*a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^3,x,8,1/24*(1-2*n)*(1-n)*(3-n)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(a^3*d*f*(1+n))+1/24*I*(5-2*n)*(2-n)*n*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(a^3*d^2*f*(2+n))+1/6*(d*tan(e+f*x))^(1+n)/(d*f*(a+I*a*tan(e+f*x))^3)+1/24*(7-2*n)*(d*tan(e+f*x))^(1+n)/(a*d*f*(a+I*a*tan(e+f*x))^2)+1/24*(5-2*n)*(2-n)*(d*tan(e+f*x))^(1+n)/(d*f*(a^3+I*a^3*tan(e+f*x)))],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^4,x,9,1/48*(1-n)*(3-n)*(1-4*n+n^2)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(a^4*d*f*(1+n))+1/48*(13-7*n+n^2)*(d*tan(e+f*x))^(1+n)/(a^4*d*f*(1+I*tan(e+f*x))^2)+1/48*(2-n)^2*(4-n)*(d*tan(e+f*x))^(1+n)/(a^4*d*f*(1+I*tan(e+f*x)))+1/48*I*(2-n)^2*(4-n)*n*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(a^4*d^2*f*(2+n))+1/8*(d*tan(e+f*x))^(1+n)/(d*f*(a+I*a*tan(e+f*x))^4)+1/24*(5-n)*(d*tan(e+f*x))^(1+n)/(a*d*f*(a+I*a*tan(e+f*x))^3)],
[(d*tan(e+f*x))^n*(a-I*a*tan(e+f*x)),x,2,a*hypergeom([1,1+n],[2+n],-I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(d*tan(e+f*x))^n/(a-I*a*tan(e+f*x)),x,6,1/2*(1-n)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(a*d*f*(1+n))-1/2*I*n*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(a*d^2*f*(2+n))+1/2*(d*tan(e+f*x))^(1+n)/(d*f*(a-I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^(3/2),x,3,a*AppellF1(1+n,-1/2,1,2+n,-I*tan(e+f*x),I*tan(e+f*x))*sqrt(a+I*a*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*sqrt(1+I*tan(e+f*x)))],
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^(1/2),x,3,a*AppellF1(1+n,1/2,1,2+n,-I*tan(e+f*x),I*tan(e+f*x))*sqrt(1+I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*sqrt(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^(1/2),x,3,AppellF1(1+n,3/2,1,2+n,-I*tan(e+f*x),I*tan(e+f*x))*sqrt(1+I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*sqrt(a+I*a*tan(e+f*x)))],
[(d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^(3/2),x,3,AppellF1(1+n,5/2,1,2+n,-I*tan(e+f*x),I*tan(e+f*x))*sqrt(1+I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)/(a*d*f*(1+n)*sqrt(a+I*a*tan(e+f*x)))],

# Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n with m synbolic
[(d*tan(e+f*x))^n*(a+I*a*tan(e+f*x))^m,x,3,AppellF1(1+n,1-m,1,2+n,-I*tan(e+f*x),I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)*(a+I*a*tan(e+f*x))^m/(d*f*(1+n)*(1+I*tan(e+f*x))^m)],
[tan(c+d*x)^4*(a+I*a*tan(c+d*x))^m,x,6,2*I*(a+I*a*tan(c+d*x))^m/(d*(6+5*m+m^2))-1/2*I*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)-I*m*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^m/(d*(6+5*m+m^2))+tan(c+d*x)^3*(a+I*a*tan(c+d*x))^m/(d*(3+m))+I*(6+3*m+m^2)*(a+I*a*tan(c+d*x))^(1+m)/(a*d*(3+m)*(2+3*m+m^2))],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^m,x,5,-2*(a+I*a*tan(c+d*x))^m/(d*m*(2+m))+1/2*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)+tan(c+d*x)^2*(a+I*a*tan(c+d*x))^m/(d*(2+m))-m*(a+I*a*tan(c+d*x))^(1+m)/(a*d*(2+3*m+m^2))],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^m,x,3,1/2*I*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)-I*(a+I*a*tan(c+d*x))^(1+m)/(a*d*(1+m))],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^m,x,3,(a+I*a*tan(c+d*x))^m/(d*m)-1/2*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)],
[(a+I*a*tan(c+d*x))^m,x,2,-1/2*I*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^m,x,5,1/2*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)-hypergeom([1,m],[1+m],1+I*tan(c+d*x))*(a+I*a*tan(c+d*x))^m/(d*m)],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^m,x,6,-cot(c+d*x)*(a+I*a*tan(c+d*x))^m/d+1/2*I*hypergeom([1,m],[1+m],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^m/(d*m)-I*hypergeom([1,m],[1+m],1+I*tan(c+d*x))*(a+I*a*tan(c+d*x))^m/d],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^m,x,4,2/5*AppellF1(5/2,1-m,1,7/2,-I*tan(c+d*x),I*tan(c+d*x))*tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^m/(d*(1+I*tan(c+d*x))^m)],
[tan(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^m,x,4,2/3*AppellF1(3/2,1-m,1,5/2,-I*tan(c+d*x),I*tan(c+d*x))*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^m/(d*(1+I*tan(c+d*x))^m)],
[(a+I*a*tan(c+d*x))^m/tan(c+d*x)^(1/2),x,4,2*AppellF1(1/2,1-m,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^m/(d*(1+I*tan(c+d*x))^m)],
[(a+I*a*tan(c+d*x))^m/tan(c+d*x)^(3/2),x,4,-2*AppellF1(-1/2,1-m,1,1/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^m/(d*sqrt(tan(c+d*x))*(1+I*tan(c+d*x))^m)],

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

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

# m>0
[(d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x)),x,5,a*d^(5/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/f-2*a*d^2*sqrt(d*tan(e+f*x))/f+2/3*a*d*(d*tan(e+f*x))^(3/2)/f+2/5*a*(d*tan(e+f*x))^(5/2)/f],
[(d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x)),x,4,a*d^(3/2)*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/f+2*a*d*sqrt(d*tan(e+f*x))/f+2/3*a*(d*tan(e+f*x))^(3/2)/f],
[(d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x)),x,3,-a*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)*sqrt(d)/f+2*a*sqrt(d*tan(e+f*x))/f],
[(a+a*tan(e+f*x))/(d*tan(e+f*x))^(1/2),x,2,-a*arctan(sqrt(d)*(1-tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(f*sqrt(d))],
[(a+a*tan(e+f*x))/(d*tan(e+f*x))^(3/2),x,3,a*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(3/2)*f)-2*a/(d*f*sqrt(d*tan(e+f*x)))],
[(a+a*tan(e+f*x))/(d*tan(e+f*x))^(5/2),x,4,a*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(5/2)*f)-2*a/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*a/(d*f*(d*tan(e+f*x))^(3/2))],
[(a+a*tan(e+f*x))/(d*tan(e+f*x))^(7/2),x,5,-a*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(7/2)*f)+2*a/(d^3*f*sqrt(d*tan(e+f*x)))-2/5*a/(d*f*(d*tan(e+f*x))^(5/2))-2/3*a/(d^2*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x))^2,x,16,-a^2*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+a^2*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))-a^2*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/f+a^2*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/f-4*a^2*d^2*sqrt(d*tan(e+f*x))/f+4/5*a^2*(d*tan(e+f*x))^(5/2)/f+2/7*a^2*(d*tan(e+f*x))^(7/2)/(d*f)],
[(d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))^2,x,15,-a^2*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+a^2*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+a^2*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/f-a^2*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/f+4/3*a^2*(d*tan(e+f*x))^(3/2)/f+2/5*a^2*(d*tan(e+f*x))^(5/2)/(d*f)],
[(d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x))^2,x,15,a^2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))-a^2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))+a^2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)*sqrt(d)/f-a^2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)*sqrt(d)/f+4*a^2*sqrt(d*tan(e+f*x))/f+2/3*a^2*(d*tan(e+f*x))^(3/2)/(d*f)],
[(a+a*tan(e+f*x))^2/(d*tan(e+f*x))^(1/2),x,14,a^2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2)*sqrt(d))-a^2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2)*sqrt(d))-a^2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(f*sqrt(d))+a^2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(f*sqrt(d))+2*a^2*sqrt(d*tan(e+f*x))/(d*f)],
[(a+a*tan(e+f*x))^2/(d*tan(e+f*x))^(3/2),x,13,-a^2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(d^(3/2)*f*sqrt(2))+a^2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(d^(3/2)*f*sqrt(2))-a^2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(d^(3/2)*f)+a^2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(d^(3/2)*f)-2*a^2/(d*f*sqrt(d*tan(e+f*x)))],
[(a+a*tan(e+f*x))^2/(d*tan(e+f*x))^(5/2),x,14,-a^2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(d^(5/2)*f*sqrt(2))+a^2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(d^(5/2)*f*sqrt(2))+a^2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(d^(5/2)*f)-a^2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(2)/(d^(5/2)*f)-4*a^2/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*a^2/(d*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(7/2)*(a+a*tan(e+f*x))^3,x,8,-2*a^3*d^(7/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/f+4*a^3*d^3*sqrt(d*tan(e+f*x))/f-4/3*a^3*d^2*(d*tan(e+f*x))^(3/2)/f-4/5*a^3*d*(d*tan(e+f*x))^(5/2)/f+4/7*a^3*(d*tan(e+f*x))^(7/2)/f+16/33*a^3*(d*tan(e+f*x))^(9/2)/(d*f)+2/11*(d*tan(e+f*x))^(9/2)*(a^3+a^3*tan(e+f*x))/(d*f)],
[(d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x))^3,x,7,-2*a^3*d^(5/2)*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/f-4*a^3*d^2*sqrt(d*tan(e+f*x))/f-4/3*a^3*d*(d*tan(e+f*x))^(3/2)/f+4/5*a^3*(d*tan(e+f*x))^(5/2)/f+40/63*a^3*(d*tan(e+f*x))^(7/2)/(d*f)+2/9*(d*tan(e+f*x))^(7/2)*(a^3+a^3*tan(e+f*x))/(d*f)],
[(d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))^3,x,6,2*a^3*d^(3/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/f-4*a^3*d*sqrt(d*tan(e+f*x))/f+4/3*a^3*(d*tan(e+f*x))^(3/2)/f+32/35*a^3*(d*tan(e+f*x))^(5/2)/(d*f)+2/7*(d*tan(e+f*x))^(5/2)*(a^3+a^3*tan(e+f*x))/(d*f)],
[(d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x))^3,x,5,2*a^3*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)*sqrt(d)/f+4*a^3*sqrt(d*tan(e+f*x))/f+8/5*a^3*(d*tan(e+f*x))^(3/2)/(d*f)+2/5*(d*tan(e+f*x))^(3/2)*(a^3+a^3*tan(e+f*x))/(d*f)],
[(a+a*tan(e+f*x))^3/(d*tan(e+f*x))^(1/2),x,4,-2*a^3*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(f*sqrt(d))+16/3*a^3*sqrt(d*tan(e+f*x))/(d*f)+2/3*sqrt(d*tan(e+f*x))*(a^3+a^3*tan(e+f*x))/(d*f)],
[(a+a*tan(e+f*x))^3/(d*tan(e+f*x))^(3/2),x,4,-2*a^3*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(3/2)*f)+4*a^3*sqrt(d*tan(e+f*x))/(d^2*f)-2*(a^3+a^3*tan(e+f*x))/(d*f*sqrt(d*tan(e+f*x)))],
[(a+a*tan(e+f*x))^3/(d*tan(e+f*x))^(5/2),x,4,2*a^3*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(5/2)*f)-16/3*a^3/(d^2*f*sqrt(d*tan(e+f*x)))-2/3*(a^3+a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(3/2))],
[(a+a*tan(e+f*x))^3/(d*tan(e+f*x))^(7/2),x,5,2*a^3*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(7/2)*f)-4*a^3/(d^3*f*sqrt(d*tan(e+f*x)))-8/5*a^3/(d^2*f*(d*tan(e+f*x))^(3/2))-2/5*(a^3+a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(5/2))],
[(a+a*tan(e+f*x))^3/(d*tan(e+f*x))^(9/2),x,6,-2*a^3*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(2)/(d^(9/2)*f)+4*a^3/(d^4*f*sqrt(d*tan(e+f*x)))-32/35*a^3/(d^2*f*(d*tan(e+f*x))^(5/2))-4/3*a^3/(d^3*f*(d*tan(e+f*x))^(3/2))-2/7*(a^3+a^3*tan(e+f*x))/(d*f*(d*tan(e+f*x))^(7/2))],

# m<0
[(d*tan(e+f*x))^(5/2)/(a+a*tan(e+f*x)),x,7,-d^(5/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a*f)+d^(5/2)*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a*f*sqrt(2))+2*d^2*sqrt(d*tan(e+f*x))/(a*f)],
[(d*tan(e+f*x))^(3/2)/(a+a*tan(e+f*x)),x,6,d^(3/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a*f)-d^(3/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a*f*sqrt(2))],
[(d*tan(e+f*x))^(1/2)/(a+a*tan(e+f*x)),x,6,-arctan(sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a*f)-arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(d)/(a*f*sqrt(2))],
[1/((d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x))),x,6,arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a*f*sqrt(d))+arctanh(sqrt(d)*(1+tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a*f*sqrt(2)*sqrt(d))],
[1/((d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))),x,7,-arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(3/2)*f)+arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a*d^(3/2)*f*sqrt(2))+(-2)/(a*d*f*sqrt(d*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x))),x,10,arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a*d^(5/2)*f)-arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a*d^(5/2)*f*sqrt(2))+2/(a*d^2*f*sqrt(d*tan(e+f*x)))+(-2/3)/(a*d*f*(d*tan(e+f*x))^(3/2))],
[(d*tan(e+f*x))^(5/2)/(a+a*tan(e+f*x))^2,x,17,3/2*d^(5/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f)+1/2*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))-1/2*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+1/4*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))-1/4*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))-1/2*d^2*sqrt(d*tan(e+f*x))/(f*(a^2+a^2*tan(e+f*x)))],
[(d*tan(e+f*x))^(3/2)/(a+a*tan(e+f*x))^2,x,18,-1/2*d^(3/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f)-1/2*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+1/2*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2))+1/4*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))-1/4*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2))+1/2*d*sqrt(d*tan(e+f*x))/(f*(a^2+a^2*tan(e+f*x)))],
[(d*tan(e+f*x))^(1/2)/(a+a*tan(e+f*x))^2,x,17,-1/2*arctan(sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^2*f)-1/2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^2*f*sqrt(2))+1/2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^2*f*sqrt(2))-1/4*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^2*f*sqrt(2))+1/4*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(a^2*f*sqrt(2))-1/2*sqrt(d*tan(e+f*x))/(f*(a^2+a^2*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x))^2),x,18,3/2*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(d))+1/2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2)*sqrt(d))-1/2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*f*sqrt(2)*sqrt(d))-1/4*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2)*sqrt(d))+1/4*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*f*sqrt(2)*sqrt(d))+1/2*sqrt(d*tan(e+f*x))/(d*f*(a^2+a^2*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))^2),x,18,-5/2*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(3/2)*f)+1/2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(3/2)*f*sqrt(2))-1/2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(3/2)*f*sqrt(2))+1/4*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(3/2)*f*sqrt(2))-1/4*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(3/2)*f*sqrt(2))+(-5/2)/(a^2*d*f*sqrt(d*tan(e+f*x)))+1/2/(d*f*sqrt(d*tan(e+f*x))*(a^2+a^2*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x))^2),x,20,7/2*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(5/2)*f)-1/2*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(5/2)*f*sqrt(2))+1/2*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(a^2*d^(5/2)*f*sqrt(2))+1/4*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(5/2)*f*sqrt(2))-1/4*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(a^2*d^(5/2)*f*sqrt(2))+9/2/(a^2*d^2*f*sqrt(d*tan(e+f*x)))+(-7/6)/(a^2*d*f*(d*tan(e+f*x))^(3/2))+1/2/(d*f*(d*tan(e+f*x))^(3/2)*(a^2+a^2*tan(e+f*x)))],
[(d*tan(e+f*x))^(9/2)/(a+a*tan(e+f*x))^3,x,9,-31/8*d^(9/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f)+1/2*d^(9/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*f*sqrt(2))+27/8*d^4*sqrt(d*tan(e+f*x))/(a^3*f)-9/8*d^3*(d*tan(e+f*x))^(3/2)/(a^3*f*(1+tan(e+f*x)))-1/4*d^2*(d*tan(e+f*x))^(5/2)/(a*f*(a+a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(7/2)/(a+a*tan(e+f*x))^3,x,8,11/8*d^(7/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f)+1/2*d^(7/2)*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*f*sqrt(2))-7/8*d^3*sqrt(d*tan(e+f*x))/(a^3*f*(1+tan(e+f*x)))-1/4*d^2*(d*tan(e+f*x))^(3/2)/(a*f*(a+a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(5/2)/(a+a*tan(e+f*x))^3,x,8,1/8*d^(5/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f)-1/2*d^(5/2)*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*f*sqrt(2))+5/8*d^2*sqrt(d*tan(e+f*x))/(a^3*f*(1+tan(e+f*x)))-1/4*d^2*sqrt(d*tan(e+f*x))/(a*f*(a+a*tan(e+f*x))^2)],
[(d*tan(e+f*x))^(3/2)/(a+a*tan(e+f*x))^3,x,8,-5/8*d^(3/2)*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f)-1/2*d^(3/2)*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*f*sqrt(2))+1/4*d*sqrt(d*tan(e+f*x))/(a*f*(a+a*tan(e+f*x))^2)-1/8*d*sqrt(d*tan(e+f*x))/(f*(a^3+a^3*tan(e+f*x)))],
[(d*tan(e+f*x))^(1/2)/(a+a*tan(e+f*x))^3,x,8,1/8*arctan(sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(a^3*f)+1/2*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))*sqrt(d)/(a^3*f*sqrt(2))-1/4*sqrt(d*tan(e+f*x))/(a*f*(a+a*tan(e+f*x))^2)-3/8*sqrt(d*tan(e+f*x))/(f*(a^3+a^3*tan(e+f*x)))],
[1/((d*tan(e+f*x))^(1/2)*(a+a*tan(e+f*x))^3),x,8,11/8*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*f*sqrt(d))+1/2*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*f*sqrt(2)*sqrt(d))+7/8*sqrt(d*tan(e+f*x))/(a^3*d*f*(1+tan(e+f*x)))+1/4*sqrt(d*tan(e+f*x))/(a*d*f*(a+a*tan(e+f*x))^2)],
[1/((d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))^3),x,9,-31/8*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*d^(3/2)*f)-1/2*arctanh((sqrt(d)+sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*d^(3/2)*f*sqrt(2))+(-27/8)/(a^3*d*f*sqrt(d*tan(e+f*x)))+9/8/(a^3*d*f*sqrt(d*tan(e+f*x))*(1+tan(e+f*x)))+1/4/(a*d*f*sqrt(d*tan(e+f*x))*(a+a*tan(e+f*x))^2)],
[1/((d*tan(e+f*x))^(5/2)*(a+a*tan(e+f*x))^3),x,10,59/8*arctan(sqrt(d*tan(e+f*x))/sqrt(d))/(a^3*d^(5/2)*f)-1/2*arctan((sqrt(d)-sqrt(d)*tan(e+f*x))/(sqrt(2)*sqrt(d*tan(e+f*x))))/(a^3*d^(5/2)*f*sqrt(2))+63/8/(a^3*d^2*f*sqrt(d*tan(e+f*x)))+(-55/24)/(a^3*d*f*(d*tan(e+f*x))^(3/2))+11/8/(a^3*d*f*(d*tan(e+f*x))^(3/2)*(1+tan(e+f*x)))+1/4/(a*d*f*(d*tan(e+f*x))^(3/2)*(a+a*tan(e+f*x))^2)],

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

# m>0
[sqrt(1+tan(e+f*x))*tan(e+f*x)^5,x,11,-arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f-arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f+2*sqrt(1+tan(e+f*x))/f+52/315*(1+tan(e+f*x))^(3/2)/f-26/105*tan(e+f*x)*(1+tan(e+f*x))^(3/2)/f-4/21*tan(e+f*x)^2*(1+tan(e+f*x))^(3/2)/f+2/9*tan(e+f*x)^3*(1+tan(e+f*x))^(3/2)/f],
[sqrt(1+tan(e+f*x))*tan(e+f*x)^3,x,9,arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f+arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f-2*sqrt(1+tan(e+f*x))/f-4/15*(1+tan(e+f*x))^(3/2)/f+2/5*tan(e+f*x)*(1+tan(e+f*x))^(3/2)/f],
[sqrt(1+tan(e+f*x))*tan(e+f*x),x,6,-arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f-arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f+2*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)*sqrt(1+tan(e+f*x)),x,9,-2*arctanh(sqrt(1+tan(e+f*x)))/f+arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f+arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f],
[cot(e+f*x)^3*sqrt(1+tan(e+f*x)),x,11,9/4*arctanh(sqrt(1+tan(e+f*x)))/f-arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f-arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f-1/4*cot(e+f*x)*sqrt(1+tan(e+f*x))/f-1/2*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^5*sqrt(1+tan(e+f*x)),x,13,-139/64*arctanh(sqrt(1+tan(e+f*x)))/f+arctan(1/2*(4-3*sqrt(2)+(2-sqrt(2))*tan(e+f*x))/(sqrt(-7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(-1+sqrt(2)))/f+arctanh(1/2*(4+3*sqrt(2)+(2+sqrt(2))*tan(e+f*x))/(sqrt(7+5*sqrt(2))*sqrt(1+tan(e+f*x))))*sqrt(1/2*(1+sqrt(2)))/f+11/64*cot(e+f*x)*sqrt(1+tan(e+f*x))/f+53/96*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f-1/24*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f-1/4*cot(e+f*x)^4*sqrt(1+tan(e+f*x))/f],
[sqrt(1+tan(e+f*x))*tan(e+f*x)^4,x,14,-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))-18/35*(1+tan(e+f*x))^(3/2)/f-8/35*tan(e+f*x)*(1+tan(e+f*x))^(3/2)/f+2/7*tan(e+f*x)^2*(1+tan(e+f*x))^(3/2)/f],
[sqrt(1+tan(e+f*x))*tan(e+f*x)^2,x,12,arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f-arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f-1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))+1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))+2/3*(1+tan(e+f*x))^(3/2)/f],
[sqrt(1+tan(e+f*x)),x,11,-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))],
[cot(e+f*x)^2*sqrt(1+tan(e+f*x)),x,16,-arctanh(sqrt(1+tan(e+f*x)))/f+arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f-arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f-1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))+1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))-cot(e+f*x)*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^4*sqrt(1+tan(e+f*x)),x,19,7/8*arctanh(sqrt(1+tan(e+f*x)))/f-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))/f+1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(2*(1+sqrt(2))))+9/8*cot(e+f*x)*sqrt(1+tan(e+f*x))/f-1/12*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f-1/3*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f],
[tan(e+f*x)^5*(1+tan(e+f*x))^(3/2),x,19,1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+2*sqrt(1+tan(e+f*x))/f+2/3*(1+tan(e+f*x))^(3/2)/f+20/231*(1+tan(e+f*x))^(5/2)/f-50/231*tan(e+f*x)*(1+tan(e+f*x))^(5/2)/f-4/33*tan(e+f*x)^2*(1+tan(e+f*x))^(5/2)/f+2/11*tan(e+f*x)^3*(1+tan(e+f*x))^(5/2)/f],
[tan(e+f*x)^3*(1+tan(e+f*x))^(3/2),x,17,-1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-2*sqrt(1+tan(e+f*x))/f-2/3*(1+tan(e+f*x))^(3/2)/f-4/35*(1+tan(e+f*x))^(5/2)/f+2/7*tan(e+f*x)*(1+tan(e+f*x))^(5/2)/f],
[tan(e+f*x)*(1+tan(e+f*x))^(3/2),x,14,1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+2*sqrt(1+tan(e+f*x))/f+2/3*(1+tan(e+f*x))^(3/2)/f],
[cot(e+f*x)*(1+tan(e+f*x))^(3/2),x,16,-2*arctanh(sqrt(1+tan(e+f*x)))/f-1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f],
[cot(e+f*x)^3*(1+tan(e+f*x))^(3/2),x,18,5/4*arctanh(sqrt(1+tan(e+f*x)))/f+1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-5/4*cot(e+f*x)*sqrt(1+tan(e+f*x))/f-1/2*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^5*(1+tan(e+f*x))^(3/2),x,20,-83/64*arctanh(sqrt(1+tan(e+f*x)))/f-1/2*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/2*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+83/64*cot(e+f*x)*sqrt(1+tan(e+f*x))/f+15/32*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f-3/8*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f-1/4*cot(e+f*x)^4*sqrt(1+tan(e+f*x))/f],
[tan(e+f*x)^4*(1+tan(e+f*x))^(3/2),x,10,-arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f+2*sqrt(1+tan(e+f*x))/f-22/63*(1+tan(e+f*x))^(5/2)/f-8/63*tan(e+f*x)*(1+tan(e+f*x))^(5/2)/f+2/9*tan(e+f*x)^2*(1+tan(e+f*x))^(5/2)/f],
[tan(e+f*x)^2*(1+tan(e+f*x))^(3/2),x,8,arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f+arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f-2*sqrt(1+tan(e+f*x))/f+2/5*(1+tan(e+f*x))^(5/2)/f],
[(1+tan(e+f*x))^(3/2),x,7,-arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f+2*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^2*(1+tan(e+f*x))^(3/2),x,11,-3*arctanh(sqrt(1+tan(e+f*x)))/f+arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f+arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f-cot(e+f*x)*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^4*(1+tan(e+f*x))^(3/2),x,13,25/8*arctanh(sqrt(1+tan(e+f*x)))/f-arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f+7/8*cot(e+f*x)*sqrt(1+tan(e+f*x))/f-7/12*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f-1/3*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f],

# m<0
[tan(e+f*x)^5/sqrt(1+tan(e+f*x)),x,10,-1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f+44/105*sqrt(1+tan(e+f*x))/f-22/105*sqrt(1+tan(e+f*x))*tan(e+f*x)/f-12/35*sqrt(1+tan(e+f*x))*tan(e+f*x)^2/f+2/7*sqrt(1+tan(e+f*x))*tan(e+f*x)^3/f],
[tan(e+f*x)^3/sqrt(1+tan(e+f*x)),x,8,1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f+1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f-4/3*sqrt(1+tan(e+f*x))/f+2/3*sqrt(1+tan(e+f*x))*tan(e+f*x)/f],
[tan(e+f*x)/sqrt(1+tan(e+f*x)),x,5,-1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f],
[cot(e+f*x)/sqrt(1+tan(e+f*x)),x,9,-2*arctanh(sqrt(1+tan(e+f*x)))/f+1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f+1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f],
[cot(e+f*x)^3/sqrt(1+tan(e+f*x)),x,12,5/4*arctanh(sqrt(1+tan(e+f*x)))/f-1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f-1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f+3/4*cot(e+f*x)*sqrt(1+tan(e+f*x))/f-1/2*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^5/sqrt(1+tan(e+f*x)),x,14,-115/64*arctanh(sqrt(1+tan(e+f*x)))/f+1/2*arctan((3-2*sqrt(2)+(1-sqrt(2))*tan(e+f*x))/(sqrt(2*(-7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(-1+sqrt(2))/f+1/2*arctanh((3+2*sqrt(2)+(1+sqrt(2))*tan(e+f*x))/(sqrt(2*(7+5*sqrt(2)))*sqrt(1+tan(e+f*x))))*sqrt(1+sqrt(2))/f-13/64*cot(e+f*x)*sqrt(1+tan(e+f*x))/f+13/96*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f+7/24*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f-1/4*cot(e+f*x)^4*sqrt(1+tan(e+f*x))/f],
[tan(e+f*x)^4/sqrt(1+tan(e+f*x)),x,14,-1/4*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/4*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+1/2*arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-14/15*sqrt(1+tan(e+f*x))/f-8/15*sqrt(1+tan(e+f*x))*tan(e+f*x)/f+2/5*sqrt(1+tan(e+f*x))*tan(e+f*x)^2/f],
[tan(e+f*x)^2/sqrt(1+tan(e+f*x)),x,12,1/4*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/4*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/2*arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-1/2*arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+2*sqrt(1+tan(e+f*x))/f],
[1/sqrt(1+tan(e+f*x)),x,11,-1/4*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/4*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+1/2*arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f],
[cot(e+f*x)^2/sqrt(1+tan(e+f*x)),x,19,arctanh(sqrt(1+tan(e+f*x)))/f+1/4*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/4*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/2*arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-1/2*arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f-cot(e+f*x)*sqrt(1+tan(e+f*x))/f],
[cot(e+f*x)^4/sqrt(1+tan(e+f*x)),x,19,-3/8*arctanh(sqrt(1+tan(e+f*x)))/f-1/4*log(1+sqrt(2)-sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))+1/4*log(1+sqrt(2)+sqrt(2*(1+sqrt(2)))*sqrt(1+tan(e+f*x))+tan(e+f*x))/(f*sqrt(1+sqrt(2)))-1/2*arctan((sqrt(2*(1+sqrt(2)))-2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+1/2*arctan((sqrt(2*(1+sqrt(2)))+2*sqrt(1+tan(e+f*x)))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))/f+3/8*cot(e+f*x)*sqrt(1+tan(e+f*x))/f+5/12*cot(e+f*x)^2*sqrt(1+tan(e+f*x))/f-1/3*cot(e+f*x)^3*sqrt(1+tan(e+f*x))/f],

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

# Integrands of the form (a+a Tan[e+f x])^m (d Tan[e+f x])^n with m symbolic
[(d*tan(e+f*x))^n*(a+a*tan(e+f*x))^m,x,7,1/2*AppellF1(1+n,-m,1,2+n,-tan(e+f*x),-I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)*(a+a*tan(e+f*x))^m/(d*f*(1+n)*(1+tan(e+f*x))^m)+1/2*AppellF1(1+n,-m,1,2+n,-tan(e+f*x),I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)*(a+a*tan(e+f*x))^m/(d*f*(1+n)*(1+tan(e+f*x))^m)],

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

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

# m>0
[tan(c+d*x)^5*(a+b*tan(c+d*x)),x,6,-b*x-a*log(cos(c+d*x))/d+b*tan(c+d*x)/d-1/2*a*tan(c+d*x)^2/d-1/3*b*tan(c+d*x)^3/d+1/4*a*tan(c+d*x)^4/d+1/5*b*tan(c+d*x)^5/d],
[tan(c+d*x)^4*(a+b*tan(c+d*x)),x,5,a*x-b*log(cos(c+d*x))/d-a*tan(c+d*x)/d-1/2*b*tan(c+d*x)^2/d+1/3*a*tan(c+d*x)^3/d+1/4*b*tan(c+d*x)^4/d],
[tan(c+d*x)^3*(a+b*tan(c+d*x)),x,4,b*x+a*log(cos(c+d*x))/d-b*tan(c+d*x)/d+1/2*a*tan(c+d*x)^2/d+1/3*b*tan(c+d*x)^3/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x)),x,3,-a*x+b*log(cos(c+d*x))/d+a*tan(c+d*x)/d+1/2*b*tan(c+d*x)^2/d],
[tan(c+d*x)*(a+b*tan(c+d*x)),x,2,-b*x-a*log(cos(c+d*x))/d+b*tan(c+d*x)/d],
[a+b*tan(c+d*x),x,2,a*x-b*log(cos(c+d*x))/d],
[cot(c+d*x)*(a+b*tan(c+d*x)),x,2,b*x+a*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x)),x,3,-a*x-a*cot(c+d*x)/d+b*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x)),x,4,-b*x-b*cot(c+d*x)/d-1/2*a*cot(c+d*x)^2/d-a*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x)),x,5,a*x+a*cot(c+d*x)/d-1/2*b*cot(c+d*x)^2/d-1/3*a*cot(c+d*x)^3/d-b*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x)),x,6,b*x+b*cot(c+d*x)/d+1/2*a*cot(c+d*x)^2/d-1/3*b*cot(c+d*x)^3/d-1/4*a*cot(c+d*x)^4/d+a*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x)),x,7,-a*x-a*cot(c+d*x)/d+1/2*b*cot(c+d*x)^2/d+1/3*a*cot(c+d*x)^3/d-1/4*b*cot(c+d*x)^4/d-1/5*a*cot(c+d*x)^5/d+b*log(sin(c+d*x))/d],
[tan(c+d*x)^4*(a+b*tan(c+d*x))^2,x,6,(a^2-b^2)*x-2*a*b*log(cos(c+d*x))/d-(a^2-b^2)*tan(c+d*x)/d-a*b*tan(c+d*x)^2/d+1/3*(a^2-b^2)*tan(c+d*x)^3/d+1/2*a*b*tan(c+d*x)^4/d+1/5*b^2*tan(c+d*x)^5/d],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^2,x,5,2*a*b*x+(a^2-b^2)*log(cos(c+d*x))/d-2*a*b*tan(c+d*x)/d+1/2*(a^2-b^2)*tan(c+d*x)^2/d+2/3*a*b*tan(c+d*x)^3/d+1/4*b^2*tan(c+d*x)^4/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^2,x,3,-(a^2-b^2)*x+2*a*b*log(cos(c+d*x))/d-b^2*tan(c+d*x)/d+1/3*(a+b*tan(c+d*x))^3/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^2,x,3,-2*a*b*x-(a^2-b^2)*log(cos(c+d*x))/d+a*b*tan(c+d*x)/d+1/2*(a+b*tan(c+d*x))^2/d],
[(a+b*tan(c+d*x))^2,x,2,(a^2-b^2)*x-2*a*b*log(cos(c+d*x))/d+b^2*tan(c+d*x)/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^2,x,3,2*a*b*x-b^2*log(cos(c+d*x))/d+a^2*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^2,x,3,-(a^2-b^2)*x-a^2*cot(c+d*x)/d+2*a*b*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^2,x,4,-2*a*b*x-2*a*b*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)^2/d-(a^2-b^2)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^2,x,5,(a^2-b^2)*x+(a^2-b^2)*cot(c+d*x)/d-a*b*cot(c+d*x)^2/d-1/3*a^2*cot(c+d*x)^3/d-2*a*b*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^2,x,6,2*a*b*x+2*a*b*cot(c+d*x)/d+1/2*(a^2-b^2)*cot(c+d*x)^2/d-2/3*a*b*cot(c+d*x)^3/d-1/4*a^2*cot(c+d*x)^4/d+(a^2-b^2)*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^2,x,7,-(a^2-b^2)*x-(a^2-b^2)*cot(c+d*x)/d+a*b*cot(c+d*x)^2/d+1/3*(a^2-b^2)*cot(c+d*x)^3/d-1/2*a*b*cot(c+d*x)^4/d-1/5*a^2*cot(c+d*x)^5/d+2*a*b*log(sin(c+d*x))/d],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^3,x,7,b*(3*a^2-b^2)*x+a*(a^2-3*b^2)*log(cos(c+d*x))/d-b*(a^2-b^2)*tan(c+d*x)/d-1/2*a*(a+b*tan(c+d*x))^2/d-1/3*(a+b*tan(c+d*x))^3/d-1/20*a*(a+b*tan(c+d*x))^4/(b^2*d)+1/5*tan(c+d*x)*(a+b*tan(c+d*x))^4/(b*d)],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^3,x,4,-a*(a^2-3*b^2)*x+b*(3*a^2-b^2)*log(cos(c+d*x))/d-2*a*b^2*tan(c+d*x)/d-1/2*b*(a+b*tan(c+d*x))^2/d+1/4*(a+b*tan(c+d*x))^4/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^3,x,4,-b*(3*a^2-b^2)*x-a*(a^2-3*b^2)*log(cos(c+d*x))/d+b*(a^2-b^2)*tan(c+d*x)/d+1/2*a*(a+b*tan(c+d*x))^2/d+1/3*(a+b*tan(c+d*x))^3/d],
[(a+b*tan(c+d*x))^3,x,3,a*(a^2-3*b^2)*x-b*(3*a^2-b^2)*log(cos(c+d*x))/d+2*a*b^2*tan(c+d*x)/d+1/2*b*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^3,x,4,b*(3*a^2-b^2)*x-3*a*b^2*log(cos(c+d*x))/d+a^3*log(sin(c+d*x))/d+b^2*(a+b*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^3,x,4,-a*(a^2-3*b^2)*x-b^3*log(cos(c+d*x))/d+3*a^2*b*log(sin(c+d*x))/d-a^2*cot(c+d*x)*(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^3,x,4,-b*(3*a^2-b^2)*x-5/2*a^2*b*cot(c+d*x)/d-a*(a^2-3*b^2)*log(sin(c+d*x))/d-1/2*a^2*cot(c+d*x)^2*(a+b*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^3,x,5,a*(a^2-3*b^2)*x+a*(a^2-3*b^2)*cot(c+d*x)/d-7/6*a^2*b*cot(c+d*x)^2/d-b*(3*a^2-b^2)*log(sin(c+d*x))/d-1/3*a^2*cot(c+d*x)^3*(a+b*tan(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^3,x,6,b*(3*a^2-b^2)*x+b*(3*a^2-b^2)*cot(c+d*x)/d+1/2*a*(a^2-3*b^2)*cot(c+d*x)^2/d-3/4*a^2*b*cot(c+d*x)^3/d+a*(a^2-3*b^2)*log(sin(c+d*x))/d-1/4*a^2*cot(c+d*x)^4*(a+b*tan(c+d*x))/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^3,x,7,-a*(a^2-3*b^2)*x-a*(a^2-3*b^2)*cot(c+d*x)/d+1/2*b*(3*a^2-b^2)*cot(c+d*x)^2/d+1/3*a*(a^2-3*b^2)*cot(c+d*x)^3/d-11/20*a^2*b*cot(c+d*x)^4/d+b*(3*a^2-b^2)*log(sin(c+d*x))/d-1/5*a^2*cot(c+d*x)^5*(a+b*tan(c+d*x))/d],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^4,x,8,4*a*b*(a^2-b^2)*x+(a^4-6*a^2*b^2+b^4)*log(cos(c+d*x))/d-a*b*(a^2-3*b^2)*tan(c+d*x)/d-1/2*(a^2-b^2)*(a+b*tan(c+d*x))^2/d-1/3*a*(a+b*tan(c+d*x))^3/d-1/4*(a+b*tan(c+d*x))^4/d-1/30*a*(a+b*tan(c+d*x))^5/(b^2*d)+1/6*tan(c+d*x)*(a+b*tan(c+d*x))^5/(b*d)],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^4,x,5,-(a^4-6*a^2*b^2+b^4)*x+4*a*b*(a^2-b^2)*log(cos(c+d*x))/d-b^2*(3*a^2-b^2)*tan(c+d*x)/d-a*b*(a+b*tan(c+d*x))^2/d-1/3*b*(a+b*tan(c+d*x))^3/d+1/5*(a+b*tan(c+d*x))^5/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^4,x,5,-4*a*b*(a^2-b^2)*x-(a^4-6*a^2*b^2+b^4)*log(cos(c+d*x))/d+a*b*(a^2-3*b^2)*tan(c+d*x)/d+1/2*(a^2-b^2)*(a+b*tan(c+d*x))^2/d+1/3*a*(a+b*tan(c+d*x))^3/d+1/4*(a+b*tan(c+d*x))^4/d],
[(a+b*tan(c+d*x))^4,x,4,(a^4-6*a^2*b^2+b^4)*x-4*a*b*(a^2-b^2)*log(cos(c+d*x))/d+b^2*(3*a^2-b^2)*tan(c+d*x)/d+a*b*(a+b*tan(c+d*x))^2/d+1/3*b*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^4,x,5,4*a*b*(a^2-b^2)*x-b^2*(6*a^2-b^2)*log(cos(c+d*x))/d+a^4*log(sin(c+d*x))/d+3*a*b^3*tan(c+d*x)/d+1/2*b^2*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^4,x,5,-(a^4-6*a^2*b^2+b^4)*x-4*a*b^3*log(cos(c+d*x))/d+4*a^3*b*log(sin(c+d*x))/d+b^2*(a^2+b^2)*tan(c+d*x)/d-a^2*cot(c+d*x)*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^4,x,5,-4*a*b*(a^2-b^2)*x-3*a^3*b*cot(c+d*x)/d-b^4*log(cos(c+d*x))/d-a^2*(a^2-6*b^2)*log(sin(c+d*x))/d-1/2*a^2*cot(c+d*x)^2*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^4,x,5,(a^4-6*a^2*b^2+b^4)*x+1/3*a^2*(3*a^2-17*b^2)*cot(c+d*x)/d-4/3*a^3*b*cot(c+d*x)^2/d-4*a*b*(a^2-b^2)*log(sin(c+d*x))/d-1/3*a^2*cot(c+d*x)^3*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^4,x,6,4*a*b*(a^2-b^2)*x+4*a*b*(a^2-b^2)*cot(c+d*x)/d+1/4*a^2*(2*a^2-11*b^2)*cot(c+d*x)^2/d-5/6*a^3*b*cot(c+d*x)^3/d+(a^4-6*a^2*b^2+b^4)*log(sin(c+d*x))/d-1/4*a^2*cot(c+d*x)^4*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^4,x,7,-(a^4-6*a^2*b^2+b^4)*x-(a^4-6*a^2*b^2+b^4)*cot(c+d*x)/d+2*a*b*(a^2-b^2)*cot(c+d*x)^2/d+1/15*a^2*(5*a^2-27*b^2)*cot(c+d*x)^3/d-3/5*a^3*b*cot(c+d*x)^4/d+4*a*b*(a^2-b^2)*log(sin(c+d*x))/d-1/5*a^2*cot(c+d*x)^5*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^7*(a+b*tan(c+d*x))^4,x,8,-4*a*b*(a^2-b^2)*x-4*a*b*(a^2-b^2)*cot(c+d*x)/d-1/2*(a^4-6*a^2*b^2+b^4)*cot(c+d*x)^2/d+4/3*a*b*(a^2-b^2)*cot(c+d*x)^3/d+1/12*a^2*(3*a^2-16*b^2)*cot(c+d*x)^4/d-7/15*a^3*b*cot(c+d*x)^5/d-(a^4-6*a^2*b^2+b^4)*log(sin(c+d*x))/d-1/6*a^2*cot(c+d*x)^6*(a+b*tan(c+d*x))^2/d],

# m<0
[tan(c+d*x)^6/(a+b*tan(c+d*x)),x,8,-a*x/(a^2+b^2)-b*log(cos(c+d*x))/((a^2+b^2)*d)+a^6*log(a+b*tan(c+d*x))/(b^5*(a^2+b^2)*d)-a*(a^2-b^2)*tan(c+d*x)/(b^4*d)+1/2*(a^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)],
[tan(c+d*x)^5/(a+b*tan(c+d*x)),x,7,b*x/(a^2+b^2)-a*log(cos(c+d*x))/((a^2+b^2)*d)-a^5*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)*d)+(a^2-b^2)*tan(c+d*x)/(b^3*d)-1/2*a*tan(c+d*x)^2/(b^2*d)+1/3*tan(c+d*x)^3/(b*d)],
[tan(c+d*x)^4/(a+b*tan(c+d*x)),x,6,a*x/(a^2+b^2)+b*log(cos(c+d*x))/((a^2+b^2)*d)+a^4*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)-a*tan(c+d*x)/(b^2*d)+1/2*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)^3/(a+b*tan(c+d*x)),x,5,-b*x/(a^2+b^2)+a*log(cos(c+d*x))/((a^2+b^2)*d)-a^3*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)+tan(c+d*x)/(b*d)],
[tan(c+d*x)^2/(a+b*tan(c+d*x)),x,4,-a*x/(a^2+b^2)-log(cos(c+d*x))/(b*d)+a^2*log(a*cos(c+d*x)+b*sin(c+d*x))/(b*(a^2+b^2)*d),-a*x/b^2+a^3*x/(b^2*(a^2+b^2))-log(cos(c+d*x))/(b*d)+a^2*log(a*cos(c+d*x)+b*sin(c+d*x))/(b*(a^2+b^2)*d)],
[tan(c+d*x)/(a+b*tan(c+d*x)),x,2,b*x/(a^2+b^2)-a*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[1/(a+b*tan(c+d*x)),x,2,a*x/(a^2+b^2)+b*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[cot(c+d*x)/(a+b*tan(c+d*x)),x,3,-b*x/(a^2+b^2)+log(sin(c+d*x))/(a*d)-b^2*log(a*cos(c+d*x)+b*sin(c+d*x))/(a*(a^2+b^2)*d)],
[cot(c+d*x)^2/(a+b*tan(c+d*x)),x,4,-a*x/(a^2+b^2)-cot(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)+b^3*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)*d)],
[cot(c+d*x)^3/(a+b*tan(c+d*x)),x,5,b*x/(a^2+b^2)+b*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)^2/(a*d)-(a^2-b^2)*log(sin(c+d*x))/(a^3*d)-b^4*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)*d)],
[cot(c+d*x)^4/(a+b*tan(c+d*x)),x,6,a*x/(a^2+b^2)+(a^2-b^2)*cot(c+d*x)/(a^3*d)+1/2*b*cot(c+d*x)^2/(a^2*d)-1/3*cot(c+d*x)^3/(a*d)+b*(a^2-b^2)*log(sin(c+d*x))/(a^4*d)+b^5*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)*d)],
[tan(c+d*x)^6/(a+b*tan(c+d*x))^2,x,8,-(a^2-b^2)*x/(a^2+b^2)^2-2*a*b*log(cos(c+d*x))/((a^2+b^2)^2*d)-2*a^5*(2*a^2+3*b^2)*log(a+b*tan(c+d*x))/(b^5*(a^2+b^2)^2*d)+(4*a^4+2*a^2*b^2-b^4)*tan(c+d*x)/(b^4*(a^2+b^2)*d)-a*(2*a^2+b^2)*tan(c+d*x)^2/(b^3*(a^2+b^2)*d)+1/3*(4*a^2+b^2)*tan(c+d*x)^3/(b^2*(a^2+b^2)*d)-a^2*tan(c+d*x)^4/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^5/(a+b*tan(c+d*x))^2,x,7,2*a*b*x/(a^2+b^2)^2-(a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^2*d)+a^4*(3*a^2+5*b^2)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^2*d)-a*(3*a^2+2*b^2)*tan(c+d*x)/(b^3*(a^2+b^2)*d)+1/2*(3*a^2+b^2)*tan(c+d*x)^2/(b^2*(a^2+b^2)*d)-a^2*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^4/(a+b*tan(c+d*x))^2,x,6,(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*log(cos(c+d*x))/((a^2+b^2)^2*d)-2*a^3*(a^2+2*b^2)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^2*d)+(2*a^2+b^2)*tan(c+d*x)/(b^2*(a^2+b^2)*d)-a^2*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3/(a+b*tan(c+d*x))^2,x,5,-2*a*b*x/(a^2+b^2)^2+(a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^2*d)+a^2*(a^2+3*b^2)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)^2*d)+a^3/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x))),-2*a*b*x/(a^2+b^2)^2+(a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^2*d)+a^2*(a^2+3*b^2)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)^2*d)-a^2*tan(c+d*x)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2/(a+b*tan(c+d*x))^2,x,3,-(a^2-b^2)*x/(a^2+b^2)^2-2*a*b*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)-a^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)/(a+b*tan(c+d*x))^2,x,3,2*a*b*x/(a^2+b^2)^2-(a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+a/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[1/(a+b*tan(c+d*x))^2,x,3,(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)-b/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)/(a+b*tan(c+d*x))^2,x,4,-2*a*b*x/(a^2+b^2)^2+log(sin(c+d*x))/(a^2*d)-b^2*(3*a^2+b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)^2*d)+b^2/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2/(a+b*tan(c+d*x))^2,x,5,-(a^2-b^2)*x/(a^2+b^2)^2-2*b*log(sin(c+d*x))/(a^3*d)+2*b^3*(2*a^2+b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^2*d)-b*(a^2+2*b^2)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x)))-cot(c+d*x)/(a*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3/(a+b*tan(c+d*x))^2,x,6,2*a*b*x/(a^2+b^2)^2-(a^2-3*b^2)*log(sin(c+d*x))/(a^4*d)-b^4*(5*a^2+3*b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^2*d)+b^2*(2*a^2+3*b^2)/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x)))+3/2*b*cot(c+d*x)/(a^2*d*(a+b*tan(c+d*x)))-1/2*cot(c+d*x)^2/(a*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^6/(a+b*tan(c+d*x))^3,x,8,-a*(a^2-3*b^2)*x/(a^2+b^2)^3-b*(3*a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a^4*(6*a^4+17*a^2*b^2+15*b^4)*log(a+b*tan(c+d*x))/(b^5*(a^2+b^2)^3*d)-a*(6*a^4+11*a^2*b^2+3*b^4)*tan(c+d*x)/(b^4*(a^2+b^2)^2*d)+1/2*(6*a^4+11*a^2*b^2+b^4)*tan(c+d*x)^2/(b^3*(a^2+b^2)^2*d)-1/2*a^2*tan(c+d*x)^4/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-2*a^2*(a^2+2*b^2)*tan(c+d*x)^3/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^5/(a+b*tan(c+d*x))^3,x,7,b*(3*a^2-b^2)*x/(a^2+b^2)^3-a*(a^2-3*b^2)*log(cos(c+d*x))/((a^2+b^2)^3*d)-a^3*(3*a^4+9*a^2*b^2+10*b^4)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^3*d)+(3*a^4+6*a^2*b^2+b^4)*tan(c+d*x)/(b^3*(a^2+b^2)^2*d)-1/2*a^2*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/2*a^2*(3*a^2+7*b^2)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^4/(a+b*tan(c+d*x))^3,x,6,a*(a^2-3*b^2)*x/(a^2+b^2)^3+b*(3*a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a^2*(a^4+3*a^2*b^2+6*b^4)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^3*d)-1/2*a^2*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+a^3*(a^2+3*b^2)/(b^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3/(a+b*tan(c+d*x))^3,x,4,-b*(3*a^2-b^2)*x/(a^2+b^2)^3+a*(a^2-3*b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)-1/2*a^2*tan(c+d*x)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/2*a^2*(a^2+5*b^2)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2/(a+b*tan(c+d*x))^3,x,4,-a*(a^2-3*b^2)*x/(a^2+b^2)^3-b*(3*a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)-1/2*a^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+2*a*b/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)/(a+b*tan(c+d*x))^3,x,4,b*(3*a^2-b^2)*x/(a^2+b^2)^3-a*(a^2-3*b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*a/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(a^2-b^2)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[1/(a+b*tan(c+d*x))^3,x,4,a*(a^2-3*b^2)*x/(a^2+b^2)^3+b*(3*a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)-1/2*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)-2*a*b/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)/(a+b*tan(c+d*x))^3,x,5,-b*(3*a^2-b^2)*x/(a^2+b^2)^3+log(sin(c+d*x))/(a^3*d)-b^2*(6*a^4+3*a^2*b^2+b^4)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^3*d)+1/2*b^2/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+b^2*(3*a^2+b^2)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2/(a+b*tan(c+d*x))^3,x,6,-a*(a^2-3*b^2)*x/(a^2+b^2)^3-3*b*log(sin(c+d*x))/(a^4*d)+b^3*(10*a^4+9*a^2*b^2+3*b^4)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^3*d)-1/2*b*(2*a^2+3*b^2)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^2)-b*(a^4+6*a^2*b^2+3*b^4)/(a^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^6/(a+b*tan(c+d*x))^4,x,8,-(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-4*a*b*(a^2-b^2)*log(cos(c+d*x))/((a^2+b^2)^4*d)-4*a^3*(a^6+4*a^4*b^2+6*a^2*b^4+5*b^6)*log(a+b*tan(c+d*x))/(b^5*(a^2+b^2)^4*d)+(4*a^6+12*a^4*b^2+13*a^2*b^4+b^6)*tan(c+d*x)/(b^4*(a^2+b^2)^3*d)-1/3*a^2*tan(c+d*x)^4/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)-1/3*a^2*(2*a^2+5*b^2)*tan(c+d*x)^3/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-2*a^2*(a^4+3*a^2*b^2+4*b^4)*tan(c+d*x)^2/(b^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^5/(a+b*tan(c+d*x))^4,x,7,4*a*b*(a^2-b^2)*x/(a^2+b^2)^4-(a^4-6*a^2*b^2+b^4)*log(cos(c+d*x))/((a^2+b^2)^4*d)+a^2*(a^6+4*a^4*b^2+5*a^2*b^4+10*b^6)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^4*d)-1/3*a^2*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)-1/2*a^2*(a^2+3*b^2)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+a^3*(a^4+3*a^2*b^2+6*b^4)/(b^4*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^4/(a+b*tan(c+d*x))^4,x,5,(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4+4*a*b*(a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)-1/3*a^2*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/3*a^3*(a^2+4*b^2)/(b^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-1/3*a^2*(2*a^4+7*a^2*b^2+17*b^4)/(b^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3/(a+b*tan(c+d*x))^4,x,5,-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+(a^4-6*a^2*b^2+b^4)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)-1/3*a^2*tan(c+d*x)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)-1/6*a^2*(a^2+7*b^2)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-a*(a^2-3*b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2/(a+b*tan(c+d*x))^4,x,5,-(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-4*a*b*(a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)-1/3*a^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+a*b/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+b*(3*a^2-b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)/(a+b*tan(c+d*x))^4,x,5,4*a*b*(a^2-b^2)*x/(a^2+b^2)^4-(a^4-6*a^2*b^2+b^4)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+1/3*a/((a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*(a^2-b^2)/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+a*(a^2-3*b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[1/(a+b*tan(c+d*x))^4,x,5,(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4+4*a*b*(a^2-b^2)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)-1/3*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^3)-a*b/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-b*(3*a^2-b^2)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)/(a+b*tan(c+d*x))^4,x,6,-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+log(sin(c+d*x))/(a^4*d)-b^2*(10*a^6+5*a^4*b^2+4*a^2*b^4+b^6)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^4*d)+1/3*b^2/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*b^2*(3*a^2+b^2)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+b^2*(6*a^4+3*a^2*b^2+b^4)/(a^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2/(a+b*tan(c+d*x))^4,x,7,-(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-4*b*log(sin(c+d*x))/(a^5*d)+4*b^3*(5*a^6+6*a^4*b^2+4*a^2*b^4+b^6)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^5*(a^2+b^2)^4*d)-1/3*b*(3*a^2+4*b^2)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)-cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^3)-b*(a^4+4*a^2*b^2+2*b^4)/(a^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-b*(a^6+13*a^4*b^2+12*a^2*b^4+4*b^6)/(a^4*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[1/(3+5*tan(c+d*x)),x,2,3/34*x+5/34*log(3*cos(c+d*x)+5*sin(c+d*x))/d],
[1/(3+5*tan(c+d*x))^2,x,3,-4/289*x+15/578*log(3*cos(c+d*x)+5*sin(c+d*x))/d+(-5/34)/(d*(3+5*tan(c+d*x)))],
[1/(3+5*tan(c+d*x))^3,x,4,-99/19652*x+5/19652*log(3*cos(c+d*x)+5*sin(c+d*x))/d+(-5/68)/(d*(3+5*tan(c+d*x))^2)+(-15/578)/(d*(3+5*tan(c+d*x)))],
[1/(3+5*tan(c+d*x))^4,x,5,-161/334084*x-60/83521*log(3*cos(c+d*x)+5*sin(c+d*x))/d+(-5/102)/(d*(3+5*tan(c+d*x))^3)+(-15/1156)/(d*(3+5*tan(c+d*x))^2)+(-5/19652)/(d*(3+5*tan(c+d*x)))],
[1/(5+3*tan(c+d*x)),x,2,5/34*x+3/34*log(5*cos(c+d*x)+3*sin(c+d*x))/d],
[1/(5+3*tan(c+d*x))^2,x,3,4/289*x+15/578*log(5*cos(c+d*x)+3*sin(c+d*x))/d+(-3/34)/(d*(5+3*tan(c+d*x)))],
[1/(5+3*tan(c+d*x))^3,x,4,-5/19652*x+99/19652*log(5*cos(c+d*x)+3*sin(c+d*x))/d+(-3/68)/(d*(5+3*tan(c+d*x))^2)+(-15/578)/(d*(5+3*tan(c+d*x)))],
[1/(5+3*tan(c+d*x))^4,x,5,-161/334084*x+60/83521*log(5*cos(c+d*x)+3*sin(c+d*x))/d+(-1/34)/(d*(5+3*tan(c+d*x))^3)+(-15/1156)/(d*(5+3*tan(c+d*x))^2)+(-99/19652)/(d*(5+3*tan(c+d*x)))],

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

# m>0
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^4,x,14,b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+2/105*(8*a^2-35*b^2)*(a+b*tan(c+d*x))^(3/2)/(b^3*d)-8/35*a*tan(c+d*x)*(a+b*tan(c+d*x))^(3/2)/(b^2*d)+2/7*tan(c+d*x)^2*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^3,x,11,arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-2*sqrt(a+b*tan(c+d*x))/d-4/15*a*(a+b*tan(c+d*x))^(3/2)/(b^2*d)+2/5*tan(c+d*x)*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2,x,12,-b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+2/3*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*tan(c+d*x),x,8,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d+2*sqrt(a+b*tan(c+d*x))/d],
[sqrt(a+b*tan(c+d*x)),x,11,b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))],
[cot(c+d*x)*sqrt(a+b*tan(c+d*x)),x,11,-2*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d],
[cot(c+d*x)^2*sqrt(a+b*tan(c+d*x)),x,16,-b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))-b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))-cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*sqrt(a+b*tan(c+d*x)),x,13,1/4*(8*a^2+b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-1/4*b*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)-1/2*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d],
[tan(c+d*x)^4*(a+b*tan(c+d*x))^(3/2),x,11,-I*(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+I*(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b*sqrt(a+b*tan(c+d*x))/d+2/315*(8*a^2-63*b^2)*(a+b*tan(c+d*x))^(5/2)/(b^3*d)-8/63*a*tan(c+d*x)*(a+b*tan(c+d*x))^(5/2)/(b^2*d)+2/9*tan(c+d*x)^2*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^(3/2),x,12,(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*a*sqrt(a+b*tan(c+d*x))/d-2/3*(a+b*tan(c+d*x))^(3/2)/d-4/35*a*(a+b*tan(c+d*x))^(5/2)/(b^2*d)+2/7*tan(c+d*x)*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^(3/2),x,9,I*(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-I*(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*b*sqrt(a+b*tan(c+d*x))/d+2/5*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^(3/2),x,9,-(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*a*sqrt(a+b*tan(c+d*x))/d+2/3*(a+b*tan(c+d*x))^(3/2)/d],
[(a+b*tan(c+d*x))^(3/2),x,8,-I*(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+I*(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^(3/2),x,11,-2*a^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^(3/2),x,12,I*(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-I*(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-3*b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d-a*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^(3/2),x,13,-(a-I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/4*(8*a^2-3*b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))-5/4*b*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-1/2*a*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^(5/2),x,13,(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*(a^2-b^2)*sqrt(a+b*tan(c+d*x))/d-2/3*a*(a+b*tan(c+d*x))^(3/2)/d-2/5*(a+b*tan(c+d*x))^(5/2)/d-4/63*a*(a+b*tan(c+d*x))^(7/2)/(b^2*d)+2/9*tan(c+d*x)*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^(5/2),x,10,I*(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-I*(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-4*a*b*sqrt(a+b*tan(c+d*x))/d-2/3*b*(a+b*tan(c+d*x))^(3/2)/d+2/7*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^(5/2),x,10,-(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*(a^2-b^2)*sqrt(a+b*tan(c+d*x))/d+2/3*a*(a+b*tan(c+d*x))^(3/2)/d+2/5*(a+b*tan(c+d*x))^(5/2)/d],
[(a+b*tan(c+d*x))^(5/2),x,9,-I*(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+I*(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+4*a*b*sqrt(a+b*tan(c+d*x))/d+2/3*b*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^(5/2),x,12,-2*a^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b^2*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^(5/2),x,12,-5*a^(3/2)*b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+I*(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-I*(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-a^2*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^(5/2),x,13,-(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/4*(8*a^2-15*b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d-9/4*a*b*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-1/2*a^2*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^(5/2),x,14,-I*(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+I*(a+I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+5/8*b*(8*a^2-b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+1/8*(8*a^2-11*b^2)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-13/12*a*b*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d-1/3*a^2*cot(c+d*x)^3*sqrt(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^(7/2),x,10,-I*(a-I*b)^(7/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+I*(a+I*b)^(7/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b*(3*a^2-b^2)*sqrt(a+b*tan(c+d*x))/d+4/3*a*b*(a+b*tan(c+d*x))^(3/2)/d+2/5*b*(a+b*tan(c+d*x))^(5/2)/d],

# m<0
[tan(c+d*x)^5/sqrt(a+b*tan(c+d*x)),x,12,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))-4/105*a*(24*a^2-35*b^2)*sqrt(a+b*tan(c+d*x))/(b^4*d)+2/105*(24*a^2-35*b^2)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^3*d)-12/35*a*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2/(b^2*d)+2/7*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^3/(b*d)],
[tan(c+d*x)^4/sqrt(a+b*tan(c+d*x)),x,14,b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))+2/15*(8*a^2-15*b^2)*sqrt(a+b*tan(c+d*x))/(b^3*d)-8/15*a*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)^3/sqrt(a+b*tan(c+d*x)),x,10,arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))-4/3*a*sqrt(a+b*tan(c+d*x))/(b^2*d)+2/3*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b*d)],
[tan(c+d*x)^2/sqrt(a+b*tan(c+d*x)),x,12,-b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))+b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))+2*sqrt(a+b*tan(c+d*x))/(b*d)],
[tan(c+d*x)/sqrt(a+b*tan(c+d*x)),x,7,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[1/sqrt(a+b*tan(c+d*x)),x,11,b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))],
[cot(c+d*x)/sqrt(a+b*tan(c+d*x)),x,11,-2*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[cot(c+d*x)^2/sqrt(a+b*tan(c+d*x)),x,17,b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))+b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))-cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^3/sqrt(a+b*tan(c+d*x)),x,14,1/4*(8*a^2-3*b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))+3/4*b*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a^2*d)-1/2*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/(a*d)],
[tan(c+d*x)^5/(a+b*tan(c+d*x))^(3/2),x,11,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2/5*(16*a^4+6*a^2*b^2-5*b^4)*sqrt(a+b*tan(c+d*x))/(b^4*(a^2+b^2)*d)-2/5*a*(8*a^2+3*b^2)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^3*(a^2+b^2)*d)+2/5*(6*a^2+b^2)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2/(b^2*(a^2+b^2)*d)-2*a^2*tan(c+d*x)^3/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^4/(a+b*tan(c+d*x))^(3/2),x,10,-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2/3*a*(8*a^2+5*b^2)*sqrt(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)+2/3*(4*a^2+b^2)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^2*(a^2+b^2)*d)-2*a^2*tan(c+d*x)^2/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^3/(a+b*tan(c+d*x))^(3/2),x,9,arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*(2*a^2+b^2)*sqrt(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)-2*a^2*tan(c+d*x)/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^2/(a+b*tan(c+d*x))^(3/2),x,8,I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2*a^2/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)/(a+b*tan(c+d*x))^(3/2),x,8,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*a/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[1/(a+b*tan(c+d*x))^(3/2),x,8,-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2*b/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)/(a+b*tan(c+d*x))^(3/2),x,12,-2*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*b^2/(a*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^2/(a+b*tan(c+d*x))^(3/2),x,13,3*b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-b*(a^2+3*b^2)/(a^2*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))-cot(c+d*x)/(a*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^3/(a+b*tan(c+d*x))^(3/2),x,14,1/4*(8*a^2-15*b^2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(7/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+1/4*b^2*(7*a^2+15*b^2)/(a^3*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))+5/4*b*cot(c+d*x)/(a^2*d*sqrt(a+b*tan(c+d*x)))-1/2*cot(c+d*x)^2/(a*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^5/(a+b*tan(c+d*x))^(5/2),x,11,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-4/3*a*(8*a^4+15*a^2*b^2+4*b^4)*sqrt(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^2*d)+2/3*(8*a^4+15*a^2*b^2+b^4)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^3*(a^2+b^2)^2*d)-4*a^2*(a^2+2*b^2)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^4/(a+b*tan(c+d*x))^(5/2),x,10,-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+4/3*a^3*(2*a^2+5*b^2)/(b^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*(4*a^2+3*b^2)*sqrt(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)-2/3*a^2*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^3/(a+b*tan(c+d*x))^(5/2),x,9,arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-4/3*a^2*(a^2+4*b^2)/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*tan(c+d*x)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^2/(a+b*tan(c+d*x))^(5/2),x,9,I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+4*a*b/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)/(a+b*tan(c+d*x))^(5/2),x,9,-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*(a^2-b^2)/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[1/(a+b*tan(c+d*x))^(5/2),x,9,-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-4*a*b/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)/(a+b*tan(c+d*x))^(5/2),x,13,-2*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*b^2*(3*a^2+b^2)/(a^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*b^2/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2/(a+b*tan(c+d*x))^(5/2),x,14,5*b*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(7/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-b*(a^4+10*a^2*b^2+5*b^4)/(a^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-1/3*b*(3*a^2+5*b^2)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))-cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^(3/2))],
[1/(a+b*tan(c+d*x))^(7/2),x,10,-I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(7/2)*d)+I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(7/2)*d)-2*b*(3*a^2-b^2)/((a^2+b^2)^3*d*sqrt(a+b*tan(c+d*x)))-2/5*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^(5/2))-4/3*a*b/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^(3/2))],

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

# m>0
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x)),x,13,(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*b*sqrt(tan(c+d*x))/d+2/3*a*tan(c+d*x)^(3/2)/d+2/5*b*tan(c+d*x)^(5/2)/d],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x)),x,12,(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*a*sqrt(tan(c+d*x))/d+2/3*b*tan(c+d*x)^(3/2)/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x)),x,11,-(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*sqrt(tan(c+d*x))/d],
[(a+b*tan(c+d*x))/sqrt(tan(c+d*x)),x,10,-(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))],
[(a+b*tan(c+d*x))/tan(c+d*x)^(3/2),x,11,(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))/tan(c+d*x)^(5/2),x,12,(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*b/(d*sqrt(tan(c+d*x)))-2/3*a/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))/tan(c+d*x)^(7/2),x,13,-(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*a/(d*sqrt(tan(c+d*x)))-2/5*a/(d*tan(c+d*x)^(5/2))-2/3*b/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2,x,14,(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-4*a*b*sqrt(tan(c+d*x))/d+2/3*(a^2-b^2)*tan(c+d*x)^(3/2)/d+4/5*a*b*tan(c+d*x)^(5/2)/d+2/7*b^2*tan(c+d*x)^(7/2)/d],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2,x,13,(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a^2-b^2)*sqrt(tan(c+d*x))/d+4/3*a*b*tan(c+d*x)^(3/2)/d+2/5*b^2*tan(c+d*x)^(5/2)/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2,x,12,-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+4*a*b*sqrt(tan(c+d*x))/d+2/3*b^2*tan(c+d*x)^(3/2)/d],
[(a+b*tan(c+d*x))^2/sqrt(tan(c+d*x)),x,11,-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b^2*sqrt(tan(c+d*x))/d],
[(a+b*tan(c+d*x))^2/tan(c+d*x)^(3/2),x,11,(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a^2/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^2/tan(c+d*x)^(5/2),x,12,(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-4*a*b/(d*sqrt(tan(c+d*x)))-2/3*a^2/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^2/tan(c+d*x)^(7/2),x,13,-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a^2-b^2)/(d*sqrt(tan(c+d*x)))-2/5*a^2/(d*tan(c+d*x)^(5/2))-4/3*a*b/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3,x,15,(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*b*(3*a^2-b^2)*sqrt(tan(c+d*x))/d+2/3*a*(a^2-3*b^2)*tan(c+d*x)^(3/2)/d+2/5*b*(3*a^2-b^2)*tan(c+d*x)^(5/2)/d+40/63*a*b^2*tan(c+d*x)^(7/2)/d+2/9*b^2*tan(c+d*x)^(7/2)*(a+b*tan(c+d*x))/d],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3,x,14,(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*a*(a^2-3*b^2)*sqrt(tan(c+d*x))/d+2/3*b*(3*a^2-b^2)*tan(c+d*x)^(3/2)/d+32/35*a*b^2*tan(c+d*x)^(5/2)/d+2/7*b^2*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^3,x,13,-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*(3*a^2-b^2)*sqrt(tan(c+d*x))/d+8/5*a*b^2*tan(c+d*x)^(3/2)/d+2/5*b^2*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^3/sqrt(tan(c+d*x)),x,12,-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+16/3*a*b^2*sqrt(tan(c+d*x))/d+2/3*b^2*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^3/tan(c+d*x)^(3/2),x,12,(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*(a^2+b^2)*sqrt(tan(c+d*x))/d-2*a^2*(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^3/tan(c+d*x)^(5/2),x,12,(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-16/3*a^2*b/(d*sqrt(tan(c+d*x)))-2/3*a^2*(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^3/tan(c+d*x)^(7/2),x,13,-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*a*(a^2-3*b^2)/(d*sqrt(tan(c+d*x)))-8/5*a^2*b/(d*tan(c+d*x)^(3/2))-2/5*a^2*(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))],
[(a+b*tan(c+d*x))^3/tan(c+d*x)^(9/2),x,14,-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*(3*a^2-b^2)/(d*sqrt(tan(c+d*x)))-32/35*a^2*b/(d*tan(c+d*x)^(5/2))+2/3*a*(a^2-3*b^2)/(d*tan(c+d*x)^(3/2))-2/7*a^2*(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))],
[(a+b*tan(c+d*x))^3/tan(c+d*x)^(11/2),x,15,(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a*(a^2-3*b^2)/(d*sqrt(tan(c+d*x)))-40/63*a^2*b/(d*tan(c+d*x)^(7/2))+2/5*a*(a^2-3*b^2)/(d*tan(c+d*x)^(5/2))+2/3*b*(3*a^2-b^2)/(d*tan(c+d*x)^(3/2))-2/9*a^2*(a+b*tan(c+d*x))/(d*tan(c+d*x)^(9/2))],
[(a+b*tan(c+d*x))/sqrt(tan(c+d*x)),x,10,-(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))],
[(a+b*tan(c+d*x))/sqrt(-tan(c+d*x)),x,10,(a-b)*arctan(1-sqrt(2)*sqrt(-tan(c+d*x)))/(d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(-tan(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1-sqrt(2)*sqrt(-tan(c+d*x))-tan(c+d*x))/(d*sqrt(2))-1/2*(a+b)*log(1+sqrt(2)*sqrt(-tan(c+d*x))-tan(c+d*x))/(d*sqrt(2))],
[(a+b*tan(c+d*x))/sqrt(e*tan(c+d*x)),x,10,-(a+b)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))+(a+b)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))-1/2*(a-b)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))+1/2*(a-b)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))],
[(a+b*tan(c+d*x))/sqrt(-e*tan(c+d*x)),x,10,(a-b)*arctan(1-sqrt(2)*sqrt(-e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))-(a-b)*arctan(1+sqrt(2)*sqrt(-e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))+1/2*(a+b)*log(sqrt(e)-sqrt(2)*sqrt(-e*tan(c+d*x))-sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))-1/2*(a+b)*log(sqrt(e)+sqrt(2)*sqrt(-e*tan(c+d*x))-sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))],

# m<0
[tan(c+d*x)^(9/2)/(a+b*tan(c+d*x)),x,17,-2*a^(9/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(7/2)*(a^2+b^2)*d)-(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*(a^2-b^2)*sqrt(tan(c+d*x))/(b^3*d)-2/3*a*tan(c+d*x)^(3/2)/(b^2*d)+2/5*tan(c+d*x)^(5/2)/(b*d)],
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x)),x,16,2*a^(7/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(5/2)*(a^2+b^2)*d)-(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*a*sqrt(tan(c+d*x))/(b^2*d)+2/3*tan(c+d*x)^(3/2)/(b*d)],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x)),x,15,-2*a^(5/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(3/2)*(a^2+b^2)*d)+(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*sqrt(tan(c+d*x))/(b*d)],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x)),x,14,(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*a^(3/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)*d*sqrt(b))],
[sqrt(tan(c+d*x))/(a+b*tan(c+d*x)),x,14,-(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)*sqrt(b)/((a^2+b^2)*d)],
[1/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))),x,14,-(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*b^(3/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)*d*sqrt(a))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,15,-2*b^(5/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(3/2)*(a^2+b^2)*d)+(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+(-2)/(a*d*sqrt(tan(c+d*x)))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,16,2*b^(7/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(5/2)*(a^2+b^2)*d)+(a-b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*b/(a^2*d*sqrt(tan(c+d*x)))+(-2/3)/(a*d*tan(c+d*x)^(3/2))],
[1/(tan(c+d*x)^(7/2)*(a+b*tan(c+d*x))),x,17,-2*b^(9/2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(7/2)*(a^2+b^2)*d)-(a+b)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*(a^2-b^2)/(a^3*d*sqrt(tan(c+d*x)))+(-2/5)/(a*d*tan(c+d*x)^(5/2))+2/3*b/(a^2*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(9/2)/(a+b*tan(c+d*x))^2,x,17,a^(7/2)*(5*a^2+9*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(7/2)*(a^2+b^2)^2*d)-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-a*(5*a^2+4*b^2)*sqrt(tan(c+d*x))/(b^3*(a^2+b^2)*d)+1/3*(5*a^2+2*b^2)*tan(c+d*x)^(3/2)/(b^2*(a^2+b^2)*d)-a^2*tan(c+d*x)^(5/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x))^2,x,16,-a^(5/2)*(3*a^2+7*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(5/2)*(a^2+b^2)^2*d)-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(3*a^2+2*b^2)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)*d)-a^2*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x))^2,x,15,a^(3/2)*(a^2+5*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(3/2)*(a^2+b^2)^2*d)+(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-a^2*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x))^2,x,15,(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(a^2-3*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)/((a^2+b^2)^2*d*sqrt(b))+a*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[sqrt(tan(c+d*x))/(a+b*tan(c+d*x))^2,x,15,-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-(3*a^2-b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(b)/((a^2+b^2)^2*d*sqrt(a))-b*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[1/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2),x,15,b^(3/2)*(5*a^2+b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(3/2)*(a^2+b^2)^2*d)-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+b^2*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2),x,16,-b^(5/2)*(7*a^2+3*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(5/2)*(a^2+b^2)^2*d)+(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(-2*a^2-3*b^2)/(a^2*(a^2+b^2)*d*sqrt(tan(c+d*x)))+b^2/(a*(a^2+b^2)*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2),x,17,b^(7/2)*(9*a^2+5*b^2)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(7/2)*(a^2+b^2)^2*d)+(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+b*(4*a^2+5*b^2)/(a^3*(a^2+b^2)*d*sqrt(tan(c+d*x)))+1/3*(-2*a^2-5*b^2)/(a^2*(a^2+b^2)*d*tan(c+d*x)^(3/2))+b^2/(a*(a^2+b^2)*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(11/2)/(a+b*tan(c+d*x))^3,x,18,1/4*a^(7/2)*(35*a^4+102*a^2*b^2+99*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(9/2)*(a^2+b^2)^3*d)+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/4*a*(35*a^4+67*a^2*b^2+24*b^4)*sqrt(tan(c+d*x))/(b^4*(a^2+b^2)^2*d)+1/12*(35*a^4+67*a^2*b^2+8*b^4)*tan(c+d*x)^(3/2)/(b^3*(a^2+b^2)^2*d)-1/2*a^2*tan(c+d*x)^(7/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*a^2*(7*a^2+15*b^2)*tan(c+d*x)^(5/2)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(9/2)/(a+b*tan(c+d*x))^3,x,17,-1/4*a^(5/2)*(15*a^4+46*a^2*b^2+63*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(7/2)*(a^2+b^2)^3*d)-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(15*a^4+31*a^2*b^2+8*b^4)*sqrt(tan(c+d*x))/(b^3*(a^2+b^2)^2*d)-1/2*a^2*tan(c+d*x)^(5/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*a^2*(5*a^2+13*b^2)*tan(c+d*x)^(3/2)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x))^3,x,16,1/4*a^(3/2)*(3*a^4+6*a^2*b^2+35*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(5/2)*(a^2+b^2)^3*d)-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*a^2*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*a^2*(3*a^2+11*b^2)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x))^3,x,16,(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(a^4+18*a^2*b^2-15*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)/(b^(3/2)*(a^2+b^2)^3*d)-1/2*a^2*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*a*(a^2+9*b^2)*sqrt(tan(c+d*x))/(b*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x))^3,x,16,(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(3*a^4-26*a^2*b^2+3*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)^3*d*sqrt(a)*sqrt(b))+1/2*a*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*(3*a^2-5*b^2)*sqrt(tan(c+d*x))/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[sqrt(tan(c+d*x))/(a+b*tan(c+d*x))^3,x,16,-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/4*(15*a^4-18*a^2*b^2-b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(b)/(a^(3/2)*(a^2+b^2)^3*d)-1/2*b*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*b*(7*a^2-b^2)*sqrt(tan(c+d*x))/(a*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[1/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^3),x,16,1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(5/2)*(a^2+b^2)^3*d)-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*b^2*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*b^2*(11*a^2+3*b^2)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3),x,17,-1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(7/2)*(a^2+b^2)^3*d)+(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(-8*a^4-31*a^2*b^2-15*b^4)/(a^3*(a^2+b^2)^2*d*sqrt(tan(c+d*x)))+1/2*b^2/(a*(a^2+b^2)*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2)+1/4*b^2*(13*a^2+5*b^2)/(a^2*(a^2+b^2)^2*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3),x,18,1/4*b^(7/2)*(99*a^4+102*a^2*b^2+35*b^4)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(9/2)*(a^2+b^2)^3*d)+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*b*(24*a^4+67*a^2*b^2+35*b^4)/(a^4*(a^2+b^2)^2*d*sqrt(tan(c+d*x)))+1/12*(-8*a^4-67*a^2*b^2-35*b^4)/(a^3*(a^2+b^2)^2*d*tan(c+d*x)^(3/2))+1/2*b^2/(a*(a^2+b^2)*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2)+1/4*b^2*(15*a^2+7*b^2)/(a^2*(a^2+b^2)^2*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x)))],

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

# m>0
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(1/2),x,14,-1/4*(a^2+8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d-1/4*a*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)+1/2*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(1/2),x,13,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(1/2),x,11,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d],
[(a+b*tan(c+d*x))^(1/2)/tan(c+d*x)^(1/2),x,7,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d],
[(a+b*tan(c+d*x))^(1/2)/tan(c+d*x)^(3/2),x,8,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d-2*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^(1/2)/tan(c+d*x)^(5/2),x,10,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d-2/3*b*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/3*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(1/2)/tan(c+d*x)^(7/2),x,10,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+2/15*(15*a^2+2*b^2)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/5*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*b*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2),x,15,I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-1/8*a*(a^2+24*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-1/8*(a^2+8*b^2)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)-1/12*a*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)+1/3*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2),x,14,(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/4*(3*a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))+3/4*a*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+1/2*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/d],
[tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(3/2),x,13,-I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+3*a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d+b*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^(3/2)/tan(c+d*x)^(1/2),x,12,-(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2*b^(3/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d],
[(a+b*tan(c+d*x))^(3/2)/tan(c+d*x)^(3/2),x,8,I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2*a*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^(3/2)/tan(c+d*x)^(5/2),x,9,(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-8/3*b*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(3/2)/tan(c+d*x)^(7/2),x,10,-I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/5*(5*a^2-b^2)*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/5*a*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-4/5*b*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(3/2)/tan(c+d*x)^(9/2),x,11,-(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+4/105*b*(70*a^2+3*b^2)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/7*a*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-16/35*b*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/105*(35*a^2-3*b^2)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2),x,16,(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-1/64*(5*a^4+240*a^2*b^2-128*b^4)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-1/64*a*(5*a^2+112*b^2)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)-1/96*(5*a^2+48*b^2)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)-1/24*a*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)/(b*d)+1/4*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2),x,15,-I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+5/8*a*(a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))+1/8*(11*a^2-8*b^2)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+13/12*a*b*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)/d+1/3*b^2*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(5/2)/d],
[tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(5/2),x,14,-(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/4*(15*a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d+9/4*a*b*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+1/2*b^2*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(1/2),x,13,I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+5*a*b^(3/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+b^2*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(3/2),x,13,(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2*b^(5/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2*a^2*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(5/2),x,9,-I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-14/3*a*b*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a^2*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(7/2),x,10,-(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/15*(15*a^2-23*b^2)*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/5*a^2*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-22/15*a*b*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(9/2),x,11,I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/21*b*(49*a^2-3*b^2)*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/7*a^2*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-6/7*a*b*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/21*(7*a^2-9*b^2)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(5/2)/tan(c+d*x)^(11/2),x,12,(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2/315*(315*a^4-483*a^2*b^2-10*b^4)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/9*a^2*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(9/2))-38/63*a*b*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+2/105*(21*a^2-25*b^2)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/315*b*(231*a^2-5*b^2)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],

# m<0
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x))^(1/2),x,14,1/4*(3*a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(5/2)*d)+arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))-3/4*a*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b^2*d)+1/2*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)/(b*d)],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x))^(1/2),x,13,-a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x))^(1/2),x,12,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[tan(c+d*x)^(1/2)/(a+b*tan(c+d*x))^(1/2),x,7,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[1/(tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(1/2)),x,7,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(1/2)),x,9,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))-2*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(1/2)),x,10,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+4/3*b*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/3*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[1/(tan(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(1/2)),x,11,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+2/15*(15*a^2-8*b^2)*sqrt(a+b*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))-2/5*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+8/15*b*sqrt(a+b*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x))^(3/2),x,14,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-3*a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(5/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+(3*a^2+b^2)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)-2*a^2*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x))^(3/2),x,13,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)-2*a^2*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x))^(3/2),x,8,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+2*a*sqrt(tan(c+d*x))/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^(1/2)/(a+b*tan(c+d*x))^(3/2),x,8,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)-2*b*sqrt(tan(c+d*x))/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(3/2)),x,8,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+2*b^2*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,9,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+(-2)/(a*d*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x)))-2*b*(a^2+2*b^2)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2)),x,10,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+8/3*b/(a^2*d*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*b^2*(5*a^2+8*b^2)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))+(-2/3)/(a*d*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(9/2)/(a+b*tan(c+d*x))^(5/2),x,15,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-5*a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(7/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+(5*a^4+10*a^2*b^2+b^4)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^2*d)-2/3*a^2*(5*a^2+11*b^2)*tan(c+d*x)^(3/2)/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*tan(c+d*x)^(5/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(7/2)/(a+b*tan(c+d*x))^(5/2),x,14,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(5/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)-2*a^2*(a^2+3*b^2)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(5/2)/(a+b*tan(c+d*x))^(5/2),x,9,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+2/3*a*(a^2+7*b^2)*sqrt(tan(c+d*x))/(b*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)/(a+b*tan(c+d*x))^(5/2),x,9,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+4/3*(a^2-2*b^2)*sqrt(tan(c+d*x))/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(1/2)/(a+b*tan(c+d*x))^(5/2),x,9,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)-2/3*b*(5*a^2-b^2)*sqrt(tan(c+d*x))/(a*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*b*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(5/2)),x,9,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+4/3*b^2*(4*a^2+b^2)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*b^2*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)),x,10,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)-2/3*b*(3*a^4+17*a^2*b^2+8*b^4)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+(-2)/(a*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2))-2/3*b*(3*a^2+4*b^2)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[1/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2)),x,11,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+4/3*b^2*(4*a^4+15*a^2*b^2+8*b^4)*sqrt(tan(c+d*x))/(a^4*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+4*b/(a^2*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2))+2/3*b^2*(7*a^2+8*b^2)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))+(-2/3)/(a*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2))],
[1/(sqrt(tan(c+d*x))*sqrt(2+3*tan(c+d*x))),x,7,arctanh(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(2+3*tan(c+d*x)))/(d*sqrt(3-2*I))+arctanh(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(2+3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[1/(sqrt(tan(c+d*x))*sqrt(-2+3*tan(c+d*x))),x,7,arctanh(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(-2+3*tan(c+d*x)))/(d*sqrt(3-2*I))+arctanh(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(-2+3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[1/(sqrt(2-3*tan(c+d*x))*sqrt(tan(c+d*x))),x,7,arctan(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(2-3*tan(c+d*x)))/(d*sqrt(3-2*I))+arctan(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(2-3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[1/(sqrt(-2-3*tan(c+d*x))*sqrt(tan(c+d*x))),x,7,arctan(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(-2-3*tan(c+d*x)))/(d*sqrt(3-2*I))+arctan(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(-2-3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[1/(sqrt(tan(c+d*x))*sqrt(3+2*tan(c+d*x))),x,7,arctanh(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(3+2*tan(c+d*x)))/(d*sqrt(2-3*I))+arctanh(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(3+2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[1/(sqrt(3-2*tan(c+d*x))*sqrt(tan(c+d*x))),x,7,arctan(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(3-2*tan(c+d*x)))/(d*sqrt(2-3*I))+arctan(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(3-2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[1/(sqrt(tan(c+d*x))*sqrt(-3+2*tan(c+d*x))),x,7,arctanh(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(-3+2*tan(c+d*x)))/(d*sqrt(2-3*I))+arctanh(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(-3+2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[1/(sqrt(-3-2*tan(c+d*x))*sqrt(tan(c+d*x))),x,7,arctan(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(-3-2*tan(c+d*x)))/(d*sqrt(2-3*I))+arctan(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(-3-2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[sqrt(tan(c+d*x))/sqrt(2+3*tan(c+d*x)),x,7,I*arctanh(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(2+3*tan(c+d*x)))/(d*sqrt(3-2*I))-I*arctanh(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(2+3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[sqrt(tan(c+d*x))/sqrt(-2+3*tan(c+d*x)),x,7,-I*arctanh(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(-2+3*tan(c+d*x)))/(d*sqrt(3-2*I))+I*arctanh(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(-2+3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[sqrt(tan(c+d*x))/sqrt(2-3*tan(c+d*x)),x,7,-I*arctan(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(2-3*tan(c+d*x)))/(d*sqrt(3-2*I))+I*arctan(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(2-3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[sqrt(tan(c+d*x))/sqrt(-2-3*tan(c+d*x)),x,7,I*arctan(sqrt(3-2*I)*sqrt(tan(c+d*x))/sqrt(-2-3*tan(c+d*x)))/(d*sqrt(3-2*I))-I*arctan(sqrt(3+2*I)*sqrt(tan(c+d*x))/sqrt(-2-3*tan(c+d*x)))/(d*sqrt(3+2*I))],
[sqrt(tan(c+d*x))/sqrt(3+2*tan(c+d*x)),x,7,I*arctanh(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(3+2*tan(c+d*x)))/(d*sqrt(2-3*I))-I*arctanh(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(3+2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[sqrt(tan(c+d*x))/sqrt(3-2*tan(c+d*x)),x,7,-I*arctan(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(3-2*tan(c+d*x)))/(d*sqrt(2-3*I))+I*arctan(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(3-2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[sqrt(tan(c+d*x))/sqrt(-3+2*tan(c+d*x)),x,7,-I*arctanh(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(-3+2*tan(c+d*x)))/(d*sqrt(2-3*I))+I*arctanh(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(-3+2*tan(c+d*x)))/(d*sqrt(2+3*I))],
[sqrt(tan(c+d*x))/sqrt(-3-2*tan(c+d*x)),x,7,I*arctan(sqrt(2-3*I)*sqrt(tan(c+d*x))/sqrt(-3-2*tan(c+d*x)))/(d*sqrt(2-3*I))-I*arctan(sqrt(2+3*I)*sqrt(tan(c+d*x))/sqrt(-3-2*tan(c+d*x)))/(d*sqrt(2+3*I))],

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

# m>0

# m<0
[tan(c+d*x)^(5/3)/(a+b*tan(c+d*x)),x,32,-1/2*b*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)+1/2*b*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)+b*arctan(tan(c+d*x)^(1/3))/((a^2+b^2)*d)-3/2*a^(5/3)*log(a^(1/3)+b^(1/3)*tan(c+d*x)^(1/3))/(b^(2/3)*(a^2+b^2)*d)-1/2*a*log(1+tan(c+d*x)^(2/3))/((a^2+b^2)*d)+1/2*a^(5/3)*log(a+b*tan(c+d*x))/(b^(2/3)*(a^2+b^2)*d)+1/4*a*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/((a^2+b^2)*d)-a^(5/3)*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x)^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(b^(2/3)*(a^2+b^2)*d)+1/2*a*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))*sqrt(3)/((a^2+b^2)*d)+1/4*b*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)-1/4*b*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)],
[tan(c+d*x)^(1/3)/(a+b*tan(c+d*x)),x,30,-1/2*b*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)+1/2*b*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)+b*arctan(tan(c+d*x)^(1/3))/((a^2+b^2)*d)-3/2*a^(1/3)*b^(2/3)*log(a^(1/3)+b^(1/3)*tan(c+d*x)^(1/3))/((a^2+b^2)*d)-1/2*a*log(1+tan(c+d*x)^(2/3))/((a^2+b^2)*d)+1/2*a^(1/3)*b^(2/3)*log(a+b*tan(c+d*x))/((a^2+b^2)*d)+1/4*a*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/((a^2+b^2)*d)+a^(1/3)*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x)^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/((a^2+b^2)*d)-1/2*a*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))*sqrt(3)/((a^2+b^2)*d)-1/4*b*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)+1/4*b*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)],
[1/(tan(c+d*x)^(1/3)*(a+b*tan(c+d*x))),x,28,1/2*b*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)-1/2*b*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)-b*arctan(tan(c+d*x)^(1/3))/((a^2+b^2)*d)-3/2*b^(4/3)*log(a^(1/3)+b^(1/3)*tan(c+d*x)^(1/3))/(a^(1/3)*(a^2+b^2)*d)+1/2*a*log(1+tan(c+d*x)^(2/3))/((a^2+b^2)*d)+1/2*b^(4/3)*log(a+b*tan(c+d*x))/(a^(1/3)*(a^2+b^2)*d)-1/4*a*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/((a^2+b^2)*d)-b^(4/3)*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x)^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(1/3)*(a^2+b^2)*d)-1/2*a*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))*sqrt(3)/((a^2+b^2)*d)-1/4*b*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)+1/4*b*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)],
[1/(tan(c+d*x)^(5/3)*(a+b*tan(c+d*x))),x,30,1/2*b*arctan(sqrt(3)-2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)-1/2*b*arctan(sqrt(3)+2*tan(c+d*x)^(1/3))/((a^2+b^2)*d)-b*arctan(tan(c+d*x)^(1/3))/((a^2+b^2)*d)-3/2*b^(8/3)*log(a^(1/3)+b^(1/3)*tan(c+d*x)^(1/3))/(a^(5/3)*(a^2+b^2)*d)+1/2*a*log(1+tan(c+d*x)^(2/3))/((a^2+b^2)*d)+1/2*b^(8/3)*log(a+b*tan(c+d*x))/(a^(5/3)*(a^2+b^2)*d)-1/4*a*log(1-tan(c+d*x)^(2/3)+tan(c+d*x)^(4/3))/((a^2+b^2)*d)+b^(8/3)*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x)^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(5/3)*(a^2+b^2)*d)+1/2*a*arctan((1-2*tan(c+d*x)^(2/3))/sqrt(3))*sqrt(3)/((a^2+b^2)*d)+1/4*b*log(1-sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)-1/4*b*log(1+sqrt(3)*tan(c+d*x)^(1/3)+tan(c+d*x)^(2/3))*sqrt(3)/((a^2+b^2)*d)-3/2*a/((a^2+b^2)*d*tan(c+d*x)^(2/3))-3/2*b^2/(a*(a^2+b^2)*d*tan(c+d*x)^(2/3))],

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

# m>0

# m<0
[tan(c+d*x)^(4/3)/sqrt(a+b*tan(c+d*x)),x,9,3/14*AppellF1(7/3,1,1/2,10/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(7/3)/(d*sqrt(a+b*tan(c+d*x)))+3/14*AppellF1(7/3,1,1/2,10/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(7/3)/(d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^(2/3)/sqrt(a+b*tan(c+d*x)),x,9,3/10*AppellF1(5/3,1,1/2,8/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(5/3)/(d*sqrt(a+b*tan(c+d*x)))+3/10*AppellF1(5/3,1,1/2,8/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(5/3)/(d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^(1/3)/sqrt(a+b*tan(c+d*x)),x,9,3/8*AppellF1(4/3,1,1/2,7/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(4/3)/(d*sqrt(a+b*tan(c+d*x)))+3/8*AppellF1(4/3,1,1/2,7/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(4/3)/(d*sqrt(a+b*tan(c+d*x)))],
[1/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1/3)),x,9,3/4*AppellF1(2/3,1,1/2,5/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(2/3)/(d*sqrt(a+b*tan(c+d*x)))+3/4*AppellF1(2/3,1,1/2,5/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(2/3)/(d*sqrt(a+b*tan(c+d*x)))],
[1/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(2/3)),x,9,3/2*AppellF1(1/3,1,1/2,4/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1/3)/(d*sqrt(a+b*tan(c+d*x)))+3/2*AppellF1(1/3,1,1/2,4/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1/3)/(d*sqrt(a+b*tan(c+d*x)))],
[1/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(4/3)),x,9,-3/2*AppellF1(-1/3,1,1/2,2/3,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)/(d*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1/3))-3/2*AppellF1(-1/3,1,1/2,2/3,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(1+b*tan(c+d*x)/a)/(d*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1/3))],

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

# m>0
[tan(e+f*x)^4*(c+d*tan(e+f*x))^(1/3),x,16,-1/4*x*(c-sqrt(-d^2))^(1/3)+1/4*log(cos(e+f*x))*(c-sqrt(-d^2))^(1/3)*sqrt(-d^2)/(d*f)+3/4*log((c-sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c-sqrt(-d^2))^(1/3)*sqrt(-d^2)/(d*f)-1/2*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c-sqrt(-d^2))^(1/3)*sqrt(-d^2)/(d*f)-1/4*x*(c+sqrt(-d^2))^(1/3)-1/4*log(cos(e+f*x))*sqrt(-d^2)*(c+sqrt(-d^2))^(1/3)/(d*f)-3/4*log((c+sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*sqrt(-d^2)*(c+sqrt(-d^2))^(1/3)/(d*f)+1/2*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*sqrt(-d^2)*(c+sqrt(-d^2))^(1/3)/(d*f)+3/140*(9*c^2-35*d^2)*(c+d*tan(e+f*x))^(4/3)/(d^3*f)-9/35*c*tan(e+f*x)*(c+d*tan(e+f*x))^(4/3)/(d^2*f)+3/10*tan(e+f*x)^2*(c+d*tan(e+f*x))^(4/3)/(d*f)],
[tan(e+f*x)^3*(c+d*tan(e+f*x))^(1/3),x,15,-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*log(cos(e+f*x))/f-1/4*(c+I*d)^(1/3)*log(cos(e+f*x))/f-3/4*(c-I*d)^(1/3)*log((c-I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-3/4*(c+I*d)^(1/3)*log((c+I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f+1/2*(c-I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-I*d)^(1/3))/sqrt(3))*sqrt(3)/f+1/2*(c+I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+I*d)^(1/3))/sqrt(3))*sqrt(3)/f-3*(c+d*tan(e+f*x))^(1/3)/f-9/28*c*(c+d*tan(e+f*x))^(4/3)/(d^2*f)+3/7*tan(e+f*x)*(c+d*tan(e+f*x))^(4/3)/(d*f)],
[tan(e+f*x)^2*(c+d*tan(e+f*x))^(1/3),x,14,1/4*x*(c-sqrt(-d^2))^(1/3)+1/4*d*log(cos(e+f*x))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+3/4*d*log((c-sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+1/4*x*(c+sqrt(-d^2))^(1/3)-1/4*d*log(cos(e+f*x))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-3/4*d*log((c+sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+3/4*(c+d*tan(e+f*x))^(4/3)/(d*f)],
[tan(e+f*x)*(c+d*tan(e+f*x))^(1/3),x,12,1/4*I*(c-I*d)^(1/3)*x-1/4*I*(c+I*d)^(1/3)*x+1/4*(c-I*d)^(1/3)*log(cos(e+f*x))/f+1/4*(c+I*d)^(1/3)*log(cos(e+f*x))/f+3/4*(c-I*d)^(1/3)*log((c-I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f+3/4*(c+I*d)^(1/3)*log((c+I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-1/2*(c-I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-I*d)^(1/3))/sqrt(3))*sqrt(3)/f-1/2*(c+I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+I*d)^(1/3))/sqrt(3))*sqrt(3)/f+3*(c+d*tan(e+f*x))^(1/3)/f],
[(c+d*tan(e+f*x))^(1/3),x,13,-1/4*x*(c-sqrt(-d^2))^(1/3)-1/4*d*log(cos(e+f*x))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-3/4*d*log((c-sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-1/4*x*(c+sqrt(-d^2))^(1/3)+1/4*d*log(cos(e+f*x))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+3/4*d*log((c+sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))],
[cot(e+f*x)*(c+d*tan(e+f*x))^(1/3),x,19,-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*log(cos(e+f*x))/f-1/4*(c+I*d)^(1/3)*log(cos(e+f*x))/f-1/2*c^(1/3)*log(tan(e+f*x))/f+3/2*c^(1/3)*log(c^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-3/4*(c-I*d)^(1/3)*log((c-I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-3/4*(c+I*d)^(1/3)*log((c+I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-c^(1/3)*arctan((c^(1/3)+2*(c+d*tan(e+f*x))^(1/3))/(c^(1/3)*sqrt(3)))*sqrt(3)/f+1/2*(c-I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-I*d)^(1/3))/sqrt(3))*sqrt(3)/f+1/2*(c+I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+I*d)^(1/3))/sqrt(3))*sqrt(3)/f],
[cot(e+f*x)^2*(c+d*tan(e+f*x))^(1/3),x,20,-1/6*d*log(tan(e+f*x))/(c^(2/3)*f)+1/2*d*log(c^(1/3)-(c+d*tan(e+f*x))^(1/3))/(c^(2/3)*f)-d*arctan((c^(1/3)+2*(c+d*tan(e+f*x))^(1/3))/(c^(1/3)*sqrt(3)))/(c^(2/3)*f*sqrt(3))+1/4*x*(c-sqrt(-d^2))^(1/3)+1/4*d*log(cos(e+f*x))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+3/4*d*log((c-sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c-sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+1/4*x*(c+sqrt(-d^2))^(1/3)-1/4*d*log(cos(e+f*x))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-3/4*d*log((c+sqrt(-d^2))^(1/3)-(c+d*tan(e+f*x))^(1/3))*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))+1/2*d*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+sqrt(-d^2))^(1/3))/sqrt(3))*sqrt(3)*(c+sqrt(-d^2))^(1/3)/(f*sqrt(-d^2))-cot(e+f*x)*(c+d*tan(e+f*x))^(1/3)/f],
[(a+b*tan(c+d*x))^(5/3),x,12,-1/4*(a-I*b)^(5/3)*x-1/4*(a+I*b)^(5/3)*x+1/4*I*(a-I*b)^(5/3)*log(cos(c+d*x))/d-1/4*I*(a+I*b)^(5/3)*log(cos(c+d*x))/d+3/4*I*(a-I*b)^(5/3)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-3/4*I*(a+I*b)^(5/3)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d+1/2*I*(a-I*b)^(5/3)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/d-1/2*I*(a+I*b)^(5/3)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/d+3/2*b*(a+b*tan(c+d*x))^(2/3)/d],
[(a+b*tan(c+d*x))^(4/3),x,12,-1/4*(a-I*b)^(4/3)*x-1/4*(a+I*b)^(4/3)*x+1/4*I*(a-I*b)^(4/3)*log(cos(c+d*x))/d-1/4*I*(a+I*b)^(4/3)*log(cos(c+d*x))/d+3/4*I*(a-I*b)^(4/3)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-3/4*I*(a+I*b)^(4/3)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-1/2*I*(a-I*b)^(4/3)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/d+1/2*I*(a+I*b)^(4/3)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/d+3*b*(a+b*tan(c+d*x))^(1/3)/d],
[(a+b*tan(c+d*x))^(2/3),x,13,-1/4*x*(a-sqrt(-b^2))^(2/3)-1/4*b*log(cos(c+d*x))*(a-sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))-3/4*b*log((a-sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))*(a-sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))-1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)*(a-sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))-1/4*x*(a+sqrt(-b^2))^(2/3)+1/4*b*log(cos(c+d*x))*(a+sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))+3/4*b*log((a+sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))*(a+sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))+1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)*(a+sqrt(-b^2))^(2/3)/(d*sqrt(-b^2))],
[(a+b*tan(c+d*x))^(1/3),x,13,-1/4*x*(a-sqrt(-b^2))^(1/3)-1/4*b*log(cos(c+d*x))*(a-sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))-3/4*b*log((a-sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))*(a-sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))+1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)*(a-sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))-1/4*x*(a+sqrt(-b^2))^(1/3)+1/4*b*log(cos(c+d*x))*(a+sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))+3/4*b*log((a+sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))*(a+sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))-1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)*(a+sqrt(-b^2))^(1/3)/(d*sqrt(-b^2))],

# m<0
[1/(a+b*tan(c+d*x))^(1/3),x,11,-1/4*x/(a-sqrt(-b^2))^(1/3)-1/4*b*log(cos(c+d*x))/(d*(a-sqrt(-b^2))^(1/3)*sqrt(-b^2))-3/4*b*log((a-sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))/(d*(a-sqrt(-b^2))^(1/3)*sqrt(-b^2))-1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)/(d*(a-sqrt(-b^2))^(1/3)*sqrt(-b^2))-1/4*x/(a+sqrt(-b^2))^(1/3)+1/4*b*log(cos(c+d*x))/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(1/3))+3/4*b*log((a+sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(1/3))+1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(1/3))],
[1/(a+b*tan(c+d*x))^(2/3),x,11,-1/4*x/(a-sqrt(-b^2))^(2/3)-1/4*b*log(cos(c+d*x))/(d*(a-sqrt(-b^2))^(2/3)*sqrt(-b^2))-3/4*b*log((a-sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))/(d*(a-sqrt(-b^2))^(2/3)*sqrt(-b^2))+1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)/(d*(a-sqrt(-b^2))^(2/3)*sqrt(-b^2))-1/4*x/(a+sqrt(-b^2))^(2/3)+1/4*b*log(cos(c+d*x))/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(2/3))+3/4*b*log((a+sqrt(-b^2))^(1/3)-(a+b*tan(c+d*x))^(1/3))/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(2/3))-1/2*b*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+sqrt(-b^2))^(1/3))/sqrt(3))*sqrt(3)/(d*sqrt(-b^2)*(a+sqrt(-b^2))^(2/3))],
[1/(a+b*tan(c+d*x))^(4/3),x,12,-1/4*x/(a-I*b)^(4/3)-1/4*x/(a+I*b)^(4/3)+1/4*I*log(cos(c+d*x))/((a-I*b)^(4/3)*d)-1/4*I*log(cos(c+d*x))/((a+I*b)^(4/3)*d)+3/4*I*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a-I*b)^(4/3)*d)-3/4*I*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a+I*b)^(4/3)*d)+1/2*I*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/((a-I*b)^(4/3)*d)-1/2*I*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/((a+I*b)^(4/3)*d)-3*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^(1/3))],
[1/(a+b*tan(c+d*x))^(5/3),x,12,-1/4*x/(a-I*b)^(5/3)-1/4*x/(a+I*b)^(5/3)+1/4*I*log(cos(c+d*x))/((a-I*b)^(5/3)*d)-1/4*I*log(cos(c+d*x))/((a+I*b)^(5/3)*d)+3/4*I*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a-I*b)^(5/3)*d)-3/4*I*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a+I*b)^(5/3)*d)-1/2*I*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/((a-I*b)^(5/3)*d)+1/2*I*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/((a+I*b)^(5/3)*d)-3/2*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^(2/3))],

# Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n with n symbolic
[(d*tan(e+f*x))^n*(a+b*tan(e+f*x))^4,x,8,-b^2*(b^2*(3+n)-a^2*(17+5*n))*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*(3+n))+(a^4-6*a^2*b^2+b^4)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+2*a*b^3*(4+n)*tan(e+f*x)*(d*tan(e+f*x))^(1+n)/(d*f*(2+n)*(3+n))+4*a*b*(a^2-b^2)*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(d^2*f*(2+n))+b^2*(d*tan(e+f*x))^(1+n)*(a+b*tan(e+f*x))^2/(d*f*(3+n))],
[(d*tan(e+f*x))^n*(a+b*tan(e+f*x))^3,x,7,a*b^2*(5+2*n)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n)*(2+n))+a*(a^2-3*b^2)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+b*(3*a^2-b^2)*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(d^2*f*(2+n))+b^2*(d*tan(e+f*x))^(1+n)*(a+b*tan(e+f*x))/(d*f*(2+n))],
[(d*tan(e+f*x))^n*(a+b*tan(e+f*x))^2,x,6,b^2*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+(a^2-b^2)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+2*a*b*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(d^2*f*(2+n))],
[(d*tan(e+f*x))^n*(a+b*tan(e+f*x)),x,5,a*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+b*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/(d^2*f*(2+n))],
[(d*tan(e+f*x))^n/(a+b*tan(e+f*x)),x,8,a*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/((a^2+b^2)*d*f*(1+n))+b^2*hypergeom([1,1+n],[2+n],-b*tan(e+f*x)/a)*(d*tan(e+f*x))^(1+n)/(a*(a^2+b^2)*d*f*(1+n))-b*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/((a^2+b^2)*d^2*f*(2+n))],
[(d*tan(e+f*x))^n/(a+b*tan(e+f*x))^2,x,9,(a^2-b^2)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/((a^2+b^2)^2*d*f*(1+n))+b^2*(a^2*(2-n)-b^2*n)*hypergeom([1,1+n],[2+n],-b*tan(e+f*x)/a)*(d*tan(e+f*x))^(1+n)/(a^2*(a^2+b^2)^2*d*f*(1+n))-2*a*b*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(2+n)/((a^2+b^2)^2*d^2*f*(2+n))+b^2*(d*tan(e+f*x))^(1+n)/(a*(a^2+b^2)*d*f*(a+b*tan(e+f*x)))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^(3/2),x,7,1/2*a*AppellF1(1+m,-3/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))+1/2*a*AppellF1(1+m,-3/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^(1/2),x,7,1/2*AppellF1(1+m,-1/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))+1/2*AppellF1(1+m,-1/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))],
[tan(c+d*x)^m/(a+b*tan(c+d*x))^(1/2),x,7,1/2*AppellF1(1+m,1/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+b*tan(c+d*x)))+1/2*AppellF1(1+m,1/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^m/(a+b*tan(c+d*x))^(3/2),x,7,1/2*AppellF1(1+m,3/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a*d*(1+m)*sqrt(a+b*tan(c+d*x)))+1/2*AppellF1(1+m,3/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a*d*(1+m)*sqrt(a+b*tan(c+d*x)))],

# Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n with m symbolic
[(d*tan(e+f*x))^n*(a+b*tan(e+f*x))^m,x,7,1/2*AppellF1(1+n,-m,1,2+n,-b*tan(e+f*x)/a,-I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)*(a+b*tan(e+f*x))^m/(d*f*(1+n)*(1+b*tan(e+f*x)/a)^m)+1/2*AppellF1(1+n,-m,1,2+n,-b*tan(e+f*x)/a,I*tan(e+f*x))*(d*tan(e+f*x))^(1+n)*(a+b*tan(e+f*x))^m/(d*f*(1+n)*(1+b*tan(e+f*x)/a)^m)],
[tan(c+d*x)^4*(a+b*tan(c+d*x))^n,x,8,(2*a^2-b^2*(2+n)*(3+n))*(a+b*tan(c+d*x))^(1+n)/(b^3*d*(1+n)*(2+n)*(3+n))-1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*sqrt(-b^2)*(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n)*(a-sqrt(-b^2)))+1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*sqrt(-b^2)*(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n)*(a+sqrt(-b^2)))-2*a*tan(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b^2*d*(2+n)*(3+n))+tan(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)/(b*d*(3+n))],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^n,x,8,-a*(a+b*tan(c+d*x))^(1+n)/(b^2*d*(1+n)*(2+n))+1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))+1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))+tan(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b*d*(2+n))],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^n,x,6,(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n))-1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*(a-sqrt(-b^2))*sqrt(-b^2))+1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*sqrt(-b^2)*(a+sqrt(-b^2)))],
[tan(c+d*x)*(a+b*tan(c+d*x))^n,x,5,-1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))-1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))],
[(a+b*tan(c+d*x))^n,x,5,1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*(a-sqrt(-b^2))*sqrt(-b^2))-1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*sqrt(-b^2)*(a+sqrt(-b^2)))],
[cot(c+d*x)*(a+b*tan(c+d*x))^n,x,8,1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))+1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))-hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a*d*(1+n))],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^n,x,10,-cot(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(a*d)-b*n*hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a^2*d*(1+n))-1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*(a-sqrt(-b^2))*sqrt(-b^2))+1/2*b*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+sqrt(-b^2)))*(a+b*tan(c+d*x))^(1+n)/(d*(1+n)*sqrt(-b^2)*(a+sqrt(-b^2)))],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^n,x,11,1/2*b*(1-n)*cot(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(a^2*d)-1/2*cot(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)/(a*d)-1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))-1/2*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))+1/2*(2*a^2+b^2*(1-n)*n)*hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a^3*d*(1+n))],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n,x,9,1/5*AppellF1(5/2,1,-n,7/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+1/5*AppellF1(5/2,1,-n,7/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^n,x,9,1/3*AppellF1(3/2,1,-n,5/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+1/3*AppellF1(3/2,1,-n,5/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n/tan(c+d*x)^(1/2),x,9,AppellF1(1/2,1,-n,3/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+AppellF1(1/2,1,-n,3/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n/tan(c+d*x)^(3/2),x,9,-AppellF1(-1/2,1,-n,1/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(tan(c+d*x))*(1+b*tan(c+d*x)/a)^n)-AppellF1(-1/2,1,-n,1/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(tan(c+d*x))*(1+b*tan(c+d*x)/a)^n)],

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

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

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

# m>0
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x)),x,5,-2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a*cot(c+d*x)^(3/2)/d-2*I*a*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x)),x,4,-2*(-1)^(1/4)*a*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2*a*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x)),x,3,2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d],
[(a+I*a*tan(c+d*x))/sqrt(cot(c+d*x)),x,4,2*(-1)^(1/4)*a*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2*I*a/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))/cot(c+d*x)^(3/2),x,5,-2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/3*I*a/(d*cot(c+d*x)^(3/2))+2*a/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^2,x,6,4*(-1)^(1/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-4/3*I*a^2*cot(c+d*x)^(3/2)/d-2/5*a^2*cot(c+d*x)^(5/2)/d+4*a^2*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2,x,5,-4*(-1)^(3/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a^2*cot(c+d*x)^(3/2)/d-4*I*a^2*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2,x,4,-4*(-1)^(1/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2*a^2*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^2,x,4,4*(-1)^(3/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2*a^2/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^2/sqrt(cot(c+d*x)),x,5,4*(-1)^(1/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a^2/(d*cot(c+d*x)^(3/2))+4*I*a^2/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^2/cot(c+d*x)^(3/2),x,6,-4*(-1)^(3/4)*a^2*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/5*a^2/(d*cot(c+d*x)^(5/2))+4/3*I*a^2/(d*cot(c+d*x)^(3/2))+4*a^2/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^3,x,6,8*(-1)^(1/4)*a^3*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-8/5*I*a^3*cot(c+d*x)^(3/2)/d-2/5*cot(c+d*x)^(3/2)*(I*a^3+a^3*cot(c+d*x))/d+8*a^3*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^3,x,5,-8*(-1)^(3/4)*a^3*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-16/3*I*a^3*sqrt(cot(c+d*x))/d-2/3*(I*a^3+a^3*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3,x,5,-8*(-1)^(1/4)*a^3*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2*(I*a^3+a^3*cot(c+d*x))/(d*sqrt(cot(c+d*x)))],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^3,x,5,8*(-1)^(3/4)*a^3*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*(I*a^3+a^3*cot(c+d*x))/(d*cot(c+d*x)^(3/2))-16/3*a^3/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^3/sqrt(cot(c+d*x)),x,6,8*(-1)^(1/4)*a^3*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-8/5*a^3/(d*cot(c+d*x)^(3/2))-2/5*(I*a^3+a^3*cot(c+d*x))/(d*cot(c+d*x)^(5/2))+8*I*a^3/(d*sqrt(cot(c+d*x)))],

# m<0
[cot(c+d*x)^(3/2)/(a+I*a*tan(c+d*x)),x,13,1/2*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x)))+(-5/4-3/4*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(5/4+3/4*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-5/8+3/8*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(5/8-3/8*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))-5/2*sqrt(cot(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))/(a+I*a*tan(c+d*x)),x,12,(3/4-1/4*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-3/4+1/4*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-3/8-1/8*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(3/8+1/8*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/2*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))),x,4,1/2*(-1)^(1/4)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/(a*d)+1/2*I*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))),x,12,(1/4-3/4*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/4+3/4*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/8-3/8*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(1/8+3/8*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))-1/2*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))),x,13,(3/4+5/4*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-3/4-5/4*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(3/8-5/8*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-3/8+5/8*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-5/2*I)/(a*d*sqrt(cot(c+d*x)))+(-1/2)/(d*(I*a+a*cot(c+d*x))*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(3/2)/(a+I*a*tan(c+d*x))^2,x,14,7/8*cot(c+d*x)^(3/2)/(a^2*d*(I+cot(c+d*x)))+1/4*cot(c+d*x)^(5/2)/(d*(I*a+a*cot(c+d*x))^2)+(-25/16-21/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(25/16+21/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-25/32+21/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(25/32-21/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))-25/8*sqrt(cot(c+d*x))/(a^2*d)],
[sqrt(cot(c+d*x))/(a+I*a*tan(c+d*x))^2,x,13,1/4*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x))^2)+(9/16-5/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-9/16+5/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-9/32-5/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(9/32+5/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+5/8*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^2),x,13,(1/16-3/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-1/16+3/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32+3/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-1/32-3/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+3/8*I*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))+1/4*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^2)],
[1/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2),x,13,(-1/16-3/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/16+3/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32-3/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-1/32+3/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/8*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))+1/4*I*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^2)],
[1/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2),x,13,(-9/16-5/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(9/16+5/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-9/32+5/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(9/32-5/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+5/8*I*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))-1/4*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^2)],
[1/(cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^2),x,14,(25/16-21/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-25/16+21/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-25/32-21/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(25/32+21/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-25/8)/(a^2*d*sqrt(cot(c+d*x)))+7/8*I/(a^2*d*(I+cot(c+d*x))*sqrt(cot(c+d*x)))+(-1/4)/(d*(I*a+a*cot(c+d*x))^2*sqrt(cot(c+d*x)))],
[sqrt(cot(c+d*x))/(a+I*a*tan(c+d*x))^3,x,14,1/6*cot(c+d*x)^(5/2)/(d*(I*a+a*cot(c+d*x))^3)+1/3*cot(c+d*x)^(3/2)/(a*d*(I*a+a*cot(c+d*x))^2)+(7/16-5/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(-7/16+5/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(-7/32-5/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(7/32+5/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+5/8*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^3),x,16,1/6*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x))^3)-1/8*I*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/8*I*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*I*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/16*I*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/4*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+1/4*I*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3),x,7,-1/8*(-1)^(3/4)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/(a^3*d)+1/6*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^3)+1/6*I*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+1/8*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^3),x,15,-1/8*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/8*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/16*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/6*I*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^3)+1/12*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)],
[1/(cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^3),x,14,(-5/16+7/16*I)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(5/16-7/16*I)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(5/32+7/32*I)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(-5/32-7/32*I)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/6*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^3)+1/3*I*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+5/8*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],

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

# m>0
[cot(c+d*x)^(7/2)*sqrt(a+I*a*tan(c+d*x)),x,7,(-1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*I*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/5*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d+26/15*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x)),x,6,(-1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/3*I*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x)),x,4,(1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)),x,3,(1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x))/sqrt(cot(c+d*x)),x,8,-2*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-1-I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x))/cot(c+d*x)^(3/2),x,10,-(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-1+I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(3/2),x,8,(-2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/5*I*a^2*cot(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-2/5*a^2*cot(c+d*x)^(5/2)/(d*sqrt(a+I*a*tan(c+d*x)))-4/5*I*a*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d+12/5*a*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2),x,5,(-2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*I*a*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/3*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2),x,4,(2+2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2),x,8,2*(-1)^(1/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^(3/2)/sqrt(cot(c+d*x)),x,10,-3*(-1)^(3/4)*a^(3/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-2-2*I)*a^(3/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-a^2/(d*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x)))+I*a^2/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+I*a*tan(c+d*x))^(5/2),x,8,(4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+32/21*a^2*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-6/7*I*a^2*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d-2/7*a^2*cot(c+d*x)^(7/2)*sqrt(a+I*a*tan(c+d*x))/d+104/21*I*a^2*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(5/2),x,6,(-4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+4*a^2*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/3*I*a*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d-2/5*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2),x,5,(-4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-4*I*a^2*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/3*a*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2),x,9,2*(-1)^(3/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(4+4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a^2*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2),x,9,5*(-1)^(1/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^(5/2)/sqrt(cot(c+d*x)),x,10,-23/4*(-1)^(3/4)*a^(5/2)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-4-4*I)*a^(5/2)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-1/2*a^2*sqrt(a+I*a*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+9/4*I*a^2*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],

# m<0
[cot(c+d*x)^(5/2)/sqrt(a+I*a*tan(c+d*x)),x,7,(-1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+cot(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-5/3*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/(a*d)+7/3*I*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(3/2)/sqrt(a+I*a*tan(c+d*x)),x,6,(1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+sqrt(cot(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))-3*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,5,(1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+1/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[1/(sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))),x,4,(-1/2-1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+I/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))),x,9,-2*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(-1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(-1)/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))),x,10,-(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(1/2+1/2*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(-1)/(d*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x)))-2*I*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)/(a+I*a*tan(c+d*x))^(3/2),x,8,(-1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+5/2*cot(c+d*x)^(3/2)/(a*d*sqrt(a+I*a*tan(c+d*x)))-7/2*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+13/2*I*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*cot(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(3/2)/(a+I*a*tan(c+d*x))^(3/2),x,7,(1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+11/6*sqrt(cot(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))-25/6*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*sqrt(cot(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,6,(1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+7/6/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)),x,5,(-1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/2*I/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3/(d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)),x,5,(-1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/2/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3*I/(d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)),x,10,2*(-1)^(3/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+(1/4+1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+3/2*I/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+(-1/3)/(d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(3/2)),x,11,-3*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+(1/4-1/4*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+13/6*I/(a*d*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x)))-7/2*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(cot(c+d*x)))+(-1/3)/(d*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(5/2)/(a+I*a*tan(c+d*x))^(5/2),x,9,(-1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+89/20*cot(c+d*x)^(3/2)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-361/60*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+707/60*I*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*cot(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+7/10*cot(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(3/2)/(a+I*a*tan(c+d*x))^(5/2),x,8,(1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+151/60*sqrt(cot(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-317/60*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*sqrt(cot(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+17/30*sqrt(cot(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,7,(1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+67/60/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+13/30/(a*d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)),x,7,(-1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+(-1/20*I)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*I/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+1/10*I/(a*d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)),x,6,(-1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/4/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5/(d*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2))+1/6*I/(a*d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)),x,6,(1/8+1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+(-1/4*I)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*I/(d*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2))+1/6/(a*d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(5/2)),x,11,2*(-1)^(1/4)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+(1/8-1/8*I)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+7/4/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+(-1/5)/(d*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2))+1/2*I/(a*d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^n with n symbolic
[(d*cot(e+f*x))^n*(a+I*a*tan(e+f*x))^3,x,6,I*a^3*d^2*(1-2*n)*(d*cot(e+f*x))^(-2+n)/(f*(2-3*n+n^2))+d^2*(d*cot(e+f*x))^(-2+n)*(I*a^3+a^3*cot(e+f*x))/(f*(1-n))-4*I*a^3*d^2*(d*cot(e+f*x))^(-2+n)*hypergeom([1,-2+n],[-1+n],-I*cot(e+f*x))/(f*(2-n))],
[(d*cot(e+f*x))^n*(a+I*a*tan(e+f*x))^2,x,5,a^2*d*(d*cot(e+f*x))^(-1+n)/(f*(1-n))-2*a^2*d*(d*cot(e+f*x))^(-1+n)*hypergeom([1,-1+n],[n],-I*cot(e+f*x))/(f*(1-n))],
[(d*cot(e+f*x))^n*(a+I*a*tan(e+f*x)),x,3,-I*a*(d*cot(e+f*x))^n*hypergeom([1,n],[1+n],-I*cot(e+f*x))/(f*n)],
[(d*cot(e+f*x))^n/(a+I*a*tan(e+f*x)),x,7,-1/2*(d*cot(e+f*x))^(2+n)/(d^2*f*(I*a+a*cot(e+f*x)))-1/2*I*n*(d*cot(e+f*x))^(2+n)*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-cot(e+f*x)^2)/(a*d^2*f*(2+n))+1/2*(1+n)*(d*cot(e+f*x))^(3+n)*hypergeom([1,1/2*(3+n)],[1/2*(5+n)],-cot(e+f*x)^2)/(a*d^3*f*(3+n))],
[(d*cot(e+f*x))^n/(a+I*a*tan(e+f*x))^2,x,8,-1/4*I*n*(d*cot(e+f*x))^(3+n)/(a^2*d^3*f*(I+cot(e+f*x)))-1/4*(d*cot(e+f*x))^(3+n)/(d^3*f*(I*a+a*cot(e+f*x))^2)+1/4*(1+n)^2*(d*cot(e+f*x))^(3+n)*hypergeom([1,1/2*(3+n)],[1/2*(5+n)],-cot(e+f*x)^2)/(a^2*d^3*f*(3+n))+1/4*I*n*(2+n)*(d*cot(e+f*x))^(4+n)*hypergeom([1,1/2*(4+n)],[1/2*(6+n)],-cot(e+f*x)^2)/(a^2*d^4*f*(4+n))],

# Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^n with m symbolic
[(d*cot(e+f*x))^n*(a+I*a*tan(e+f*x))^m,x,4,AppellF1(1-n,1-m,1,2-n,-I*tan(e+f*x),I*tan(e+f*x))*(d*cot(e+f*x))^n*tan(e+f*x)*(a+I*a*tan(e+f*x))^m/(f*(1-n)*(1+I*tan(e+f*x))^m)],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n,x,5,-2*AppellF1(-1/2,1-n,1,1/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)],
[cot(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^n,x,5,2*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n/cot(c+d*x)^(1/2),x,5,2/3*AppellF1(3/2,1-n,1,5/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*cot(c+d*x)^(3/2)*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n/cot(c+d*x)^(3/2),x,5,2/5*AppellF1(5/2,1-n,1,7/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*cot(c+d*x)^(5/2)*(1+I*tan(c+d*x))^n)],

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

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

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

# m>0
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x)),x,14,-2/3*b*cot(c+d*x)^(3/2)/d-2/5*a*cot(c+d*x)^(5/2)/d+(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*a*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x)),x,13,-2/3*a*cot(c+d*x)^(3/2)/d-(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*b*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x)),x,12,-(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*a*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x)),x,11,(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))],
[(a+b*tan(c+d*x))/sqrt(cot(c+d*x)),x,12,(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))/cot(c+d*x)^(3/2),x,13,2/3*b/(d*cot(c+d*x)^(3/2))-(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*a/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))/cot(c+d*x)^(5/2),x,14,2/5*b/(d*cot(c+d*x)^(5/2))+2/3*a/(d*cot(c+d*x)^(3/2))-(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*b/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^2,x,15,2/3*(a^2-b^2)*cot(c+d*x)^(3/2)/d-4/5*a*b*cot(c+d*x)^(5/2)/d-2/7*a^2*cot(c+d*x)^(7/2)/d+(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+4*a*b*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^2,x,14,-4/3*a*b*cot(c+d*x)^(3/2)/d-2/5*a^2*cot(c+d*x)^(5/2)/d+(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(a^2-b^2)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2,x,13,-2/3*a^2*cot(c+d*x)^(3/2)/d-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-4*a*b*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2,x,12,-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*a^2*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^2,x,12,(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b^2/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^2/sqrt(cot(c+d*x)),x,13,2/3*b^2/(d*cot(c+d*x)^(3/2))+(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+4*a*b/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^2/cot(c+d*x)^(3/2),x,14,2/5*b^2/(d*cot(c+d*x)^(5/2))+4/3*a*b/(d*cot(c+d*x)^(3/2))-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(a^2-b^2)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^2/cot(c+d*x)^(5/2),x,15,2/7*b^2/(d*cot(c+d*x)^(7/2))+4/5*a*b/(d*cot(c+d*x)^(5/2))+2/3*(a^2-b^2)/(d*cot(c+d*x)^(3/2))-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-4*a*b/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^3,x,15,2/3*a*(a^2-3*b^2)*cot(c+d*x)^(3/2)/d-32/35*a^2*b*cot(c+d*x)^(5/2)/d-2/7*a^2*cot(c+d*x)^(5/2)*(b+a*cot(c+d*x))/d+(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b*(3*a^2-b^2)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^3,x,14,-8/5*a^2*b*cot(c+d*x)^(3/2)/d-2/5*a^2*cot(c+d*x)^(3/2)*(b+a*cot(c+d*x))/d+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*a*(a^2-3*b^2)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3,x,13,-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-16/3*a^2*b*sqrt(cot(c+d*x))/d-2/3*a^2*(b+a*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3,x,13,-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b^2*(b+a*cot(c+d*x))/(d*sqrt(cot(c+d*x)))-2*a*(a^2+b^2)*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^3,x,13,2/3*b^2*(b+a*cot(c+d*x))/(d*cot(c+d*x)^(3/2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+16/3*a*b^2/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^3/sqrt(cot(c+d*x)),x,14,8/5*a*b^2/(d*cot(c+d*x)^(3/2))+2/5*b^2*(b+a*cot(c+d*x))/(d*cot(c+d*x)^(5/2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b*(3*a^2-b^2)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^3/cot(c+d*x)^(3/2),x,15,32/35*a*b^2/(d*cot(c+d*x)^(5/2))+2/3*b*(3*a^2-b^2)/(d*cot(c+d*x)^(3/2))+2/7*b^2*(b+a*cot(c+d*x))/(d*cot(c+d*x)^(7/2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*a*(a^2-3*b^2)/(d*sqrt(cot(c+d*x)))],

# m<0
[cot(c+d*x)^(5/2)/(a+b*tan(c+d*x)),x,17,-2*b^(7/2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(5/2)*(a^2+b^2)*d)-2/3*cot(c+d*x)^(3/2)/(a*d)-(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*b*sqrt(cot(c+d*x))/(a^2*d)],
[cot(c+d*x)^(3/2)/(a+b*tan(c+d*x)),x,16,2*b^(5/2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(3/2)*(a^2+b^2)*d)-(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-2*sqrt(cot(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))/(a+b*tan(c+d*x)),x,15,(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-2*b^(3/2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/((a^2+b^2)*d*sqrt(a))],
[1/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))),x,15,(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)*sqrt(b)/((a^2+b^2)*d)],
[1/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,15,-(a-b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a-b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-2*a^(3/2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/((a^2+b^2)*d*sqrt(b))],
[1/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,16,2*a^(5/2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(3/2)*(a^2+b^2)*d)-(a+b)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2/(b*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)/(a+b*tan(c+d*x))^2,x,18,-b^(7/2)*(9*a^2+5*b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(7/2)*(a^2+b^2)^2*d)-1/3*(2*a^2+5*b^2)*cot(c+d*x)^(3/2)/(a^2*(a^2+b^2)*d)+b^2*cot(c+d*x)^(5/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x)))-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+b*(4*a^2+5*b^2)*sqrt(cot(c+d*x))/(a^3*(a^2+b^2)*d)],
[cot(c+d*x)^(3/2)/(a+b*tan(c+d*x))^2,x,17,b^(5/2)*(7*a^2+3*b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(5/2)*(a^2+b^2)^2*d)+b^2*cot(c+d*x)^(3/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x)))-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(2*a^2+3*b^2)*sqrt(cot(c+d*x))/(a^2*(a^2+b^2)*d)],
[sqrt(cot(c+d*x))/(a+b*tan(c+d*x))^2,x,16,-b^(3/2)*(5*a^2+b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(3/2)*(a^2+b^2)^2*d)+(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+b^2*sqrt(cot(c+d*x))/(a*(a^2+b^2)*d*(b+a*cot(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^2),x,16,(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(3*a^2-b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(b)/((a^2+b^2)^2*d*sqrt(a))-b*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2),x,16,-(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2-3*b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)/((a^2+b^2)^2*d*sqrt(b))+a*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2),x,16,-a^(3/2)*(a^2+5*b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(3/2)*(a^2+b^2)^2*d)-(a^2+2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2+2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2-2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-a^2*sqrt(cot(c+d*x))/(b*(a^2+b^2)*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^2),x,17,a^(5/2)*(3*a^2+7*b^2)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(5/2)*(a^2+b^2)^2*d)+(a^2-2*a*b-b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2-2*a*b-b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2+2*a*b-b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(3*a^2+2*b^2)/(b^2*(a^2+b^2)*d*sqrt(cot(c+d*x)))-a^2/(b*(a^2+b^2)*d*(b+a*cot(c+d*x))*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)/(a+b*tan(c+d*x))^3,x,19,-1/4*b^(7/2)*(99*a^4+102*a^2*b^2+35*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(9/2)*(a^2+b^2)^3*d)-1/12*(8*a^4+67*a^2*b^2+35*b^4)*cot(c+d*x)^(3/2)/(a^3*(a^2+b^2)^2*d)+1/2*b^2*cot(c+d*x)^(7/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)+1/4*b^2*(15*a^2+7*b^2)*cot(c+d*x)^(5/2)/(a^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/4*b*(24*a^4+67*a^2*b^2+35*b^4)*sqrt(cot(c+d*x))/(a^4*(a^2+b^2)^2*d)],
[cot(c+d*x)^(3/2)/(a+b*tan(c+d*x))^3,x,18,1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(7/2)*(a^2+b^2)^3*d)+1/2*b^2*cot(c+d*x)^(5/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)+1/4*b^2*(13*a^2+5*b^2)*cot(c+d*x)^(3/2)/(a^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(8*a^4+31*a^2*b^2+15*b^4)*sqrt(cot(c+d*x))/(a^3*(a^2+b^2)^2*d)],
[sqrt(cot(c+d*x))/(a+b*tan(c+d*x))^3,x,17,-1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(5/2)*(a^2+b^2)^3*d)+1/2*b^2*cot(c+d*x)^(3/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/4*b^2*(11*a^2+3*b^2)*sqrt(cot(c+d*x))/(a^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^3),x,17,(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/4*(15*a^4-18*a^2*b^2-b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(b)/(a^(3/2)*(a^2+b^2)^3*d)+1/2*b^2*sqrt(cot(c+d*x))/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)-1/4*b*(9*a^2+b^2)*sqrt(cot(c+d*x))/(a*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3),x,17,-(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(3*a^4-26*a^2*b^2+3*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/((a^2+b^2)^3*d*sqrt(a)*sqrt(b))-1/2*b*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x))^2)+1/4*(5*a^2-3*b^2)*sqrt(cot(c+d*x))/((a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3),x,17,-(a-b)*(a^2+4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a-b)*(a^2+4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a+b)*(a^2-4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(a^4+18*a^2*b^2-15*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)/(b^(3/2)*(a^2+b^2)^3*d)+1/2*a*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x))^2)-1/4*a*(a^2-7*b^2)*sqrt(cot(c+d*x))/(b*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[1/(cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^3),x,17,-1/4*a^(3/2)*(3*a^4+6*a^2*b^2+35*b^4)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(5/2)*(a^2+b^2)^3*d)+(a+b)*(a^2-4*a*b+b^2)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a+b)*(a^2-4*a*b+b^2)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a-b)*(a^2+4*a*b+b^2)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*a^2*sqrt(cot(c+d*x))/(b*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)-1/4*a^2*(3*a^2+11*b^2)*sqrt(cot(c+d*x))/(b^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],

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

# m>0
[cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x)),x,11,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*b*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)-2/5*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/15*(15*a^2+2*b^2)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x)),x,11,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/3*b*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x)),x,9,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)),x,8,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+b*tan(c+d*x))/sqrt(cot(c+d*x)),x,12,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+b*tan(c+d*x))/cot(c+d*x)^(3/2),x,14,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^(3/2),x,12,-(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/105*(35*a^2-3*b^2)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)-16/35*b*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d-2/7*a*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d+4/105*b*(70*a^2+3*b^2)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(3/2),x,11,-I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-4/5*b*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/5*a*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/5*(5*a^2-b^2)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2),x,10,(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*a*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-8/3*b*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2),x,9,I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2),x,13,-(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2*b^(3/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[(a+b*tan(c+d*x))^(3/2)/sqrt(cot(c+d*x)),x,14,-I*(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+3*a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+b*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(3/2)/cot(c+d*x)^(3/2),x,15,(I*a-b)^(3/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(3/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/4*(3*a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))+3/4*a*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))+1/2*(a+b*tan(c+d*x))^(3/2)/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(11/2)*(a+b*tan(c+d*x))^(5/2),x,13,(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/315*b*(231*a^2-5*b^2)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)+2/105*(21*a^2-25*b^2)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d-38/63*a*b*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d-2/9*a^2*cot(c+d*x)^(9/2)*sqrt(a+b*tan(c+d*x))/d-2/315*(315*a^4-483*a^2*b^2-10*b^4)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^(5/2),x,12,I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/21*(7*a^2-9*b^2)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-6/7*a*b*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d-2/7*a^2*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d+2/21*b*(49*a^2-3*b^2)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(5/2),x,11,-(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-22/15*a*b*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/5*a^2*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/15*(15*a^2-23*b^2)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2),x,10,-I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*a^2*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-14/3*a*b*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2),x,14,(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2*b^(5/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a^2*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(5/2),x,14,I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+5*a*b^(3/2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+b^2*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)/sqrt(cot(c+d*x)),x,15,-(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/4*(15*a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/2*b^2*sqrt(a+b*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+9/4*a*b*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)/cot(c+d*x)^(3/2),x,16,-I*(I*a-b)^(5/2)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-I*(I*a+b)^(5/2)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+5/8*a*(a^2-8*b^2)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))+1/3*b^2*sqrt(a+b*tan(c+d*x))/(d*cot(c+d*x)^(5/2))+13/12*a*b*sqrt(a+b*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+1/8*(11*a^2-8*b^2)*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],

# m<0
[cot(c+d*x)^(5/2)/sqrt(a+b*tan(c+d*x)),x,11,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))-2/3*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)+4/3*b*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(3/2)/sqrt(a+b*tan(c+d*x)),x,10,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))-2*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))/sqrt(a+b*tan(c+d*x)),x,8,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[1/(sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))),x,8,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[1/(cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))),x,13,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[1/(cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))),x,14,-a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))+sqrt(a+b*tan(c+d*x))/(b*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)/(a+b*tan(c+d*x))^(3/2),x,11,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2/3*cot(c+d*x)^(3/2)/(a*d*sqrt(a+b*tan(c+d*x)))+2/3*b^2*(5*a^2+8*b^2)/(a^3*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+8/3*b*sqrt(cot(c+d*x))/(a^2*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^(3/2)/(a+b*tan(c+d*x))^(3/2),x,10,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*b*(a^2+2*b^2)/(a^2*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2*sqrt(cot(c+d*x))/(a*d*sqrt(a+b*tan(c+d*x)))],
[sqrt(cot(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,9,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)+2*b^2/(a*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[1/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2)),x,9,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*b/((a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[1/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,9,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)+2*a/((a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[1/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2)),x,14,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+2*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*a^2/(b*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[1/(cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(3/2)),x,15,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-3*a*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(5/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*a^2/(b*(a^2+b^2)*d*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x)))+(3*a^2+b^2)*sqrt(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)/(a+b*tan(c+d*x))^(5/2),x,12,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+4/3*b^2*(4*a^4+15*a^2*b^2+8*b^4)/(a^4*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*cot(c+d*x)^(3/2)/(a*d*(a+b*tan(c+d*x))^(3/2))+2/3*b^2*(7*a^2+8*b^2)/(a^3*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))+4*b*sqrt(cot(c+d*x))/(a^2*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(3/2)/(a+b*tan(c+d*x))^(5/2),x,11,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)-2/3*b*(3*a^4+17*a^2*b^2+8*b^4)/(a^3*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*b*(3*a^2+4*b^2)/(a^2*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))-2*sqrt(cot(c+d*x))/(a*d*(a+b*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,10,-arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)-arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+4/3*b^2*(4*a^2+b^2)/(a^2*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*b^2/(a*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[1/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(5/2)),x,10,-I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)-2/3*b*(5*a^2-b^2)/(a*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*b/((a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)),x,10,arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+4/3*(a^2-2*b^2)/((a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*a/((a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[1/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2)),x,10,I*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)-I*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+2/3*a*(a^2+7*b^2)/(b*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*a^2/(b*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],

# Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n with n symbolic
[(d*cot(e+f*x))^n*(a+b*tan(e+f*x))^3,x,8,a^2*b*d^2*(1-2*n)*(d*cot(e+f*x))^(-2+n)/(f*(2-3*n+n^2))+a^2*d^2*(d*cot(e+f*x))^(-2+n)*(b+a*cot(e+f*x))/(f*(1-n))-b*(3*a^2-b^2)*d^2*(d*cot(e+f*x))^(-2+n)*hypergeom([1,1/2*(-2+n)],[1/2*n],-cot(e+f*x)^2)/(f*(2-n))-a*(a^2-3*b^2)*d*(d*cot(e+f*x))^(-1+n)*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],-cot(e+f*x)^2)/(f*(1-n))],
[(d*cot(e+f*x))^n*(a+b*tan(e+f*x))^2,x,7,a^2*d*(d*cot(e+f*x))^(-1+n)/(f*(1-n))-(a^2-b^2)*d*(d*cot(e+f*x))^(-1+n)*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],-cot(e+f*x)^2)/(f*(1-n))-2*a*b*(d*cot(e+f*x))^n*hypergeom([1,1/2*n],[1/2*(2+n)],-cot(e+f*x)^2)/(f*n)],
[(d*cot(e+f*x))^n*(a+b*tan(e+f*x)),x,6,-b*(d*cot(e+f*x))^n*hypergeom([1,1/2*n],[1/2*(2+n)],-cot(e+f*x)^2)/(f*n)-a*(d*cot(e+f*x))^(1+n)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-cot(e+f*x)^2)/(d*f*(1+n))],
[(d*cot(e+f*x))^n/(a+b*tan(e+f*x)),x,9,-b*(d*cot(e+f*x))^(2+n)*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],-cot(e+f*x)^2)/((a^2+b^2)*d^2*f*(2+n))-a^2*(d*cot(e+f*x))^(2+n)*hypergeom([1,2+n],[3+n],-a*cot(e+f*x)/b)/(b*(a^2+b^2)*d^2*f*(2+n))+a*(d*cot(e+f*x))^(3+n)*hypergeom([1,1/2*(3+n)],[1/2*(5+n)],-cot(e+f*x)^2)/((a^2+b^2)*d^3*f*(3+n))],
[(d*cot(e+f*x))^n/(a+b*tan(e+f*x))^2,x,10,-a^2*(d*cot(e+f*x))^(3+n)/(b*(a^2+b^2)*d^3*f*(b+a*cot(e+f*x)))+(a^2-b^2)*(d*cot(e+f*x))^(3+n)*hypergeom([1,1/2*(3+n)],[1/2*(5+n)],-cot(e+f*x)^2)/((a^2+b^2)^2*d^3*f*(3+n))+a^2*(b^2*n+a^2*(2+n))*(d*cot(e+f*x))^(3+n)*hypergeom([1,3+n],[4+n],-a*cot(e+f*x)/b)/(b^2*(a^2+b^2)^2*d^3*f*(3+n))+2*a*b*(d*cot(e+f*x))^(4+n)*hypergeom([1,1/2*(4+n)],[1/2*(6+n)],-cot(e+f*x)^2)/((a^2+b^2)^2*d^4*f*(4+n))],

# Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n with m symbolic
[(d*cot(e+f*x))^n*(a+b*tan(e+f*x))^m,x,8,1/2*AppellF1(1-n,-m,1,2-n,-b*tan(e+f*x)/a,-I*tan(e+f*x))*(d*cot(e+f*x))^n*tan(e+f*x)*(a+b*tan(e+f*x))^m/(f*(1-n)*(1+b*tan(e+f*x)/a)^m)+1/2*AppellF1(1-n,-m,1,2-n,-b*tan(e+f*x)/a,I*tan(e+f*x))*(d*cot(e+f*x))^n*tan(e+f*x)*(a+b*tan(e+f*x))^m/(f*(1-n)*(1+b*tan(e+f*x)/a)^m)],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n,x,10,-AppellF1(-1/2,1,-n,1/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)-AppellF1(-1/2,1,-n,1/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n,x,10,AppellF1(1/2,1,-n,3/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+b*tan(c+d*x)/a)^n)+AppellF1(1/2,1,-n,3/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n/sqrt(cot(c+d*x)),x,10,1/3*AppellF1(3/2,1,-n,5/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(3/2)*(1+b*tan(c+d*x)/a)^n)+1/3*AppellF1(3/2,1,-n,5/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(3/2)*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n/cot(c+d*x)^(3/2),x,10,1/5*AppellF1(5/2,1,-n,7/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(5/2)*(1+b*tan(c+d*x)/a)^n)+1/5*AppellF1(5/2,1,-n,7/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(5/2)*(1+b*tan(c+d*x)/a)^n)],

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

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

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

# n > 0
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x)),x,3,-1/3*I*c*(a+I*a*tan(e+f*x))^3/f],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x)),x,4,-1/2*I*c*(a+I*a*tan(e+f*x))^2/f,1/2*I*a^2*c*sec(e+f*x)^2/f+a^2*c*tan(e+f*x)/f],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x)),x,3,a*c*tan(e+f*x)/f],
[(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,3,I*c/(f*(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,3,1/2*I*c/(f*(a+I*a*tan(e+f*x))^2)],
[(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,3,1/3*I*c/(f*(a+I*a*tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^4*(c-I*c*tan(e+f*x))^2,x,4,-1/2*I*c^2*(a+I*a*tan(e+f*x))^4/f+1/5*I*c^2*(a+I*a*tan(e+f*x))^5/(a*f)],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^2,x,4,1/4*I*a^3*c^2*sec(e+f*x)^4/f+a^3*c^2*tan(e+f*x)/f+1/3*a^3*c^2*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^2,x,3,a^2*c^2*tan(e+f*x)/f+1/3*a^2*c^2*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^2,x,4,1/2*I*a*(c-I*c*tan(e+f*x))^2/f,-1/2*I*a*c^2*sec(e+f*x)^2/f+a*c^2*tan(e+f*x)/f],
[(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x)),x,4,-c^2*x/a-I*c^2*log(cos(e+f*x))/(a*f)+2*I*c^2/(f*(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^2,x,3,c^2*tan(e+f*x)/(f*(a+I*a*tan(e+f*x))^2)],
[(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^3,x,4,2/3*I*c^2/(f*(a+I*a*tan(e+f*x))^3)-1/2*I*c^2/(a*f*(a+I*a*tan(e+f*x))^2)],
[(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^4,x,4,1/2*I*c^2/(f*(a+I*a*tan(e+f*x))^4)-1/3*I*a^2*c^2/(f*(a^2+I*a^2*tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^5*(c-I*c*tan(e+f*x))^3,x,4,-4/5*I*c^3*(a+I*a*tan(e+f*x))^5/f+2/3*I*c^3*(a+I*a*tan(e+f*x))^6/(a*f)-1/7*I*c^3*(a+I*a*tan(e+f*x))^7/(a^2*f)],
[(a+I*a*tan(e+f*x))^4*(c-I*c*tan(e+f*x))^3,x,4,1/6*I*a^4*c^3*sec(e+f*x)^6/f+a^4*c^3*tan(e+f*x)/f+2/3*a^4*c^3*tan(e+f*x)^3/f+1/5*a^4*c^3*tan(e+f*x)^5/f],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^3,x,3,a^3*c^3*tan(e+f*x)/f+2/3*a^3*c^3*tan(e+f*x)^3/f+1/5*a^3*c^3*tan(e+f*x)^5/f],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^3,x,4,-1/4*I*a^2*c^3*sec(e+f*x)^4/f+a^2*c^3*tan(e+f*x)/f+1/3*a^2*c^3*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^3,x,3,1/3*I*a*(c-I*c*tan(e+f*x))^3/f],
[(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x)),x,4,-4*c^3*x/a-4*I*c^3*log(cos(e+f*x))/(a*f)+c^3*tan(e+f*x)/(a*f)+4*I*c^3/(f*(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^2,x,4,c^3*x/a^2+I*c^3*log(cos(e+f*x))/(a^2*f)+2*I*c^3/(f*(a+I*a*tan(e+f*x))^2)-4*I*c^3/(f*(a^2+I*a^2*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^3,x,3,1/6*I*c^3*(a^2-I*a^2*tan(e+f*x))^3/(f*(a^3+I*a^3*tan(e+f*x))^3)],
[(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^4,x,4,I*c^3/(f*(a+I*a*tan(e+f*x))^4)-4/3*I*c^3/(a*f*(a+I*a*tan(e+f*x))^3)+1/2*I*c^3/(f*(a^2+I*a^2*tan(e+f*x))^2)],
[(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^5,x,4,4/5*I*c^3/(f*(a+I*a*tan(e+f*x))^5)+1/3*I*c^3/(a^2*f*(a+I*a*tan(e+f*x))^3)-I*a^3*c^3/(f*(a^2+I*a^2*tan(e+f*x))^4)],
[(a+I*a*tan(e+f*x))^5*(c-I*c*tan(e+f*x))^4,x,4,1/8*I*a^5*c^4*sec(e+f*x)^8/f+a^5*c^4*tan(e+f*x)/f+a^5*c^4*tan(e+f*x)^3/f+3/5*a^5*c^4*tan(e+f*x)^5/f+1/7*a^5*c^4*tan(e+f*x)^7/f],
[(a+I*a*tan(e+f*x))^4*(c-I*c*tan(e+f*x))^4,x,3,a^4*c^4*tan(e+f*x)/f+a^4*c^4*tan(e+f*x)^3/f+3/5*a^4*c^4*tan(e+f*x)^5/f+1/7*a^4*c^4*tan(e+f*x)^7/f],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^4,x,4,-1/6*I*a^3*c^4*sec(e+f*x)^6/f+a^3*c^4*tan(e+f*x)/f+2/3*a^3*c^4*tan(e+f*x)^3/f+1/5*a^3*c^4*tan(e+f*x)^5/f],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^4,x,4,1/2*I*a^2*(c-I*c*tan(e+f*x))^4/f-1/5*I*a^2*(c-I*c*tan(e+f*x))^5/(c*f)],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^4,x,3,1/4*I*a*(c-I*c*tan(e+f*x))^4/f],
[(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x)),x,4,-12*c^4*x/a-12*I*c^4*log(cos(e+f*x))/(a*f)+5*c^4*tan(e+f*x)/(a*f)-1/2*I*c^4*tan(e+f*x)^2/(a*f)+8*I*c^4/(f*(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^2,x,4,6*c^4*x/a^2+6*I*c^4*log(cos(e+f*x))/(a^2*f)-c^4*tan(e+f*x)/(a^2*f)+4*I*c^4/(f*(a+I*a*tan(e+f*x))^2)-12*I*c^4/(f*(a^2+I*a^2*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^3,x,4,-c^4*x/a^3-I*c^4*log(cos(e+f*x))/(a^3*f)+8/3*I*c^4/(f*(a+I*a*tan(e+f*x))^3)-6*I*c^4/(a*f*(a+I*a*tan(e+f*x))^2)+6*I*c^4/(f*(a^3+I*a^3*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^4,x,3,1/8*I*c^4*(a^2-I*a^2*tan(e+f*x))^4/(f*(a^3+I*a^3*tan(e+f*x))^4)],
[(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^5,x,4,1/10*I*c^4*(1-I*tan(e+f*x))^4/(f*(a+I*a*tan(e+f*x))^5)+1/80*I*c^4*(a-I*a*tan(e+f*x))^4/(a^5*f*(a+I*a*tan(e+f*x))^4)],

# n < 0
[(a+I*a*tan(e+f*x))^4/(c-I*c*tan(e+f*x)),x,4,-12*a^4*x/c+12*I*a^4*log(cos(e+f*x))/(c*f)+5*a^4*tan(e+f*x)/(c*f)+1/2*I*a^4*tan(e+f*x)^2/(c*f)-8*I*a^4/(f*(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x)),x,4,-4*a^3*x/c+4*I*a^3*log(cos(e+f*x))/(c*f)+a^3*tan(e+f*x)/(c*f)-4*I*a^3/(f*(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x)),x,4,-a^2*x/c+I*a^2*log(cos(e+f*x))/(c*f)-2*I*a^2/(f*(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x)),x,3,-I*a/(f*(c-I*c*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))),x,3,1/2*x/(a*c)+1/2*cos(e+f*x)*sin(e+f*x)/(a*c*f)],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))),x,5,3/8*x/(a^2*c)+1/4*I*cos(e+f*x)^4/(a^2*c*f)+3/8*cos(e+f*x)*sin(e+f*x)/(a^2*c*f)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a^2*c*f)],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))),x,5,1/4*x/(a^3*c)+(-1/16*I)/(a^3*f*(c-I*c*tan(e+f*x)))+1/12*I*c^2/(a^3*f*(c+I*c*tan(e+f*x))^3)+1/8*I*c/(a^3*f*(c+I*c*tan(e+f*x))^2)+3/16*I/(a^3*f*(c+I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^4/(c-I*c*tan(e+f*x))^2,x,4,6*a^4*x/c^2-6*I*a^4*log(cos(e+f*x))/(c^2*f)-a^4*tan(e+f*x)/(c^2*f)-4*I*a^4/(f*(c-I*c*tan(e+f*x))^2)+12*I*a^4/(f*(c^2-I*c^2*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x))^2,x,4,a^3*x/c^2-I*a^3*log(cos(e+f*x))/(c^2*f)-2*I*a^3/(f*(c-I*c*tan(e+f*x))^2)+4*I*a^3/(f*(c^2-I*c^2*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x))^2,x,3,a^2*tan(e+f*x)/(f*(c-I*c*tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x))^2,x,3,-1/2*I*a/(f*(c-I*c*tan(e+f*x))^2)],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^2),x,5,3/8*x/(a*c^2)+(-1/8*I)/(a*f*(c-I*c*tan(e+f*x))^2)+(-1/4*I)/(a*f*(c^2-I*c^2*tan(e+f*x)))+1/8*I/(a*f*(c^2+I*c^2*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^2),x,4,3/8*x/(a^2*c^2)+3/8*cos(e+f*x)*sin(e+f*x)/(a^2*c^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a^2*c^2*f)],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^2),x,6,5/16*x/(a^3*c^2)+1/6*I*cos(e+f*x)^6/(a^3*c^2*f)+5/16*cos(e+f*x)*sin(e+f*x)/(a^3*c^2*f)+5/24*cos(e+f*x)^3*sin(e+f*x)/(a^3*c^2*f)+1/6*cos(e+f*x)^5*sin(e+f*x)/(a^3*c^2*f)],
[(a+I*a*tan(e+f*x))^6/(c-I*c*tan(e+f*x))^3,x,4,-40*a^6*x/c^3+40*I*a^6*log(cos(e+f*x))/(c^3*f)+9*a^6*tan(e+f*x)/(c^3*f)+1/2*I*a^6*tan(e+f*x)^2/(c^3*f)-32/3*I*a^6/(f*(c-I*c*tan(e+f*x))^3)+40*I*a^6/(c*f*(c-I*c*tan(e+f*x))^2)-80*I*a^6/(f*(c^3-I*c^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^5/(c-I*c*tan(e+f*x))^3,x,4,-8*a^5*x/c^3+8*I*a^5*log(cos(e+f*x))/(c^3*f)+a^5*tan(e+f*x)/(c^3*f)-16/3*I*a^5/(f*(c-I*c*tan(e+f*x))^3)-24*I*a^5/(f*(c^3-I*c^3*tan(e+f*x)))+16*I*a^5*c^5/(f*(c^4-I*c^4*tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))^4/(c-I*c*tan(e+f*x))^3,x,4,-a^4*x/c^3+I*a^4*log(cos(e+f*x))/(c^3*f)-8/3*I*a^4/(f*(c-I*c*tan(e+f*x))^3)+6*I*a^4/(c*f*(c-I*c*tan(e+f*x))^2)-6*I*a^4/(f*(c^3-I*c^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x))^3,x,3,-1/6*I*a^3*(c^2+I*c^2*tan(e+f*x))^3/(f*(c^3-I*c^3*tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x))^3,x,4,-2/3*I*a^2/(f*(c-I*c*tan(e+f*x))^3)+1/2*I*a^2/(c*f*(c-I*c*tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x))^3,x,3,-1/3*I*a/(f*(c-I*c*tan(e+f*x))^3)],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^3),x,5,1/4*x/(a*c^3)+(-1/12*I)/(a*f*(c-I*c*tan(e+f*x))^3)+(-1/8*I)/(a*c*f*(c-I*c*tan(e+f*x))^2)+(-3/16*I)/(a*f*(c^3-I*c^3*tan(e+f*x)))+1/16*I/(a*f*(c^3+I*c^3*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^3),x,5,5/16*x/(a^2*c^3)+(-1/24*I)/(a^2*f*(c-I*c*tan(e+f*x))^3)+(-3/32*I)/(a^2*c*f*(c-I*c*tan(e+f*x))^2)+1/32*I/(a^2*c*f*(c+I*c*tan(e+f*x))^2)+(-3/16*I)/(a^2*f*(c^3-I*c^3*tan(e+f*x)))+1/8*I/(a^2*f*(c^3+I*c^3*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^3),x,5,5/16*x/(a^3*c^3)+5/16*cos(e+f*x)*sin(e+f*x)/(a^3*c^3*f)+5/24*cos(e+f*x)^3*sin(e+f*x)/(a^3*c^3*f)+1/6*cos(e+f*x)^5*sin(e+f*x)/(a^3*c^3*f)],
[(a+I*a*tan(e+f*x))^6/(c-I*c*tan(e+f*x))^4,x,4,10*a^6*x/c^4-10*I*a^6*log(cos(e+f*x))/(c^4*f)-a^6*tan(e+f*x)/(c^4*f)-8*I*a^6/(f*(c-I*c*tan(e+f*x))^4)+80/3*I*a^6/(c*f*(c-I*c*tan(e+f*x))^3)-40*I*a^6/(f*(c^2-I*c^2*tan(e+f*x))^2)+40*I*a^6/(f*(c^4-I*c^4*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^5/(c-I*c*tan(e+f*x))^4,x,4,a^5*x/c^4-I*a^5*log(cos(e+f*x))/(c^4*f)-4*I*a^5/(f*(c-I*c*tan(e+f*x))^4)-12*I*a^5/(f*(c^2-I*c^2*tan(e+f*x))^2)+32/3*I*a^5*c^5/(f*(c^3-I*c^3*tan(e+f*x))^3)+8*I*a^5/(f*(c^4-I*c^4*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^4/(c-I*c*tan(e+f*x))^4,x,3,-1/8*I*a^4*(c^2+I*c^2*tan(e+f*x))^4/(f*(c^3-I*c^3*tan(e+f*x))^4)],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x))^4,x,4,-I*a^3/(f*(c-I*c*tan(e+f*x))^4)+4/3*I*a^3/(c*f*(c-I*c*tan(e+f*x))^3)-1/2*I*a^3/(f*(c^2-I*c^2*tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x))^4,x,4,-1/2*I*a^2/(f*(c-I*c*tan(e+f*x))^4)+1/3*I*a^2*c^2/(f*(c^2-I*c^2*tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x))^4,x,3,-1/4*I*a/(f*(c-I*c*tan(e+f*x))^4)],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^4),x,5,5/32*x/(a*c^4)+(-1/16*I)/(a*f*(c-I*c*tan(e+f*x))^4)+(-1/12*I)/(a*c*f*(c-I*c*tan(e+f*x))^3)+(-3/32*I)/(a*f*(c^2-I*c^2*tan(e+f*x))^2)+(-1/8*I)/(a*f*(c^4-I*c^4*tan(e+f*x)))+1/32*I/(a*f*(c^4+I*c^4*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^4),x,5,15/64*x/(a^2*c^4)+(-1/32*I)/(a^2*f*(c-I*c*tan(e+f*x))^4)+(-1/16*I)/(a^2*c*f*(c-I*c*tan(e+f*x))^3)+(-3/32*I)/(a^2*f*(c^2-I*c^2*tan(e+f*x))^2)+1/64*I/(a^2*f*(c^2+I*c^2*tan(e+f*x))^2)+(-5/32*I)/(a^2*f*(c^4-I*c^4*tan(e+f*x)))+5/64*I/(a^2*f*(c^4+I*c^4*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^4),x,5,35/128*x/(a^3*c^4)+(-1/64*I)/(a^3*f*(c-I*c*tan(e+f*x))^4)+(-1/24*I)/(a^3*c*f*(c-I*c*tan(e+f*x))^3)+1/96*I/(a^3*c*f*(c+I*c*tan(e+f*x))^3)+(-5/64*I)/(a^3*f*(c^2-I*c^2*tan(e+f*x))^2)+5/128*I/(a^3*f*(c^2+I*c^2*tan(e+f*x))^2)+(-5/32*I)/(a^3*f*(c^4-I*c^4*tan(e+f*x)))+15/128*I/(a^3*f*(c^4+I*c^4*tan(e+f*x)))],

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

# n > 0
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^3,x,4,8*I*a^3*sqrt(c-I*c*tan(e+f*x))/f-8/3*I*a^3*(c-I*c*tan(e+f*x))^(3/2)/(c*f)+2/5*I*a^3*(c-I*c*tan(e+f*x))^(5/2)/(c^2*f)],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^2,x,4,4*I*a^2*sqrt(c-I*c*tan(e+f*x))/f-2/3*I*a^2*(c-I*c*tan(e+f*x))^(3/2)/(c*f)],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x)),x,3,2*I*a*sqrt(c-I*c*tan(e+f*x))/f],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,5,1/2*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a*f*sqrt(2))+1/2*I*sqrt(c-I*c*tan(e+f*x))/(a*f*(1+I*tan(e+f*x)))],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,6,3/16*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a^2*f*sqrt(2))+1/4*I*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x))^2)+3/16*I*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,7,5/64*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a^3*f*sqrt(2))+1/6*I*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x))^3)+5/48*I*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x))^2)+5/64*I*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(3/2),x,4,8/3*I*a^3*(c-I*c*tan(e+f*x))^(3/2)/f-8/5*I*a^3*(c-I*c*tan(e+f*x))^(5/2)/(c*f)+2/7*I*a^3*(c-I*c*tan(e+f*x))^(7/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2),x,4,4/3*I*a^2*(c-I*c*tan(e+f*x))^(3/2)/f-2/5*I*a^2*(c-I*c*tan(e+f*x))^(5/2)/(c*f)],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,3,2/3*I*a*(c-I*c*tan(e+f*x))^(3/2)/f],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x)),x,5,-I*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*f*sqrt(2))+I*c^2*sqrt(c-I*c*tan(e+f*x))/(a*f*(c+I*c*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^2,x,6,-1/8*I*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))+1/2*I*c^3*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(c+I*c*tan(e+f*x))^2)-1/8*I*c^2*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(c+I*c*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^3,x,7,-1/32*I*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))+1/3*I*c^4*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(c+I*c*tan(e+f*x))^3)-1/24*I*c^3*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(c+I*c*tan(e+f*x))^2)-1/32*I*c^2*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(c+I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(5/2),x,4,8/5*I*a^3*(c-I*c*tan(e+f*x))^(5/2)/f-8/7*I*a^3*(c-I*c*tan(e+f*x))^(7/2)/(c*f)+2/9*I*a^3*(c-I*c*tan(e+f*x))^(9/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2),x,4,4/5*I*a^2*(c-I*c*tan(e+f*x))^(5/2)/f-2/7*I*a^2*(c-I*c*tan(e+f*x))^(7/2)/(c*f)],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,3,2/5*I*a*(c-I*c*tan(e+f*x))^(5/2)/f],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x)),x,6,-3*I*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*f)+3*I*c^2*sqrt(c-I*c*tan(e+f*x))/(a*f)+I*c^2*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(c+I*c*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^2,x,6,3/4*I*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))+1/2*I*c^3*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*(c+I*c*tan(e+f*x))^2)-3/4*I*c^3*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(c+I*c*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^3,x,7,1/16*I*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))+1/3*I*c^4*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f*(c+I*c*tan(e+f*x))^3)-1/4*I*c^4*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(c+I*c*tan(e+f*x))^2)+1/16*I*c^3*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(c+I*c*tan(e+f*x)))],

# n < 0
[(a+I*a*tan(e+f*x))^3/sqrt(c-I*c*tan(e+f*x)),x,4,-8*I*a^3/(f*sqrt(c-I*c*tan(e+f*x)))-8*I*a^3*sqrt(c-I*c*tan(e+f*x))/(c*f)+2/3*I*a^3*(c-I*c*tan(e+f*x))^(3/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2/sqrt(c-I*c*tan(e+f*x)),x,4,-4*I*a^2/(f*sqrt(c-I*c*tan(e+f*x)))-2*I*a^2*sqrt(c-I*c*tan(e+f*x))/(c*f)],
[(a+I*a*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,3,-2*I*a/(f*sqrt(c-I*c*tan(e+f*x)))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))),x,6,3/4*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*f*sqrt(2)*sqrt(c))+(-3/4*I)/(a*f*sqrt(c-I*c*tan(e+f*x)))+1/2*I/(a*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^2),x,7,15/32*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2)*sqrt(c))+(-15/32*I)/(a^2*f*sqrt(c-I*c*tan(e+f*x)))+1/4*I/(a^2*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^2)+5/16*I/(a^2*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^3),x,8,35/128*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2)*sqrt(c))+(-35/128*I)/(a^3*f*sqrt(c-I*c*tan(e+f*x)))+1/6*I/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^3)+7/48*I/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^2)+35/192*I/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x))^(3/2),x,4,8*I*a^3/(c*f*sqrt(c-I*c*tan(e+f*x)))+2*I*a^3*sqrt(c-I*c*tan(e+f*x))/(c^2*f)-8/3*I*a^3/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x))^(3/2),x,4,2*I*a^2/(c*f*sqrt(c-I*c*tan(e+f*x)))-4/3*I*a^2/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,3,-2/3*I*a/(f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)),x,7,5/8*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*c^(3/2)*f*sqrt(2))+(-5/8*I)/(a*c*f*sqrt(c-I*c*tan(e+f*x)))+(-5/12*I)/(a*f*(c-I*c*tan(e+f*x))^(3/2))+1/2*I/(a*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2)),x,8,35/64*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*c^(3/2)*f*sqrt(2))+(-35/64*I)/(a^2*c*f*sqrt(c-I*c*tan(e+f*x)))+(-35/96*I)/(a^2*f*(c-I*c*tan(e+f*x))^(3/2))+1/4*I/(a^2*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2))+7/16*I/(a^2*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(3/2)),x,9,105/256*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*c^(3/2)*f*sqrt(2))+(-105/256*I)/(a^3*c*f*sqrt(c-I*c*tan(e+f*x)))+(-35/128*I)/(a^3*f*(c-I*c*tan(e+f*x))^(3/2))+1/6*I/(a^3*f*(1+I*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(3/2))+3/16*I/(a^3*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2))+21/64*I/(a^3*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^3/(c-I*c*tan(e+f*x))^(5/2),x,4,-2*I*a^3/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-8/5*I*a^3/(f*(c-I*c*tan(e+f*x))^(5/2))+8/3*I*a^3/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2/(c-I*c*tan(e+f*x))^(5/2),x,4,-4/5*I*a^2/(f*(c-I*c*tan(e+f*x))^(5/2))+2/3*I*a^2/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,3,-2/5*I*a/(f*(c-I*c*tan(e+f*x))^(5/2))],
[1/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)),x,8,7/16*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*f*sqrt(2))+(-7/16*I)/(a*c^2*f*sqrt(c-I*c*tan(e+f*x)))+(-7/20*I)/(a*f*(c-I*c*tan(e+f*x))^(5/2))+1/2*I/(a*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))+(-7/24*I)/(a*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2)),x,9,63/128*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*c^(5/2)*f*sqrt(2))+(-63/128*I)/(a^2*c^2*f*sqrt(c-I*c*tan(e+f*x)))+(-63/160*I)/(a^2*f*(c-I*c*tan(e+f*x))^(5/2))+1/4*I/(a^2*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2))+9/16*I/(a^2*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))+(-21/64*I)/(a^2*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(5/2)),x,10,231/512*I*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*c^(5/2)*f*sqrt(2))+(-231/512*I)/(a^3*c^2*f*sqrt(c-I*c*tan(e+f*x)))+(-231/640*I)/(a^3*f*(c-I*c*tan(e+f*x))^(5/2))+1/6*I/(a^3*f*(1+I*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(5/2))+11/48*I/(a^3*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2))+33/64*I/(a^3*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))+(-77/256*I)/(a^3*c*f*(c-I*c*tan(e+f*x))^(3/2))],

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

# n > 0
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2),x,6,-3*I*a^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/f+3/2*I*a^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f+1/2*I*a*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/f],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2),x,5,-2*I*a^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/f+I*a*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(1/2),x,4,-2*I*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)*sqrt(c)/f],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^(1/2),x,2,I*sqrt(c-I*c*tan(e+f*x))/(f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^(3/2),x,3,1/3*I*sqrt(c-I*c*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*I*sqrt(c-I*c*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(3/2))],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^(5/2),x,4,2/15*I*sqrt(c-I*c*tan(e+f*x))/(a^2*f*sqrt(a+I*a*tan(e+f*x)))+1/5*I*sqrt(c-I*c*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(5/2))+2/15*I*sqrt(c-I*c*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^(3/2))],
[sqrt(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^(7/2),x,5,2/35*I*sqrt(c-I*c*tan(e+f*x))/(a^3*f*sqrt(a+I*a*tan(e+f*x)))+1/7*I*sqrt(c-I*c*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(7/2))+3/35*I*sqrt(c-I*c*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^(5/2))+2/35*I*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2),x,6,-I*a^(5/2)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/2*a^2*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/3*I*a*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2),x,5,-I*a^(3/2)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/2*a*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f],
[(a+I*a*tan(e+f*x))^(1/2)*(c-I*c*tan(e+f*x))^(3/2),x,5,-2*I*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/f-I*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(1/2),x,5,2*I*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(a))+2*I*c*sqrt(c-I*c*tan(e+f*x))/(f*sqrt(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(3/2),x,2,1/3*I*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(5/2),x,3,1/5*I*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(5/2))+1/15*I*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(7/2),x,4,1/7*I*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(7/2))+2/35*I*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(5/2))+2/105*I*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*(a+I*a*tan(e+f*x))^(3/2))],
[(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(9/2),x,5,1/9*I*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(9/2))+1/21*I*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(7/2))+2/105*I*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*(a+I*a*tan(e+f*x))^(5/2))+2/315*I*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2),x,6,-3/4*I*a^(5/2)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+3/8*a^2*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/4*a*c*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2),x,6,-I*a^(3/2)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/2*a*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f-1/3*I*c*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[(a+I*a*tan(e+f*x))^(1/2)*(c-I*c*tan(e+f*x))^(5/2),x,6,-3*I*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/f-3/2*I*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f-1/2*I*c*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/f],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(1/2),x,6,6*I*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(a))+3*I*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a*f)+2*I*c*(c-I*c*tan(e+f*x))^(3/2)/(f*sqrt(a+I*a*tan(e+f*x)))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(3/2),x,6,-2*I*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(3/2)*f)-2*I*c^2*sqrt(c-I*c*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+2/3*I*c*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(5/2),x,2,1/5*I*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(5/2))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(7/2),x,3,1/7*I*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(7/2))+1/35*I*(c-I*c*tan(e+f*x))^(5/2)/(a*f*(a+I*a*tan(e+f*x))^(5/2))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(9/2),x,4,1/9*I*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(9/2))+2/63*I*(c-I*c*tan(e+f*x))^(5/2)/(a*f*(a+I*a*tan(e+f*x))^(7/2))+2/315*I*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f*(a+I*a*tan(e+f*x))^(5/2))],
[(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(11/2),x,5,1/11*I*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(11/2))+1/33*I*(c-I*c*tan(e+f*x))^(5/2)/(a*f*(a+I*a*tan(e+f*x))^(9/2))+2/231*I*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f*(a+I*a*tan(e+f*x))^(7/2))+2/1155*I*(c-I*c*tan(e+f*x))^(5/2)/(a^3*f*(a+I*a*tan(e+f*x))^(5/2))],

# n < 0
[(a+I*a*tan(e+f*x))^(7/2)/sqrt(c-I*c*tan(e+f*x)),x,7,15*I*a^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-15/2*I*a^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c*f)-5/2*I*a^2*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c*f)-2*I*a*(a+I*a*tan(e+f*x))^(5/2)/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(5/2)/sqrt(c-I*c*tan(e+f*x)),x,6,6*I*a^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-3*I*a^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c*f)-2*I*a*(a+I*a*tan(e+f*x))^(3/2)/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(3/2)/sqrt(c-I*c*tan(e+f*x)),x,5,2*I*a^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-2*I*a*sqrt(a+I*a*tan(e+f*x))/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(1/2)/sqrt(c-I*c*tan(e+f*x)),x,2,-I*sqrt(a+I*a*tan(e+f*x))/(f*sqrt(c-I*c*tan(e+f*x)))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(1/2)),x,2,tan(e+f*x)/(f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)),x,3,2/3*tan(e+f*x)/(a*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/3*I/(f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)),x,4,2/5*tan(e+f*x)/(a^2*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/5*I/(f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2))+1/5*I/(a*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))],
[1/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(7/2)),x,5,8/35*tan(e+f*x)/(a^3*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/7*I/(f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(7/2))+4/35*I/(a*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2))+4/35*I/(a^2*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(9/2)/(c-I*c*tan(e+f*x))^(3/2),x,8,-35*I*a^(9/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+35/2*I*a^4*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^2*f)+35/6*I*a^3*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f)+14/3*I*a^2*(a+I*a*tan(e+f*x))^(5/2)/(c*f*sqrt(c-I*c*tan(e+f*x)))-2/3*I*a*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(7/2)/(c-I*c*tan(e+f*x))^(3/2),x,7,-10*I*a^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+5*I*a^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^2*f)+10/3*I*a^2*(a+I*a*tan(e+f*x))^(3/2)/(c*f*sqrt(c-I*c*tan(e+f*x)))-2/3*I*a*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)/(c-I*c*tan(e+f*x))^(3/2),x,6,-2*I*a^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+2*I*a^2*sqrt(a+I*a*tan(e+f*x))/(c*f*sqrt(c-I*c*tan(e+f*x)))-2/3*I*a*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)/(c-I*c*tan(e+f*x))^(3/2),x,2,-1/3*I*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(1/2)/(c-I*c*tan(e+f*x))^(3/2),x,3,-1/3*I*sqrt(a+I*a*tan(e+f*x))/(c*f*sqrt(c-I*c*tan(e+f*x)))-1/3*I*sqrt(a+I*a*tan(e+f*x))/(f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(1/2)*(c-I*c*tan(e+f*x))^(3/2)),x,4,-2/3*I*sqrt(a+I*a*tan(e+f*x))/(a*c*f*sqrt(c-I*c*tan(e+f*x)))+I/(f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))-2/3*I*sqrt(a+I*a*tan(e+f*x))/(a*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)),x,3,2/3*tan(e+f*x)/(a*c*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/3*tan(e+f*x)/(f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2)),x,4,8/15*tan(e+f*x)/(a^2*c*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/5*I/(f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2))+4/15*tan(e+f*x)/(a*f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(3/2)),x,5,8/21*tan(e+f*x)/(a^3*c*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/7*I/(f*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(3/2))+1/7*I/(a*f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2))+4/21*tan(e+f*x)/(a^2*f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(11/2)/(c-I*c*tan(e+f*x))^(5/2),x,9,63*I*a^(11/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(5/2)*f)-63/2*I*a^5*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^3*f)-21/2*I*a^4*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c^3*f)-42/5*I*a^3*(a+I*a*tan(e+f*x))^(5/2)/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-2/5*I*a*(a+I*a*tan(e+f*x))^(9/2)/(f*(c-I*c*tan(e+f*x))^(5/2))+6/5*I*a^2*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(9/2)/(c-I*c*tan(e+f*x))^(5/2),x,8,14*I*a^(9/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(5/2)*f)-7*I*a^4*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^3*f)-14/3*I*a^3*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-2/5*I*a*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(5/2))+14/15*I*a^2*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(7/2)/(c-I*c*tan(e+f*x))^(5/2),x,7,2*I*a^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(5/2)*f)-2*I*a^3*sqrt(a+I*a*tan(e+f*x))/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-2/5*I*a*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(5/2))+2/3*I*a^2*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)/(c-I*c*tan(e+f*x))^(5/2),x,2,-1/5*I*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(3/2)/(c-I*c*tan(e+f*x))^(5/2),x,3,-1/5*I*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(5/2))-1/15*I*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(1/2)/(c-I*c*tan(e+f*x))^(5/2),x,4,-2/15*I*sqrt(a+I*a*tan(e+f*x))/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-1/5*I*sqrt(a+I*a*tan(e+f*x))/(f*(c-I*c*tan(e+f*x))^(5/2))-2/15*I*sqrt(a+I*a*tan(e+f*x))/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(1/2)*(c-I*c*tan(e+f*x))^(5/2)),x,5,-2/5*I*sqrt(a+I*a*tan(e+f*x))/(a*c^2*f*sqrt(c-I*c*tan(e+f*x)))+I/(f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-3/5*I*sqrt(a+I*a*tan(e+f*x))/(a*f*(c-I*c*tan(e+f*x))^(5/2))-2/5*I*sqrt(a+I*a*tan(e+f*x))/(a*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2)),x,6,-8/15*I*sqrt(a+I*a*tan(e+f*x))/(a^2*c^2*f*sqrt(c-I*c*tan(e+f*x)))+4/3*I/(a*f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-4/5*I*sqrt(a+I*a*tan(e+f*x))/(a^2*f*(c-I*c*tan(e+f*x))^(5/2))+1/3*I/(f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2))-8/15*I*sqrt(a+I*a*tan(e+f*x))/(a^2*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)),x,4,8/15*tan(e+f*x)/(a^2*c^2*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/5*tan(e+f*x)/(f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2))+4/15*tan(e+f*x)/(a*c*f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(5/2)),x,5,16/35*tan(e+f*x)/(a^3*c^2*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/7*I/(f*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(5/2))+6/35*tan(e+f*x)/(a*f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2))+8/35*tan(e+f*x)/(a^2*c*f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],

# Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n with n symbolic
[(a+I*a*tan(e+f*x))^4*(c-I*c*tan(e+f*x))^n,x,4,8*I*a^4*(c-I*c*tan(e+f*x))^n/(f*n)-12*I*a^4*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))+6*I*a^4*(c-I*c*tan(e+f*x))^(2+n)/(c^2*f*(2+n))-I*a^4*(c-I*c*tan(e+f*x))^(3+n)/(c^3*f*(3+n))],
[(a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^n,x,4,4*I*a^3*(c-I*c*tan(e+f*x))^n/(f*n)-4*I*a^3*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))+I*a^3*(c-I*c*tan(e+f*x))^(2+n)/(c^2*f*(2+n))],
[(a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^n,x,4,2*I*a^2*(c-I*c*tan(e+f*x))^n/(f*n)-I*a^2*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))],
[(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^n,x,3,I*a*(c-I*c*tan(e+f*x))^n/(f*n)],
[(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x)),x,3,1/4*I*hypergeom([2,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a*f*n)],
[(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^2,x,3,1/8*I*hypergeom([3,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a^2*f*n)],
[(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^3,x,3,1/16*I*hypergeom([4,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a^3*f*n)],

# Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n with m symbolic
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^n,x,3,1/2*I*hypergeom([1,m+n],[1+n],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^n/(f*n),-I*2^(-1+n)*hypergeom([m,1-n],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^n/(f*m*(1-I*tan(e+f*x))^n)],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^4,x,4,-8*I*c^4*(a+I*a*tan(e+f*x))^m/(f*m)+12*I*c^4*(a+I*a*tan(e+f*x))^(1+m)/(a*f*(1+m))-6*I*c^4*(a+I*a*tan(e+f*x))^(2+m)/(a^2*f*(2+m))+I*c^4*(a+I*a*tan(e+f*x))^(3+m)/(a^3*f*(3+m))],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^3,x,4,-4*I*c^3*(a+I*a*tan(e+f*x))^m/(f*m)+4*I*c^3*(a+I*a*tan(e+f*x))^(1+m)/(a*f*(1+m))-I*c^3*(a+I*a*tan(e+f*x))^(2+m)/(a^2*f*(2+m))],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^2,x,4,-2*I*c^2*(a+I*a*tan(e+f*x))^m/(f*m)+I*c^2*(a+I*a*tan(e+f*x))^(1+m)/(a*f*(1+m))],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x)),x,3,-I*c*(a+I*a*tan(e+f*x))^m/(f*m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x)),x,3,-1/4*I*hypergeom([2,m],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(c*f*m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^2,x,3,-1/8*I*hypergeom([3,m],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(c^2*f*m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^3,x,3,-1/16*I*hypergeom([4,m],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(c^3*f*m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^4,x,3,-1/32*I*hypergeom([5,m],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(c^4*f*m)],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(5/2),x,3,1/5*I*hypergeom([1,5/2+m],[7/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(5/2)/f,1/5*I*2^m*hypergeom([5/2,1-m],[7/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(5/2)/(f*(1+I*tan(e+f*x))^m)],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(3/2),x,3,1/3*I*hypergeom([1,3/2+m],[5/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(3/2)/f,1/3*I*2^m*hypergeom([3/2,1-m],[5/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(3/2)/(f*(1+I*tan(e+f*x))^m)],
[(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(1/2),x,3,I*hypergeom([1,1/2+m],[3/2],1/2*(1-I*tan(e+f*x)))*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^m/f,I*2^m*hypergeom([1/2,1-m],[3/2],1/2*(1-I*tan(e+f*x)))*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^m/(f*(1+I*tan(e+f*x))^m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^(1/2),x,3,-I*hypergeom([1,-1/2+m],[1/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*sqrt(c-I*c*tan(e+f*x))),-I*2^m*hypergeom([-1/2,1-m],[1/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^m)],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^(3/2),x,3,-1/3*I*hypergeom([1,-3/2+m],[-1/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*(c-I*c*tan(e+f*x))^(3/2)),-1/3*I*2^m*hypergeom([-3/2,1-m],[-1/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*(1+I*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^m/(c-I*c*tan(e+f*x))^(5/2),x,3,-1/5*I*hypergeom([1,-5/2+m],[-3/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*(c-I*c*tan(e+f*x))^(5/2)),-1/5*I*2^m*hypergeom([-5/2,1-m],[-3/2],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(f*(1+I*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^(5/2))],

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

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

# n > 0
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x)),x,4,4*a^3*(c-I*d)*x-4*a^3*(I*c+d)*log(cos(e+f*x))/f-2*a^3*(c-I*d)*tan(e+f*x)/f+1/2*a*(I*c+d)*(a+I*a*tan(e+f*x))^2/f+1/3*d*(a+I*a*tan(e+f*x))^3/f],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x)),x,3,2*a^2*(c-I*d)*x-2*a^2*(I*c+d)*log(cos(e+f*x))/f-a^2*(c-I*d)*tan(e+f*x)/f+1/2*d*(a+I*a*tan(e+f*x))^2/f],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x)),x,2,a*(c-I*d)*x-a*(I*c+d)*log(cos(e+f*x))/f+I*a*d*tan(e+f*x)/f],
[(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,2,1/2*(c-I*d)*x/a+1/2*(I*c-d)/(f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,3,1/4*(c-I*d)*x/a^2+1/4*(I*c-d)/(f*(a+I*a*tan(e+f*x))^2)+1/4*(I*c+d)/(f*(a^2+I*a^2*tan(e+f*x)))],
[(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,4,1/8*(c-I*d)*x/a^3+1/6*(I*c-d)/(f*(a+I*a*tan(e+f*x))^3)+1/8*(I*c+d)/(a*f*(a+I*a*tan(e+f*x))^2)+1/8*(I*c+d)/(f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^2,x,5,4*a^3*(c-I*d)^2*x-4*I*a^3*(c-I*d)^2*log(cos(e+f*x))/f-2*a^3*(c-I*d)^2*tan(e+f*x)/f+1/2*I*a*(c-I*d)^2*(a+I*a*tan(e+f*x))^2/f+2/3*c*d*(a+I*a*tan(e+f*x))^3/f-1/4*I*d^2*(a+I*a*tan(e+f*x))^4/(a*f)],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^2,x,4,2*a^2*(c-I*d)^2*x-2*I*a^2*(c-I*d)^2*log(cos(e+f*x))/f-a^2*(c-I*d)^2*tan(e+f*x)/f+c*d*(a+I*a*tan(e+f*x))^2/f-1/3*I*d^2*(a+I*a*tan(e+f*x))^3/(a*f)],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^2,x,3,a*(c-I*d)^2*x-I*a*(c-I*d)^2*log(cos(e+f*x))/f+a*d*(I*c+d)*tan(e+f*x)/f+1/2*I*a*(c+d*tan(e+f*x))^2/f],
[(c+d*tan(e+f*x))^2/(a+I*a*tan(e+f*x)),x,3,1/2*(c^2-2*I*c*d+d^2)*x/a+I*d^2*log(cos(e+f*x))/(a*f)+1/2*I*(c+I*d)^2/(f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^2,x,3,1/4*(c-I*d)^2*x/a^2+1/4*(c+I*d)*(I*c+3*d)/(a^2*f*(1+I*tan(e+f*x)))+1/4*I*(c+I*d)^2/(f*(a+I*a*tan(e+f*x))^2)],
[(c+d*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^3,x,4,1/8*(c-I*d)^2*x/a^3+1/6*I*(c+I*d)^2/(f*(a+I*a*tan(e+f*x))^3)+1/8*(c+I*d)*(I*c+3*d)/(a*f*(a+I*a*tan(e+f*x))^2)+1/8*I*(c-I*d)^2/(f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^3,x,6,4*a^3*(c-I*d)^3*x+4*a^3*(I*c+d)^3*log(cos(e+f*x))/f+4*I*a^3*(c-I*d)^2*d*tan(e+f*x)/f+2*a^3*(I*c+d)*(c+d*tan(e+f*x))^2/f+4/3*I*a^3*(c+d*tan(e+f*x))^3/f+1/20*a^3*(I*c-11*d)*(c+d*tan(e+f*x))^4/(d^2*f)-1/5*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^4/(d*f)],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^3,x,5,2*a^2*(c-I*d)^3*x+2*a^2*(I*c+d)^3*log(cos(e+f*x))/f+2*I*a^2*(c-I*d)^2*d*tan(e+f*x)/f+a^2*(I*c+d)*(c+d*tan(e+f*x))^2/f+2/3*I*a^2*(c+d*tan(e+f*x))^3/f-1/4*a^2*(c+d*tan(e+f*x))^4/(d*f)],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^3,x,4,a*(c-I*d)^3*x+a*(I*c+d)^3*log(cos(e+f*x))/f+I*a*(c-I*d)^2*d*tan(e+f*x)/f+1/2*a*(I*c+d)*(c+d*tan(e+f*x))^2/f+1/3*I*a*(c+d*tan(e+f*x))^3/f],
[(c+d*tan(e+f*x))^3/(a+I*a*tan(e+f*x)),x,3,1/2*(c^3-3*I*c^2*d+3*c*d^2+3*I*d^3)*x/a+(3*I*c-d)*d^2*log(cos(e+f*x))/(a*f)-1/2*(c+3*I*d)*d^2*tan(e+f*x)/(a*f)+1/2*(I*c-d)*(c+d*tan(e+f*x))^2/(f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^2,x,5,1/4*(c^3-3*I*c^2*d-3*c*d^2-3*I*d^3)*x/a^2+d^3*log(cos(e+f*x))/(a^2*f)+1/4*(c+I*d)^2*(I*c+3*d)/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*c-d)*(c+d*tan(e+f*x))^2/(f*(a+I*a*tan(e+f*x))^2)],
[(c+d*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^3,x,4,1/8*(c-I*d)^3*x/a^3+1/8*(c+I*d)*(c-3*I*d)*(I*c+d)/(a^3*f*(1+I*tan(e+f*x)))+1/8*(c+I*d)^2*(I*c+d)/(a*f*(a+I*a*tan(e+f*x))^2)+1/6*I*(c+d*tan(e+f*x))^3/(f*(a+I*a*tan(e+f*x))^3)],

# n < 0
[(a+I*a*tan(e+f*x))^3/(c+d*tan(e+f*x)),x,5,4*a^3*x/(c-I*d)-a^3*(I*c-3*d)*log(cos(e+f*x))/(d^2*f)-a^3*(c+I*d)^2*log(c*cos(e+f*x)+d*sin(e+f*x))/(d^2*(I*c+d)*f)+(-a^3-I*a^3*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^2/(c+d*tan(e+f*x)),x,4,-a^2*c*(c+I*d)*x/((c-I*d)*d^2)+a^2*(c+2*I*d)*x/d^2+a^2*log(cos(e+f*x))/(d*f)-a^2*(I*c-d)*log(c*cos(e+f*x)+d*sin(e+f*x))/(d*(I*c+d)*f)],
[(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x)),x,2,a*x/(c-I*d)+a*log(c*cos(e+f*x)+d*sin(e+f*x))/((I*c+d)*f)],
[1/((a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))),x,5,1/2*x/(a*(c+I*d))-c*d*x/(a*(I*c-d)*(c^2+d^2))-d^2*log(c*cos(e+f*x)+d*sin(e+f*x))/(a*(I*c-d)*(c^2+d^2)*f)+(-1/2)/((I*c-d)*f*(a+I*a*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))),x,4,1/4*(c^3+3*I*c^2*d-3*c*d^2+3*I*d^3)*x/(a^2*(c-I*d)*(c+I*d)^3)-d^3*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^2*(c-I*d)*(c+I*d)^3*f)+1/4*(I*c-3*d)/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x)))+(-1/4)/((I*c-d)*f*(a+I*a*tan(e+f*x))^2)],
[1/((a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))),x,5,1/8*(c^4+4*I*c^3*d-6*c^2*d^2-4*I*c*d^3-7*d^4)*x/(a^3*(c-I*d)*(c+I*d)^4)+d^4*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^3*(c+I*d)^4*(I*c+d)*f)+(-1/6)/((I*c-d)*f*(a+I*a*tan(e+f*x))^3)+1/8*(I*c-3*d)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2)+1/8*(c^2+4*I*c*d-7*d^2)/((I*c-d)^3*f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c+d*tan(e+f*x))^2,x,5,4*a^3*x/(c-I*d)^2+I*a^3*log(cos(e+f*x))/(d^2*f)-a^3*(I*c-d)*(c-3*I*d)*log(c*cos(e+f*x)+d*sin(e+f*x))/((c-I*d)^2*d^2*f)+(c+I*d)*(a^3+I*a^3*tan(e+f*x))/((c-I*d)*d*f*(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(c+d*tan(e+f*x))^2,x,3,2*a^2*x/(c-I*d)^2-2*I*a^2*log(c*cos(e+f*x)+d*sin(e+f*x))/((c-I*d)^2*f)+a^2*(I*c-d)/(d*(I*c+d)*f*(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^2,x,3,a*x/(c-I*d)^2-I*a*log(c*cos(e+f*x)+d*sin(e+f*x))/((c-I*d)^2*f)-a/((I*c+d)*f*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^2),x,4,1/2*(c^3+3*I*c^2*d+3*c*d^2-3*I*d^3)*x/(a*(c-I*d)^2*(c+I*d)^3)+(3*c-I*d)*d^2*log(c*cos(e+f*x)+d*sin(e+f*x))/(a*(I*c-d)^3*(c-I*d)^2*f)+1/2*(c-3*I*d)*d/(a*(c-I*d)*(c+I*d)^2*f*(c+d*tan(e+f*x)))+(-1/2)/((I*c-d)*f*(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^2),x,5,1/4*(c^4+4*I*c^3*d-6*c^2*d^2+12*I*c*d^3+9*d^4)*x/(a^2*(c-I*d)^2*(c+I*d)^4)-2*(2*c-I*d)*d^3*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^2*(c-I*d)^2*(c+I*d)^4*f)+1/4*d*(c^2+4*I*c*d+9*d^2)/(a^2*(c-I*d)*(c+I*d)^3*f*(c+d*tan(e+f*x)))+1/4*(I*c-4*d)/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x))*(c+d*tan(e+f*x)))+(-1/4)/((I*c-d)*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^2),x,6,1/8*(c^5+5*I*c^4*d-10*c^3*d^2-10*I*c^2*d^3-35*c*d^4+25*I*d^5)*x/(a^3*(c-I*d)^2*(c+I*d)^5)+(5*c-3*I*d)*d^4*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^3*(I*c-d)^5*(c-I*d)^2*f)+1/8*d*(c^3+5*I*c^2*d-11*c*d^2+25*I*d^3)/(a^3*(c-I*d)*(c+I*d)^4*f*(c+d*tan(e+f*x)))+(-1/6)/((I*c-d)*f*(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x)))+1/24*(3*I*c-11*d)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x)))+1/8*(c^2+5*I*c*d-12*d^2)/((I*c-d)^3*f*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c+d*tan(e+f*x))^3,x,4,4*a^3*x/(c-I*d)^3-4*a^3*log(c*cos(e+f*x)+d*sin(e+f*x))/((I*c+d)^3*f)-1/2*a*(a+I*a*tan(e+f*x))^2/((I*c+d)*f*(c+d*tan(e+f*x))^2)+2*a^3*(c+I*d)/((c-I*d)^2*d*f*(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(c+d*tan(e+f*x))^3,x,4,2*a^2*x/(c-I*d)^3-2*a^2*log(c*cos(e+f*x)+d*sin(e+f*x))/((I*c+d)^3*f)+1/2*a^2*(I*c-d)/(d*(I*c+d)*f*(c+d*tan(e+f*x))^2)+2*I*a^2/((c-I*d)^2*f*(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^3,x,4,a*x/(c-I*d)^3-a*log(c*cos(e+f*x)+d*sin(e+f*x))/((I*c+d)^3*f)-1/2*a/((I*c+d)*f*(c+d*tan(e+f*x))^2)+I*a/((c-I*d)^2*f*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^3),x,5,1/2*(c^4+4*I*c^3*d+6*c^2*d^2-12*I*c*d^3-3*d^4)*x/(a*(c-I*d)^3*(c+I*d)^4)+2*d^2*(3*c^2-2*I*c*d-d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/(a*(c+I*d)^4*(I*c+d)^3*f)+1/2*(c-2*I*d)*d/(a*(c-I*d)*(c+I*d)^2*f*(c+d*tan(e+f*x))^2)+(-1/2)/((I*c-d)*f*(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^2)+1/2*d*(c^2-8*I*c*d-3*d^2)/(a*(c-I*d)^2*(c+I*d)^3*f*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^3),x,6,1/4*(c^5+5*I*c^4*d-10*c^3*d^2+30*I*c^2*d^3+45*c*d^4-15*I*d^5)*x/(a^2*(c-I*d)^3*(c+I*d)^5)-2*d^3*(5*c^2-5*I*c*d-2*d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^2*(I*c-d)^5*(I*c+d)^3*f)+1/4*d*(c^2+5*I*c*d+8*d^2)/(a^2*(c-I*d)*(c+I*d)^3*f*(c+d*tan(e+f*x))^2)+1/4*(I*c-5*d)/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x))*(c+d*tan(e+f*x))^2)+(-1/4)/((I*c-d)*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^2)+1/4*(c-3*I*d)*d*(c^2+8*I*c*d+5*d^2)/(a^2*(c-I*d)^2*(c+I*d)^4*f*(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^3),x,7,1/8*(c^6+6*I*c^5*d-15*c^4*d^2-20*I*c^3*d^3-105*c^2*d^4+150*I*c*d^5+55*d^6)*x/(a^3*(c-I*d)^3*(c+I*d)^6)-d^4*(15*c^2-18*I*c*d-7*d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/(a^3*(c+I*d)^6*(I*c+d)^3*f)+1/8*d*(c^3+6*I*c^2*d-17*c*d^2+28*I*d^3)/(a^3*(c-I*d)*(c+I*d)^4*f*(c+d*tan(e+f*x))^2)+(-1/6)/((I*c-d)*f*(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^2)+1/24*(3*I*c-13*d)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^2)+1/24*(3*c^2+18*I*c*d-55*d^2)/((I*c-d)^3*f*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^2)+1/8*d*(c^4+6*I*c^3*d-16*c^2*d^2+94*I*c*d^3+55*d^4)/(a^3*(c-I*d)^2*(c+I*d)^5*f*(c+d*tan(e+f*x)))],

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

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3,x,6,-8*I*a^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+8*I*a^3*sqrt(c+d*tan(e+f*x))/f+4/15*a^3*(I*c-6*d)*(c+d*tan(e+f*x))^(3/2)/(d^2*f)-2/5*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2,x,5,-4*I*a^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+4*I*a^2*sqrt(c+d*tan(e+f*x))/f-2/3*a^2*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x)),x,4,-2*I*a*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+2*I*a*sqrt(c+d*tan(e+f*x))/f],
[sqrt(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,8,-1/2*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/(a*f)+1/2*I*c*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a*f*sqrt(c+I*d))+1/2*I*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,9,-1/8*(2*c*d-I*(2*c^2+d^2))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^2*(c+I*d)^(3/2)*f)-1/4*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/(a^2*f)+1/8*(2*I*c-d)*sqrt(c+d*tan(e+f*x))/(a^2*(c+I*d)*f*(1+I*tan(e+f*x)))+1/4*I*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^2)],
[sqrt(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,10,1/16*(2*I*c^3-4*c^2*d-I*c*d^2-2*d^3)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*(c+I*d)^(5/2)*f)-1/8*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/(a^3*f)+1/6*I*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^3)+1/24*(3*I*c-2*d)*sqrt(c+d*tan(e+f*x))/(a*(c+I*d)*f*(a+I*a*tan(e+f*x))^2)+1/16*c*(2*I*c-3*d)*sqrt(c+d*tan(e+f*x))/((c+I*d)^2*f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2),x,7,-8*I*a^3*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+8*a^3*(I*c+d)*sqrt(c+d*tan(e+f*x))/f+8/3*I*a^3*(c+d*tan(e+f*x))^(3/2)/f+4/35*a^3*(I*c-8*d)*(c+d*tan(e+f*x))^(5/2)/(d^2*f)-2/7*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2),x,6,-4*I*a^2*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+4*a^2*(I*c+d)*sqrt(c+d*tan(e+f*x))/f+4/3*I*a^2*(c+d*tan(e+f*x))^(3/2)/f-2/5*a^2*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2),x,5,-2*I*a*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+2*a*(I*c+d)*sqrt(c+d*tan(e+f*x))/f+2/3*I*a*(c+d*tan(e+f*x))^(3/2)/f],
[(c+d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x)),x,8,-1/2*I*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a*f)+1/2*(I*c+2*d)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/(a*f)+1/2*(I*c-d)*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^2,x,9,-1/4*I*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^2*f)+1/8*(2*c*d+I*(2*c^2+d^2))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^2*f*sqrt(c+I*d))+1/8*(2*I*c+3*d)*sqrt(c+d*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*c-d)*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^2)],
[(c+d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^3,x,10,-1/8*I*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^3*f)+1/16*I*c*(2*c^2+3*d^2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*(c+I*d)^(3/2)*f)+1/6*(I*c-d)*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^3)+1/24*(3*I*c+4*d)*sqrt(c+d*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^2)-1/16*(2*c^2-I*c*d+2*d^2)*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^(5/2),x,8,-8*I*a^3*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+8*I*a^3*(c-I*d)^2*sqrt(c+d*tan(e+f*x))/f+8/3*a^3*(I*c+d)*(c+d*tan(e+f*x))^(3/2)/f+8/5*I*a^3*(c+d*tan(e+f*x))^(5/2)/f+4/63*a^3*(I*c-10*d)*(c+d*tan(e+f*x))^(7/2)/(d^2*f)-2/9*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2),x,7,-4*I*a^2*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+4*I*a^2*(c-I*d)^2*sqrt(c+d*tan(e+f*x))/f+4/3*a^2*(I*c+d)*(c+d*tan(e+f*x))^(3/2)/f+4/5*I*a^2*(c+d*tan(e+f*x))^(5/2)/f-2/7*a^2*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2),x,6,-2*I*a*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+2*I*a*(c-I*d)^2*sqrt(c+d*tan(e+f*x))/f+2/3*a*(I*c+d)*(c+d*tan(e+f*x))^(3/2)/f+2/5*I*a*(c+d*tan(e+f*x))^(5/2)/f],
[(c+d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x)),x,9,-1/2*I*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a*f)+1/2*(c+I*d)^(3/2)*(I*c+4*d)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a*f)-1/2*(c+5*I*d)*d*sqrt(c+d*tan(e+f*x))/(a*f)+1/2*(I*c-d)*(c+d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^2,x,9,-1/4*I*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^2*f)+1/8*(2*I*c^2+6*c*d-7*I*d^2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/(a^2*f)+1/8*(c+I*d)*(2*I*c+5*d)*sqrt(c+d*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*c-d)*(c+d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^2)],
[(c+d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^3,x,10,-1/8*I*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^3*f)+1/16*(2*I*c^3+4*c^2*d-I*c*d^2+2*d^3)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*f*sqrt(c+I*d))+1/8*(c+I*d)*(I*c+2*d)*sqrt(c+d*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^2)+1/16*(2*I*c^2+5*c*d-4*I*d^2)*sqrt(c+d*tan(e+f*x))/(f*(a^3+I*a^3*tan(e+f*x)))+1/6*(I*c-d)*(c+d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^3)],

# n < 0
[(a+I*a*tan(e+f*x))^3/sqrt(c+d*tan(e+f*x)),x,5,-8*I*a^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+4/3*a^3*(I*c-4*d)*sqrt(c+d*tan(e+f*x))/(d^2*f)-2/3*sqrt(c+d*tan(e+f*x))*(a^3+I*a^3*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^2/sqrt(c+d*tan(e+f*x)),x,4,-4*I*a^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))-2*a^2*sqrt(c+d*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))/sqrt(c+d*tan(e+f*x)),x,3,-2*I*a*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))],
[1/(sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))),x,8,1/2*(I*c-2*d)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a*(c+I*d)^(3/2)*f)-1/2*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a*f*sqrt(c-I*d))-1/2*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a+I*a*tan(e+f*x)))],
[1/(sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2),x,9,1/8*(2*I*c^2-6*c*d-7*I*d^2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^2*(c+I*d)^(5/2)*f)-1/4*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^2*f*sqrt(c-I*d))+1/8*(2*I*c-5*d)*sqrt(c+d*tan(e+f*x))/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x)))-1/4*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a+I*a*tan(e+f*x))^2)],
[1/(sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3),x,10,1/16*(2*I*c^3-8*c^2*d-13*I*c*d^2+12*d^3)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*(c+I*d)^(7/2)*f)-1/8*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^3*f*sqrt(c-I*d))-1/6*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a+I*a*tan(e+f*x))^3)+1/24*(3*I*c-8*d)*sqrt(c+d*tan(e+f*x))/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2)+1/16*(2*c^2+7*I*c*d-10*d^2)*sqrt(c+d*tan(e+f*x))/((I*c-d)^3*f*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c+d*tan(e+f*x))^(3/2),x,5,-8*I*a^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+4*a^3*c*sqrt(c+d*tan(e+f*x))/(d^2*(I*c+d)*f)+2*(c+I*d)*(a^3+I*a^3*tan(e+f*x))/((c-I*d)*d*f*sqrt(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2/(c+d*tan(e+f*x))^(3/2),x,4,-4*I*a^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+2*a^2*(I*c-d)/(d*(I*c+d)*f*sqrt(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^(3/2),x,4,-2*I*a*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)-2*a/((I*c+d)*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)),x,9,-1/2*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a*(c-I*d)^(3/2)*f)+1/2*(I*c-4*d)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a*(c+I*d)^(5/2)*f)+1/2*(c-5*I*d)*d/(a*(c-I*d)*(c+I*d)^2*f*sqrt(c+d*tan(e+f*x)))+(-1/2)/((I*c-d)*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)),x,10,-1/4*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^2*(c-I*d)^(3/2)*f)+1/8*(2*I*c^2-10*c*d-23*I*d^2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^2*(c+I*d)^(7/2)*f)+1/8*d*(2*c^2+7*I*c*d+25*d^2)/(a^2*(c-I*d)*(c+I*d)^3*f*sqrt(c+d*tan(e+f*x)))+1/8*(2*I*c-7*d)/(a^2*(c+I*d)^2*f*sqrt(c+d*tan(e+f*x))*(1+I*tan(e+f*x)))+(-1/4)/((I*c-d)*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2)],
[1/((a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2)),x,11,-1/8*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^3*(c-I*d)^(3/2)*f)+1/16*(2*I*c^3-12*c^2*d-33*I*c*d^2+58*d^3)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*(c+I*d)^(9/2)*f)+1/16*d*(2*c^3+9*I*c^2*d-17*c*d^2+60*I*d^3)/(a^3*(c-I*d)*(c+I*d)^4*f*sqrt(c+d*tan(e+f*x)))+(-1/6)/((I*c-d)*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^3)+1/24*(3*I*c-10*d)/(a*(c+I*d)^2*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^2)+1/48*(6*c^2+27*I*c*d-56*d^2)/((I*c-d)^3*f*sqrt(c+d*tan(e+f*x))*(a^3+I*a^3*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3/(c+d*tan(e+f*x))^(5/2),x,5,-8*I*a^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+4/3*a^3*(I*c-d)*(c-4*I*d)/((c-I*d)^2*d^2*f*sqrt(c+d*tan(e+f*x)))+2/3*(c+I*d)*(a^3+I*a^3*tan(e+f*x))/((c-I*d)*d*f*(c+d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2/(c+d*tan(e+f*x))^(5/2),x,5,-4*I*a^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+4*I*a^2/((c-I*d)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*a^2*(I*c-d)/(d*(I*c+d)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^(5/2),x,5,-2*I*a*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+2*I*a/((c-I*d)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*a/((I*c+d)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)),x,10,-1/2*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a*(c-I*d)^(5/2)*f)+1/2*(I*c-6*d)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a*(c+I*d)^(7/2)*f)+1/2*d*(c^2-14*I*c*d-5*d^2)/(a*(c-I*d)^2*(c+I*d)^3*f*sqrt(c+d*tan(e+f*x)))+1/6*d*(3*I*c+7*d)/(a*(I*c-d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))+(-1/2)/((I*c-d)*f*(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)),x,11,-1/4*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^2*(c-I*d)^(5/2)*f)+1/8*(2*I*c^2-14*c*d-47*I*d^2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^2*(c+I*d)^(9/2)*f)+1/8*d*(2*c^3+9*I*c^2*d+88*c*d^2-45*I*d^3)/(a^2*(c-I*d)^2*(c+I*d)^4*f*sqrt(c+d*tan(e+f*x)))+1/24*d*(6*c^2+27*I*c*d+49*d^2)/(a^2*(c-I*d)*(c+I*d)^3*f*(c+d*tan(e+f*x))^(3/2))+1/8*(2*I*c-9*d)/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))+(-1/4)/((I*c-d)*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^(5/2)),x,12,-1/8*I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(a^3*(c-I*d)^(5/2)*f)+1/16*(2*I*c^3-16*c^2*d-61*I*c*d^2+152*d^3)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(a^3*(c+I*d)^(11/2)*f)+1/16*d*(2*c^4+11*I*c^3*d-26*c^2*d^2+253*I*c*d^3+150*d^4)/(a^3*(c-I*d)^2*(c+I*d)^5*f*sqrt(c+d*tan(e+f*x)))+1/48*d*(6*c^3+33*I*c^2*d-83*c*d^2+154*I*d^3)/(a^3*(c-I*d)*(c+I*d)^4*f*(c+d*tan(e+f*x))^(3/2))+(-1/6)/((I*c-d)*f*(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2))+1/8*(I*c-4*d)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2))+1/16*(2*c^2+11*I*c*d-30*d^2)/((I*c-d)^3*f*(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))],

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

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2),x,9,-1/4*(-1)^(1/4)*a^(5/2)*(c^2+10*I*c*d+23*d^2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-4*I*a^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(c-I*d)/f+1/4*a^2*(c+9*I*d)*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)-1/2*a^2*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2),x,9,-2*I*a^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(c-I*d)/f-(-1)^(1/4)*a^(3/2)*(I*c+3*d)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(d))+a^2*(c+I*d)*sqrt(c+d*tan(e+f*x))/(d*f*sqrt(a+I*a*tan(e+f*x)))-a^2*(c+d*tan(e+f*x))^(3/2)/(d*f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x)),x,7,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)*sqrt(c-I*d)/f-2*(-1)^(1/4)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))*sqrt(a)*sqrt(d)/f],
[sqrt(c+d*tan(e+f*x))/sqrt(a+I*a*tan(e+f*x)),x,3,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(c-I*d)/(f*sqrt(2)*sqrt(a))+I*sqrt(c+d*tan(e+f*x))/(f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^(3/2),x,4,-1/2*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(c-I*d)/(a^(3/2)*f*sqrt(2))+1/2*I*sqrt(c+d*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))-1/3*(c+d*tan(e+f*x))^(3/2)/((I*c-d)*f*(a+I*a*tan(e+f*x))^(3/2))],
[sqrt(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^(5/2),x,6,-1/4*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(c-I*d)/(a^(5/2)*f*sqrt(2))-1/60*(20*c*d-I*(15*c^2+3*d^2))*sqrt(c+d*tan(e+f*x))/(a^2*(c+I*d)^2*f*sqrt(a+I*a*tan(e+f*x)))+1/5*I*sqrt(c+d*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(5/2))+1/30*(5*I*c-3*d)*sqrt(c+d*tan(e+f*x))/(a*(c+I*d)*f*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(3/2),x,10,-1/8*(-1)^(1/4)*a^(5/2)*(c-3*I*d)*(c^2+18*I*c*d+15*d^2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-4*I*a^(5/2)*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/f+1/8*a^2*(c^2+14*I*c*d+19*d^2)*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)+1/12*a^2*(c+13*I*d)*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)-1/3*a^2*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2),x,10,-2*I*a^(3/2)*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/f-1/4*(-1)^(1/4)*a^(3/2)*(3*I*c^2+18*c*d-11*I*d^2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(d))+1/4*a*(3*I*c+5*d)*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f+1/2*a^2*(c+I*d)*(c+d*tan(e+f*x))^(3/2)/(d*f*sqrt(a+I*a*tan(e+f*x)))-1/2*a^2*(c+d*tan(e+f*x))^(5/2)/(d*f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2),x,8,-I*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)/f-(-1)^(1/4)*(3*c-I*d)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))*sqrt(a)*sqrt(d)/f+d*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f],
[(c+d*tan(e+f*x))^(3/2)/sqrt(a+I*a*tan(e+f*x)),x,8,2*(-1)^(3/4)*d^(3/2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a))-I*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(f*sqrt(2)*sqrt(a))+(I*c-d)*sqrt(c+d*tan(e+f*x))/(f*sqrt(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(3/2),x,4,-1/2*I*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/2*(I*c+d)*sqrt(c+d*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*I*(c+d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(5/2),x,5,-1/4*I*(c-I*d)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(5/2)*f*sqrt(2))+1/4*(I*c+d)*sqrt(c+d*tan(e+f*x))/(a^2*f*sqrt(a+I*a*tan(e+f*x)))+1/6*I*(c+d*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))-1/5*(c+d*tan(e+f*x))^(5/2)/((I*c-d)*f*(a+I*a*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(5/2),x,11,-1/64*(-1)^(1/4)*a^(5/2)*(5*c^4+100*I*c^3*d+690*c^2*d^2-900*I*c*d^3-363*d^4)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-4*I*a^(5/2)*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/f+1/64*a^2*(5*c^3+95*I*c^2*d+273*c*d^2-149*I*d^3)*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)+1/96*a^2*(5*c^2+90*I*c*d+107*d^2)*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)+1/24*a^2*(c+17*I*d)*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d*f)-1/4*a^2*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(5/2),x,11,-2*I*a^(3/2)*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/f-1/8*(-1)^(1/4)*a^(3/2)*(5*I*c^3+45*c^2*d-55*I*c*d^2-23*d^3)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(d))+1/8*a*(c-3*I*d)*(5*I*c+3*d)*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f+1/12*a*(5*I*c+7*d)*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/f+1/3*a^2*(c+I*d)*(c+d*tan(e+f*x))^(5/2)/(d*f*sqrt(a+I*a*tan(e+f*x)))-1/3*a^2*(c+d*tan(e+f*x))^(7/2)/(d*f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2),x,9,-I*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)/f-1/4*(-1)^(1/4)*(15*c^2-10*I*c*d-7*d^2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))*sqrt(a)*sqrt(d)/f+1/4*(7*c-I*d)*d*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f+1/2*d*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/f],
[(c+d*tan(e+f*x))^(5/2)/sqrt(a+I*a*tan(e+f*x)),x,9,(-1)^(1/4)*(5*I*c-d)*d^(3/2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a))-I*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(f*sqrt(2)*sqrt(a))-(c+2*I*d)*d*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(a*f)+(I*c-d)*(c+d*tan(e+f*x))^(3/2)/(f*sqrt(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(3/2),x,9,2*(-1)^(1/4)*d^(5/2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(a^(3/2)*f)-1/2*I*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/2*(c+I*d)*(I*c+3*d)*sqrt(c+d*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*c-d)*(c+d*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(5/2),x,5,-1/4*I*(c-I*d)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(5/2)*f*sqrt(2))+1/4*I*(c-I*d)^2*sqrt(c+d*tan(e+f*x))/(a^2*f*sqrt(a+I*a*tan(e+f*x)))+1/6*(I*c+d)*(c+d*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))+1/5*I*(c+d*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(5/2))],

# n < 0
[(a+I*a*tan(e+f*x))^(5/2)/sqrt(c+d*tan(e+f*x)),x,8,-(-1)^(1/4)*a^(5/2)*(c+5*I*d)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-4*I*a^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/(f*sqrt(c-I*d))-a^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)],
[(a+I*a*tan(e+f*x))^(3/2)/sqrt(c+d*tan(e+f*x)),x,7,-2*I*a^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/(f*sqrt(c-I*d))-2*(-1)^(3/4)*a^(3/2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(d))],
[sqrt(a+I*a*tan(e+f*x))/sqrt(c+d*tan(e+f*x)),x,2,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)/(f*sqrt(c-I*d))],
[1/(sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x))),x,4,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(f*sqrt(2)*sqrt(a)*sqrt(c-I*d))-sqrt(c+d*tan(e+f*x))/((I*c+d)*f*sqrt(a+I*a*tan(e+f*x)))+2*d*sqrt(c+d*tan(e+f*x))/((c^2+d^2)*f*sqrt(a+I*a*tan(e+f*x)))],
[1/(sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)),x,5,-1/2*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(3/2)*f*sqrt(2)*sqrt(c-I*d))+1/6*(3*I*c-7*d)*sqrt(c+d*tan(e+f*x))/(a*(c+I*d)^2*f*sqrt(a+I*a*tan(e+f*x)))-1/3*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a+I*a*tan(e+f*x))^(3/2))],
[1/(sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)),x,6,-1/4*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(5/2)*f*sqrt(2)*sqrt(c-I*d))+1/60*(15*c^2+50*I*c*d-67*d^2)*sqrt(c+d*tan(e+f*x))/(a^2*(I*c-d)^3*f*sqrt(a+I*a*tan(e+f*x)))-1/5*sqrt(c+d*tan(e+f*x))/((I*c-d)*f*(a+I*a*tan(e+f*x))^(5/2))+1/30*(5*I*c-13*d)*sqrt(c+d*tan(e+f*x))/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)/(c+d*tan(e+f*x))^(3/2),x,8,2*(-1)^(1/4)*a^(5/2)*arctanh((-1)^(3/4)*sqrt(d)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-4*I*a^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/((c-I*d)^(3/2)*f)+2*a^2*(c+I*d)*sqrt(a+I*a*tan(e+f*x))/((c-I*d)*d*f*sqrt(c+d*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(3/2)/(c+d*tan(e+f*x))^(3/2),x,3,-2*I*a^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/((c-I*d)^(3/2)*f)-2*a*sqrt(a+I*a*tan(e+f*x))/((I*c+d)*f*sqrt(c+d*tan(e+f*x)))],
[sqrt(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^(3/2),x,3,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)/((c-I*d)^(3/2)*f)-2*d*sqrt(a+I*a*tan(e+f*x))/((c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/(sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)),x,5,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/((c-I*d)^(3/2)*f*sqrt(2)*sqrt(a))+(-1)/((I*c-d)*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x)))+(c-3*I*d)*d*sqrt(a+I*a*tan(e+f*x))/(a*(c-I*d)*(c+I*d)^2*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2)),x,6,-1/2*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(3/2)*(c-I*d)^(3/2)*f*sqrt(2))+1/6*(3*I*c-11*d)/(a*(c+I*d)^2*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x)))+1/6*(3*c-5*I*d)*(c+5*I*d)*d*sqrt(a+I*a*tan(e+f*x))/(a^2*(c-I*d)*(c+I*d)^3*f*sqrt(c+d*tan(e+f*x)))+(-1/3)/((I*c-d)*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(3/2)),x,7,-1/4*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(5/2)*(c-I*d)^(3/2)*f*sqrt(2))+1/60*(15*c^2+70*I*c*d-151*d^2)/(a^2*(I*c-d)^3*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c+d*tan(e+f*x)))+1/60*d*(15*c^3+65*I*c^2*d-117*c*d^2+317*I*d^3)*sqrt(a+I*a*tan(e+f*x))/(a^3*(c-I*d)*(c+I*d)^4*f*sqrt(c+d*tan(e+f*x)))+(-1/5)/((I*c-d)*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2))+1/30*(5*I*c-17*d)/(a*(c+I*d)^2*f*sqrt(c+d*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)/(c+d*tan(e+f*x))^(5/2),x,4,-4*I*a^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/((c-I*d)^(5/2)*f)+4*I*a^2*sqrt(a+I*a*tan(e+f*x))/((c-I*d)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*a*(a+I*a*tan(e+f*x))^(3/2)/((I*c+d)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)/(c+d*tan(e+f*x))^(5/2),x,4,-2*I*a^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)/((c-I*d)^(5/2)*f)+2*I*a*sqrt(a+I*a*tan(e+f*x))/((c-I*d)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*d*(a+I*a*tan(e+f*x))^(3/2)/((c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[sqrt(a+I*a*tan(e+f*x))/(c+d*tan(e+f*x))^(5/2),x,5,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))*sqrt(2)*sqrt(a)/((c-I*d)^(5/2)*f)-2/3*(5*c+I*d)*d*sqrt(a+I*a*tan(e+f*x))/((c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*d*sqrt(a+I*a*tan(e+f*x))/((c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/(sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)),x,6,-I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/((c-I*d)^(5/2)*f*sqrt(2)*sqrt(a))+1/3*(3*c-I*d)*(c-7*I*d)*d*sqrt(a+I*a*tan(e+f*x))/(a*(c-I*d)^2*(c+I*d)^3*f*sqrt(c+d*tan(e+f*x)))+(-1)/((I*c-d)*f*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))+1/3*d*(3*I*c+5*d)*sqrt(a+I*a*tan(e+f*x))/(a*(I*c-d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(5/2)),x,7,-1/2*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(3/2)*(c-I*d)^(5/2)*f*sqrt(2))+1/6*(c-3*I*d)*d*(3*c^2+22*I*c*d+13*d^2)*sqrt(a+I*a*tan(e+f*x))/(a^2*(c-I*d)^2*(c+I*d)^4*f*sqrt(c+d*tan(e+f*x)))+1/2*(I*c-5*d)/(a*(c+I*d)^2*f*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))+1/6*d*(3*c^2+14*I*c*d+21*d^2)*sqrt(a+I*a*tan(e+f*x))/(a^2*(c-I*d)*(c+I*d)^3*f*(c+d*tan(e+f*x))^(3/2))+(-1/3)/((I*c-d)*f*(a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2))],
[1/((a+I*a*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(5/2)),x,8,-1/4*I*arctanh(sqrt(2)*sqrt(a)*sqrt(c+d*tan(e+f*x))/(sqrt(c-I*d)*sqrt(a+I*a*tan(e+f*x))))/(a^(5/2)*(c-I*d)^(5/2)*f*sqrt(2))+1/60*d*(15*c^4+80*I*c^3*d-182*c^2*d^2+1224*I*c*d^3+707*d^4)*sqrt(a+I*a*tan(e+f*x))/(a^3*(c-I*d)^2*(c+I*d)^5*f*sqrt(c+d*tan(e+f*x)))+1/20*(5*c^2+30*I*c*d-89*d^2)/(a^2*(I*c-d)^3*f*sqrt(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))+1/60*d*(15*c^3+85*I*c^2*d-221*c*d^2+361*I*d^3)*sqrt(a+I*a*tan(e+f*x))/(a^3*(c-I*d)*(c+I*d)^4*f*(c+d*tan(e+f*x))^(3/2))+(-1/5)/((I*c-d)*f*(a+I*a*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(3/2))+1/30*(5*I*c-21*d)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2))],

# Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^n with n symbolic
[(a+I*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n,x,3,-1/2*I*AppellF1(m,-n,1,1+m,-d*(1+I*tan(e+f*x))/(I*c-d),1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n/(f*m*((c+d*tan(e+f*x))/(c+I*d))^n)],
[(a+I*a*tan(e+f*x))^3*(c+d*tan(e+f*x))^n,x,4,a^3*(I*c-d*(5+2*n))*(c+d*tan(e+f*x))^(1+n)/(d^2*f*(1+n)*(2+n))+4*a^3*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/((I*c+d)*f*(1+n))-(a^3+I*a^3*tan(e+f*x))*(c+d*tan(e+f*x))^(1+n)/(d*f*(2+n))],
[(a+I*a*tan(e+f*x))^2*(c+d*tan(e+f*x))^n,x,3,-a^2*(c+d*tan(e+f*x))^(1+n)/(d*f*(1+n))+2*a^2*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/((I*c+d)*f*(1+n))],
[(a+I*a*tan(e+f*x))*(c+d*tan(e+f*x))^n,x,2,a*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/((I*c+d)*f*(1+n))],
[(c+d*tan(e+f*x))^n/(a+I*a*tan(e+f*x)),x,6,1/4*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/(a*(I*c+d)*f*(1+n))+1/4*(I*c-d+2*d*n)*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c+I*d))*(c+d*tan(e+f*x))^(1+n)/(a*(c+I*d)^2*f*(1+n))-1/2*(c+d*tan(e+f*x))^(1+n)/((I*c-d)*f*(a+I*a*tan(e+f*x)))],
[(c+d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^2,x,7,1/8*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/(a^2*(I*c+d)*f*(1+n))+1/8*(c^2+2*I*c*d*(1-n)-d^2*(1-4*n+2*n^2))*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c+I*d))*(c+d*tan(e+f*x))^(1+n)/(a^2*(I*c-d)^3*f*(1+n))+1/4*(I*c-d*(2-n))*(c+d*tan(e+f*x))^(1+n)/(a^2*(c+I*d)^2*f*(1+I*tan(e+f*x)))-1/4*(c+d*tan(e+f*x))^(1+n)/((I*c-d)*f*(a+I*a*tan(e+f*x))^2)],
[(c+d*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^3,x,8,1/16*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c-I*d))*(c+d*tan(e+f*x))^(1+n)/(a^3*(I*c+d)*f*(1+n))+1/48*(3*I*c^3-c^2*d*(9-6*n)-3*I*c*d^2*(3-6*n+2*n^2)+d^3*(3-20*n+18*n^2-4*n^3))*hypergeom([1,1+n],[2+n],(c+d*tan(e+f*x))/(c+I*d))*(c+d*tan(e+f*x))^(1+n)/(a^3*(c+I*d)^4*f*(1+n))-1/6*(c+d*tan(e+f*x))^(1+n)/((I*c-d)*f*(a+I*a*tan(e+f*x))^3)+1/24*(3*I*c-d*(7-2*n))*(c+d*tan(e+f*x))^(1+n)/(a*(c+I*d)^2*f*(a+I*a*tan(e+f*x))^2)+1/24*(3*I*c^2-3*c*d*(3-n)-I*d^2*(10-9*n+2*n^2))*(c+d*tan(e+f*x))^(1+n)/((c+I*d)^3*f*(a^3+I*a^3*tan(e+f*x)))],

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

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

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

# n > 0
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x)),x,4,(a^3*c-3*a*b^2*c-3*a^2*b*d+b^3*d)*x-(3*a^2*b*c-b^3*c+a^3*d-3*a*b^2*d)*log(cos(e+f*x))/f+b*(2*a*b*c+a^2*d-b^2*d)*tan(e+f*x)/f+1/2*(b*c+a*d)*(a+b*tan(e+f*x))^2/f+1/3*d*(a+b*tan(e+f*x))^3/f],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x)),x,3,(a^2*c-b^2*c-2*a*b*d)*x-(2*a*b*c+a^2*d-b^2*d)*log(cos(e+f*x))/f+b*(b*c+a*d)*tan(e+f*x)/f+1/2*d*(a+b*tan(e+f*x))^2/f],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x)),x,2,(a*c-b*d)*x-(b*c+a*d)*log(cos(e+f*x))/f+b*d*tan(e+f*x)/f],
[(c+d*tan(e+f*x))/(a+b*tan(e+f*x)),x,2,(a*c+b*d)*x/(a^2+b^2)+(b*c-a*d)*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)*f)],
[(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^2,x,3,(a^2*c-b^2*c+2*a*b*d)*x/(a^2+b^2)^2+(2*a*b*c-a^2*d+b^2*d)*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^2*f)+(-b*c+a*d)/((a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^3,x,4,(a^3*c-3*a*b^2*c+3*a^2*b*d-b^3*d)*x/(a^2+b^2)^3+(3*a^2*b*c-b^3*c-a^3*d+3*a*b^2*d)*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*f)+1/2*(-b*c+a*d)/((a^2+b^2)*f*(a+b*tan(e+f*x))^2)+(-2*a*b*c+a^2*d-b^2*d)/((a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^2,x,5,-(6*a^2*b*c*d-2*b^3*c*d-a^3*(c^2-d^2)+3*a*b^2*(c^2-d^2))*x-(2*a^3*c*d-6*a*b^2*c*d+3*a^2*b*(c^2-d^2)-b^3*(c^2-d^2))*log(cos(e+f*x))/f+2*b*(b*c+a*d)*(a*c-b*d)*tan(e+f*x)/f+1/2*(2*a*c*d+b*(c^2-d^2))*(a+b*tan(e+f*x))^2/f+2/3*c*d*(a+b*tan(e+f*x))^3/f+1/4*d^2*(a+b*tan(e+f*x))^4/(b*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^2,x,4,(a*c-b*c-a*d-b*d)*(a*c+b*c+a*d-b*d)*x-2*(b*c+a*d)*(a*c-b*d)*log(cos(e+f*x))/f+b*(2*a*c*d+b*(c^2-d^2))*tan(e+f*x)/f+c*d*(a+b*tan(e+f*x))^2/f+1/3*d^2*(a+b*tan(e+f*x))^3/(b*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^2,x,3,-(2*b*c*d-a*(c^2-d^2))*x-(2*a*c*d+b*(c^2-d^2))*log(cos(e+f*x))/f+d*(b*c+a*d)*tan(e+f*x)/f+1/2*b*(c+d*tan(e+f*x))^2/f],
[(c+d*tan(e+f*x))^2/(a+b*tan(e+f*x)),x,4,a*(b*c-a*d)^2*x/(b^2*(a^2+b^2))+d*(2*b*c-a*d)*x/b^2-d^2*log(cos(e+f*x))/(b*f)+(b*c-a*d)^2*log(a*cos(e+f*x)+b*sin(e+f*x))/(b*(a^2+b^2)*f)],
[(c+d*tan(e+f*x))^2/(a+b*tan(e+f*x))^2,x,3,-(b*(c-d)-a*(c+d))*(a*(c-d)+b*(c+d))*x/(a^2+b^2)^2+2*(b*c-a*d)*(a*c+b*d)*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^2*f)-(b*c-a*d)^2/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^2/(a+b*tan(e+f*x))^3,x,4,(6*a^2*b*c*d-2*b^3*c*d+a^3*(c^2-d^2)-3*a*b^2*(c^2-d^2))*x/(a^2+b^2)^3-(2*a^3*c*d-6*a*b^2*c*d-3*a^2*b*(c^2-d^2)+b^3*(c^2-d^2))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*f)-1/2*(b*c-a*d)^2/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)-2*(b*c-a*d)*(a*c+b*d)/((a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^3,x,6,-(a*c-b*d)*(8*a*b*c*d-a^2*(c^2-3*d^2)+b^2*(3*c^2-d^2))*x+(b*c+a*d)*(8*a*b*c*d+b^2*(c^2-3*d^2)-a^2*(3*c^2-d^2))*log(cos(e+f*x))/f+d*(2*a^3*c*d-6*a*b^2*c*d+3*a^2*b*(c^2-d^2)-b^3*(c^2-d^2))*tan(e+f*x)/f+1/2*(3*a^2*b*c-b^3*c+a^3*d-3*a*b^2*d)*(c+d*tan(e+f*x))^2/f+1/3*b*(3*a^2-b^2)*(c+d*tan(e+f*x))^3/f-1/20*b^2*(b*c-11*a*d)*(c+d*tan(e+f*x))^4/(d^2*f)+1/5*b^2*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^4/(d*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^3,x,5,-(b^2*c*(c^2-3*d^2)+2*a*b*d*(3*c^2-d^2)-a^2*(c^3-3*c*d^2))*x-(2*a*b*c*(c^2-3*d^2)-b^2*d*(3*c^2-d^2)+a^2*(3*c^2*d-d^3))*log(cos(e+f*x))/f+2*d*(b*c+a*d)*(a*c-b*d)*tan(e+f*x)/f+1/2*(2*a*b*c+a^2*d-b^2*d)*(c+d*tan(e+f*x))^2/f+2/3*a*b*(c+d*tan(e+f*x))^3/f+1/4*b^2*(c+d*tan(e+f*x))^4/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^3,x,4,-(b*d*(3*c^2-d^2)-a*(c^3-3*c*d^2))*x-(b*c^3+3*a*c^2*d-3*b*c*d^2-a*d^3)*log(cos(e+f*x))/f+d*(2*a*c*d+b*(c^2-d^2))*tan(e+f*x)/f+1/2*(b*c+a*d)*(c+d*tan(e+f*x))^2/f+1/3*b*(c+d*tan(e+f*x))^3/f],
[(c+d*tan(e+f*x))^3/(a+b*tan(e+f*x)),x,5,(a*c^3+3*b*c^2*d-3*a*c*d^2-b*d^3)*x/(a^2+b^2)+(b*c^3-3*a*c^2*d-3*b*c*d^2+a*d^3)*log(cos(e+f*x))/((a^2+b^2)*f)+(b*c-a*d)^3*log(a+b*tan(e+f*x))/(b^2*(a^2+b^2)*f)+d^2*(c+d*tan(e+f*x))/(b*f)],
[(c+d*tan(e+f*x))^3/(a+b*tan(e+f*x))^2,x,5,-(b^2*c*(c^2-3*d^2)-2*a*b*d*(3*c^2-d^2)-a^2*(c^3-3*c*d^2))*x/(a^2+b^2)^2+(2*a*b*c*(c^2-3*d^2)+b^2*d*(3*c^2-d^2)-a^2*(3*c^2*d-d^3))*log(cos(e+f*x))/((a^2+b^2)^2*f)+(b*c-a*d)^2*(2*a*b*c+a^2*d+3*b^2*d)*log(a+b*tan(e+f*x))/(b^2*(a^2+b^2)^2*f)-(b*c-a*d)^2*(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^3/(a+b*tan(e+f*x))^3,x,4,(a*c+b*d)*(8*a*b*c*d+a^2*(c^2-3*d^2)-b^2*(3*c^2-d^2))*x/(a^2+b^2)^3+(b*c-a*d)*(8*a*b*c*d-b^2*(c^2-3*d^2)+a^2*(3*c^2-d^2))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*f)-1/2*(b*c-a*d)^2*(4*a*b*c+a^2*d+5*b^2*d)/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))-1/2*(b*c-a*d)^2*(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)],

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

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

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^3,x,10,(I*a+b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f-(I*a-b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*b*(3*a^2-b^2)*sqrt(c+d*tan(e+f*x))/f-4/15*b^2*(b*c-6*a*d)*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+2/5*b^2*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2,x,9,-I*(a-I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+I*(a+I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+4*a*b*sqrt(c+d*tan(e+f*x))/f+2/3*b^2*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x)),x,8,-(I*a+b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+(I*a-b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*b*sqrt(c+d*tan(e+f*x))/f],
[sqrt(c+d*tan(e+f*x))/(a+b*tan(e+f*x)),x,11,arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((I*a+b)*f)-arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((I*a-b)*f)-2*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b)*sqrt(b*c-a*d)/((a^2+b^2)*f)],
[sqrt(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^2,x,12,-I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((a-I*b)^2*f)+I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((a+I*b)^2*f)-(4*a*b*c-3*a^2*d+b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b)/((a^2+b^2)^2*f*sqrt(b*c-a*d))-b*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^3,x,13,1/4*(40*a^3*b*c*d-24*a*b^3*c*d-15*a^4*d^2-6*a^2*b^2*(4*c^2-3*d^2)+b^4*(8*c^2+d^2))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b)/((a^2+b^2)^3*(b*c-a*d)^(3/2)*f)-arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((I*a+b)^3*f)+arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((I*a-b)^3*f)-1/2*b*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^2)-1/4*b*(8*a*b*c-7*a^2*d+b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2),x,11,(I*a+b)^3*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f-(I*a-b)^3*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(3*a^2*b*c-b^3*c+a^3*d-3*a*b^2*d)*sqrt(c+d*tan(e+f*x))/f+2/3*b*(3*a^2-b^2)*(c+d*tan(e+f*x))^(3/2)/f-4/35*b^2*(b*c-8*a*d)*(c+d*tan(e+f*x))^(5/2)/(d^2*f)+2/7*b^2*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2),x,10,-I*(a-I*b)^2*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+I*(a+I*b)^2*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(2*a*b*c+a^2*d-b^2*d)*sqrt(c+d*tan(e+f*x))/f+4/3*a*b*(c+d*tan(e+f*x))^(3/2)/f+2/5*b^2*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2),x,9,-(I*a+b)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(I*a-b)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(b*c+a*d)*sqrt(c+d*tan(e+f*x))/f+2/3*b*(c+d*tan(e+f*x))^(3/2)/f],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x)),x,11,(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*f)-(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*f)-2*(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*f*sqrt(b))],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x))^2,x,12,-I*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f)+I*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f)-(4*a*b*c-a^2*d+3*b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b*c-a*d)/((a^2+b^2)^2*f*sqrt(b))-(b*c-a*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x))^3,x,13,-(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)^3*f)+(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)^3*f)+1/4*(24*a^3*b*c*d-40*a*b^3*c*d-3*a^4*d^2-2*a^2*b^2*(12*c^2-13*d^2)+b^4*(8*c^2-3*d^2))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^3*f*sqrt(b)*sqrt(b*c-a*d))-1/2*(b*c-a*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^2)-1/4*(8*a*b*c-3*a^2*d+5*b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^(5/2),x,12,(I*a+b)^3*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f-(I*a-b)^3*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(2*a^3*c*d-6*a*b^2*c*d+3*a^2*b*(c^2-d^2)-b^3*(c^2-d^2))*sqrt(c+d*tan(e+f*x))/f+2/3*(3*a^2*b*c-b^3*c+a^3*d-3*a*b^2*d)*(c+d*tan(e+f*x))^(3/2)/f+2/5*b*(3*a^2-b^2)*(c+d*tan(e+f*x))^(5/2)/f-4/63*b^2*(b*c-10*a*d)*(c+d*tan(e+f*x))^(7/2)/(d^2*f)+2/9*b^2*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2),x,11,-I*(a-I*b)^2*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+I*(a+I*b)^2*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+4*(b*c+a*d)*(a*c-b*d)*sqrt(c+d*tan(e+f*x))/f+2/3*(2*a*b*c+a^2*d-b^2*d)*(c+d*tan(e+f*x))^(3/2)/f+4/5*a*b*(c+d*tan(e+f*x))^(5/2)/f+2/7*b^2*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2),x,10,-(I*a+b)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(I*a-b)*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(2*a*c*d+b*(c^2-d^2))*sqrt(c+d*tan(e+f*x))/f+2/3*(b*c+a*d)*(c+d*tan(e+f*x))^(3/2)/f+2/5*b*(c+d*tan(e+f*x))^(5/2)/f],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x)),x,12,(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*f)-(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*f)-2*(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(3/2)*(a^2+b^2)*f)+2*d^2*sqrt(c+d*tan(e+f*x))/(b*f)],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x))^2,x,12,-I*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f)+I*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f)-(b*c-a*d)^(3/2)*(4*a*b*c+a^2*d+5*b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(3/2)*(a^2+b^2)^2*f)-(b*c-a*d)^2*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x))^3,x,13,-(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)^3*f)+(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)^3*f)+1/4*(8*a^3*b*c*d-56*a*b^3*c*d+a^4*d^2+b^4*(8*c^2-15*d^2)-6*a^2*b^2*(4*c^2-3*d^2))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(b^(3/2)*(a^2+b^2)^3*f)-1/2*(b*c-a*d)^2*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)-1/4*(b*c-a*d)*(8*a*b*c+a^2*d+9*b^2*d)*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))],

# n < 0
[(a+b*tan(e+f*x))^4/sqrt(c+d*tan(e+f*x)),x,10,-I*(a-I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+I*(a+I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))-2/15*b^2*(40*a*b*c*d-87*a^2*d^2-b^2*(8*c^2-15*d^2))*sqrt(c+d*tan(e+f*x))/(d^3*f)-4/15*b^3*(2*b*c-7*a*d)*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^2*f)+2/5*b^2*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d*f)],
[(a+b*tan(e+f*x))^3/sqrt(c+d*tan(e+f*x)),x,9,(I*a+b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))-(I*a-b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))-4/3*b^2*(b*c-4*a*d)*sqrt(c+d*tan(e+f*x))/(d^2*f)+2/3*b^2*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))/(d*f)],
[(a+b*tan(e+f*x))^2/sqrt(c+d*tan(e+f*x)),x,8,-I*(a-I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+I*(a+I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))+2*b^2*sqrt(c+d*tan(e+f*x))/(d*f)],
[(a+b*tan(e+f*x))/sqrt(c+d*tan(e+f*x)),x,7,-(I*a+b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+(I*a-b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))],
[1/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))),x,11,arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*f*sqrt(c-I*d))-arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*f*sqrt(c+I*d))-2*b^(3/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*f*sqrt(b*c-a*d))],
[1/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2),x,12,-b^(3/2)*(4*a*b*c-5*a^2*d-b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^2*(b*c-a*d)^(3/2)*f)-I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f*sqrt(c-I*d))+I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f*sqrt(c+I*d))-b^2*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^4/(c+d*tan(e+f*x))^(3/2),x,10,-I*(a-I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+I*(a+I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)-2/3*b*(15*a^2*b*c*d^2-6*a^3*d^3-12*a*b^2*d*(2*c^2+d^2)+b^3*(8*c^3+5*c*d^2))*sqrt(c+d*tan(e+f*x))/(d^3*(c^2+d^2)*f)-2/3*b^2*(3*a*d*(2*b*c-a*d)-b^2*(4*c^2+d^2))*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^2*(c^2+d^2)*f)-2*(b*c-a*d)^2*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3/(c+d*tan(e+f*x))^(3/2),x,9,(I*a+b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)-(I*a-b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)-2*b*(a*d*(2*b*c-a*d)-b^2*(2*c^2+d^2))*sqrt(c+d*tan(e+f*x))/(d^2*(c^2+d^2)*f)-2*(b*c-a*d)^2*(a+b*tan(e+f*x))/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^2/(c+d*tan(e+f*x))^(3/2),x,8,-I*(a-I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+I*(a+I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)-2*(b*c-a*d)^2/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))/(c+d*tan(e+f*x))^(3/2),x,8,-(I*a+b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+(I*a-b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)+2*(b*c-a*d)/((c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)),x,12,arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*(c-I*d)^(3/2)*f)-arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*(c+I*d)^(3/2)*f)-2*b^(5/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*(b*c-a*d)^(3/2)*f)+2*d^2/((b*c-a*d)*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)),x,13,-I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*(c-I*d)^(3/2)*f)+I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*(c+I*d)^(3/2)*f)-b^(5/2)*(4*a*b*c-7*a^2*d-3*b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^2*(b*c-a*d)^(5/2)*f)-d*(2*a^2*d^2+b^2*(c^2+3*d^2))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))-b^2/((a^2+b^2)*(b*c-a*d)*f*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^4/(c+d*tan(e+f*x))^(5/2),x,10,-I*(a-I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+I*(a+I*b)^4*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)+4/3*(b*c-a*d)^3*(2*b*c^2+3*a*c*d+5*b*d^2)/(d^3*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*b^2*(a*d*(2*b*c-a*d)-b^2*(4*c^2+3*d^2))*sqrt(c+d*tan(e+f*x))/(d^3*(c^2+d^2)*f)-2/3*(b*c-a*d)^2*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^3/(c+d*tan(e+f*x))^(5/2),x,9,(I*a+b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)-(I*a-b)^3*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)-4/3*(b*c-a*d)^2*(3*a*c*d+b*(c^2+4*d^2))/(d^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(b*c-a*d)^2*(a+b*tan(e+f*x))/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^2/(c+d*tan(e+f*x))^(5/2),x,9,-I*(a-I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+I*(a+I*b)^2*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)+4*(b*c-a*d)*(a*c+b*d)/((c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(b*c-a*d)^2/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))/(c+d*tan(e+f*x))^(5/2),x,9,-(I*a+b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+(I*a-b)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)-2*(2*a*c*d-b*(c^2-d^2))/((c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*(b*c-a*d)/((c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)),x,13,arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*(c-I*d)^(5/2)*f)-arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*(c+I*d)^(5/2)*f)-2*b^(7/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*(b*c-a*d)^(5/2)*f)-2*d^2*(2*a*c*d-b*(3*c^2+d^2))/((b*c-a*d)^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*d^2/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)),x,14,-I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*(c-I*d)^(5/2)*f)+I*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*(c+I*d)^(5/2)*f)-b^(7/2)*(4*a*b*c-9*a^2*d-5*b^2*d)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^2*(b*c-a*d)^(7/2)*f)+d*(4*a^3*c*d^3+4*a*b^2*c*d^3-4*a^2*b*d^2*(2*c^2+d^2)-b^3*(c^4+10*c^2*d^2+5*d^4))/((a^2+b^2)*(b*c-a*d)^3*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-1/3*d*(2*a^2*d^2+b^2*(3*c^2+5*d^2))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))-b^2/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))],

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

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(5/2),x,14,1/4*(10*a*b*c*d+15*a^2*d^2-b^2*(c^2+8*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(b)/(d^(3/2)*f)-I*(a-I*b)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/f+I*(a+I*b)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/f-1/4*b*(b*c-9*a*d)*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)+1/2*b^2*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2),x,13,-I*(a-I*b)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/f+I*(a+I*b)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/f+(b*c+3*a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(b)/(f*sqrt(d))+b*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f],
[sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x)),x,11,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)*sqrt(c-I*d)/f+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)*sqrt(c+I*d)/f+2*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(b)*sqrt(d)/f],
[sqrt(c+d*tan(e+f*x))/sqrt(a+b*tan(e+f*x)),x,7,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/(f*sqrt(a-I*b))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/(f*sqrt(a+I*b))],
[sqrt(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^(3/2),x,8,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/((a-I*b)^(3/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/((a+I*b)^(3/2)*f)-2*b*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))/(a+b*tan(e+f*x))^(5/2),x,9,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/((a-I*b)^(5/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/((a+I*b)^(5/2)*f)-2/3*b*(6*a*b*c-5*a^2*d+b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))-2/3*b*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2),x,14,-I*(a-I*b)^(3/2)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/f+I*(a+I*b)^(3/2)*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/f+1/4*(18*a*b*c*d+3*a^2*d^2+b^2*(3*c^2-8*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(b)*sqrt(d))+1/4*(3*b*c+5*a*d)*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f+1/2*b*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/f],
[sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2),x,13,-I*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/f+I*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/f+(3*b*c+a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(d)/(f*sqrt(b))+d*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/f],
[(c+d*tan(e+f*x))^(3/2)/sqrt(a+b*tan(e+f*x)),x,12,-I*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a-I*b))+I*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a+I*b))+2*d^(3/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(b))],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x))^(3/2),x,8,-I*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*f)+I*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*f)-2*(b*c-a*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x))^(5/2),x,9,-I*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*f)+I*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*f)-4/3*(3*a*b*c-a^2*d+2*b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*f*sqrt(a+b*tan(e+f*x)))-2/3*(b*c-a*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(3/2)/(a+b*tan(e+f*x))^(7/2),x,10,-I*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(7/2)*f)+I*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(7/2)*f)+2/15*(50*a^3*b*c*d-70*a*b^3*c*d-8*a^4*d^2-a^2*b^2*(45*c^2-49*d^2)+3*b^4*(5*c^2-d^2))*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^3*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))-2/5*(b*c-a*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*f*(a+b*tan(e+f*x))^(5/2))-4/15*(5*a*b*c-2*a^2*d+3*b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*f*(a+b*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(5/2),x,15,-I*(a-I*b)^(3/2)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/f+I*(a+I*b)^(3/2)*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/f+1/8*(15*a^2*b*c*d^2-a^3*d^3+3*a*b^2*d*(15*c^2-8*d^2)+5*b^3*(c^3-8*c*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*f*sqrt(d))+1/8*(14*a*b*c*d-a^2*d^2+b^2*(11*c^2-8*d^2))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*f)+1/12*d*(13*b*c-a*d)*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2)/(b*f)+1/3*d^2*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(5/2)/(b*f)],
[sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2),x,14,-I*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/f+I*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/f+1/4*(10*a*b*c*d-a^2*d^2+b^2*(15*c^2-8*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(d)/(b^(3/2)*f)+1/4*d*(9*b*c-a*d)*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*f)+1/2*d^2*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2)/(b*f)],
[(c+d*tan(e+f*x))^(5/2)/sqrt(a+b*tan(e+f*x)),x,13,d^(3/2)*(5*b*c-a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*f)-I*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a-I*b))+I*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a+I*b))+d^2*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*f)],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x))^(3/2),x,13,-I*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*f)+I*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*f)+2*d^(5/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*f)-2*(b*c-a*d)^2*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x))^(5/2),x,9,-I*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*f)+I*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*f)-2/3*(b*c-a*d)*(6*a*b*c+a^2*d+7*b^2*d)*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*f*sqrt(a+b*tan(e+f*x)))-2/3*(b*c-a*d)^2*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(5/2)/(a+b*tan(e+f*x))^(7/2),x,10,-I*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(7/2)*f)+I*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(7/2)*f)+2/15*(20*a^3*b*c*d-100*a*b^3*c*d+2*a^4*d^2+b^4*(15*c^2-23*d^2)-3*a^2*b^2*(15*c^2-13*d^2))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^3*f*sqrt(a+b*tan(e+f*x)))-2/5*(b*c-a*d)^2*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(5/2))-2/15*(b*c-a*d)*(10*a*b*c+a^2*d+11*b^2*d)*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*f*(a+b*tan(e+f*x))^(3/2))],

# n < 0
[(a+b*tan(e+f*x))^(5/2)/sqrt(c+d*tan(e+f*x)),x,13,-b^(3/2)*(b*c-5*a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-I*(a-I*b)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(c-I*d))+I*(a+I*b)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(c+I*d))+b^2*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d*f)],
[(a+b*tan(e+f*x))^(3/2)/sqrt(c+d*tan(e+f*x)),x,12,-I*(a-I*b)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(c-I*d))+I*(a+I*b)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(c+I*d))+2*b^(3/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(d))],
[sqrt(a+b*tan(e+f*x))/sqrt(c+d*tan(e+f*x)),x,7,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/(f*sqrt(c-I*d))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/(f*sqrt(c+I*d))],
[1/(sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))),x,7,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a-I*b)*sqrt(c-I*d))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a+I*b)*sqrt(c+I*d))],
[1/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2)),x,8,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*f*sqrt(c-I*d))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*f*sqrt(c+I*d))-2*b^2*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))],
[1/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(5/2)),x,9,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*f*sqrt(c-I*d))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*f*sqrt(c+I*d))-4/3*b^2*(3*a*b*c-4*a^2*d-b^2*d)*sqrt(c+d*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^2*f*sqrt(a+b*tan(e+f*x)))-2/3*b^2*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(7/2)/(c+d*tan(e+f*x))^(3/2),x,14,-I*(a-I*b)^(7/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(3/2)*f)+I*(a+I*b)^(7/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(3/2)*f)-b^(5/2)*(3*b*c-7*a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(d^(5/2)*f)-b*(2*a*d*(2*b*c-a*d)-b^2*(3*c^2+d^2))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d^2*(c^2+d^2)*f)-2*(b*c-a*d)^2*(a+b*tan(e+f*x))^(3/2)/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^(5/2)/(c+d*tan(e+f*x))^(3/2),x,13,-I*(a-I*b)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(3/2)*f)+I*(a+I*b)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(3/2)*f)+2*b^(5/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(d^(3/2)*f)-2*(b*c-a*d)^2*sqrt(a+b*tan(e+f*x))/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^(3/2)/(c+d*tan(e+f*x))^(3/2),x,8,-I*(a-I*b)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(3/2)*f)+I*(a+I*b)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(3/2)*f)+2*(b*c-a*d)*sqrt(a+b*tan(e+f*x))/((c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[sqrt(a+b*tan(e+f*x))/(c+d*tan(e+f*x))^(3/2),x,8,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/((c-I*d)^(3/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/((c+I*d)^(3/2)*f)-2*d*sqrt(a+b*tan(e+f*x))/((c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/(sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)),x,8,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(3/2)*f*sqrt(a-I*b))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(3/2)*f*sqrt(a+I*b))+2*d^2*sqrt(a+b*tan(e+f*x))/((b*c-a*d)*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2)),x,9,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*(c-I*d)^(3/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*(c+I*d)^(3/2)*f)-2*b^2/((a^2+b^2)*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x)))-2*d*(a^2*d^2+b^2*(c^2+2*d^2))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[1/((a+b*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(3/2)),x,10,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*(c-I*d)^(3/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*(c+I*d)^(3/2)*f)-4/3*b^2*(3*a*b*c-5*a^2*d-2*b^2*d)/((a^2+b^2)^2*(b*c-a*d)^2*f*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x)))+2/3*d*(3*a^4*d^3-6*a*b^3*c*(c^2+d^2)+b^4*d*(5*c^2+8*d^2)+a^2*b^2*d*(11*c^2+17*d^2))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^3*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))-2/3*b^2/((a^2+b^2)*(b*c-a*d)*f*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(9/2)/(c+d*tan(e+f*x))^(5/2),x,15,-I*(a-I*b)^(9/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(5/2)*f)+I*(a+I*b)^(9/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(5/2)*f)-b^(7/2)*(5*b*c-9*a*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(d^(7/2)*f)+b*(4*a^3*c*d^3-4*a^2*b*d^2*(c^2-2*d^2)-4*a*b^2*c*d*(c^2+4*d^2)+b^3*(5*c^4+10*c^2*d^2+d^4))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(d^3*(c^2+d^2)^2*f)-2/3*(b*c-a*d)^2*(5*b*c^2+6*a*c*d+11*b*d^2)*(a+b*tan(e+f*x))^(3/2)/(d^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(b*c-a*d)^2*(a+b*tan(e+f*x))^(5/2)/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(7/2)/(c+d*tan(e+f*x))^(5/2),x,14,-I*(a-I*b)^(7/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(5/2)*f)+I*(a+I*b)^(7/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(5/2)*f)+2*b^(7/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(d^(5/2)*f)-2*(b*c-a*d)^2*(2*a*c*d+b*(c^2+3*d^2))*sqrt(a+b*tan(e+f*x))/(d^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(b*c-a*d)^2*(a+b*tan(e+f*x))^(3/2)/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(5/2)/(c+d*tan(e+f*x))^(5/2),x,9,-I*(a-I*b)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(5/2)*f)+I*(a+I*b)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(5/2)*f)+2/3*(b*c-a*d)*(6*a*c*d+b*(c^2+7*d^2))*sqrt(a+b*tan(e+f*x))/(d*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(b*c-a*d)^2*sqrt(a+b*tan(e+f*x))/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(3/2)/(c+d*tan(e+f*x))^(5/2),x,9,-I*(a-I*b)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(5/2)*f)+I*(a+I*b)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(5/2)*f)+4/3*(b*c^2-3*a*c*d-2*b*d^2)*sqrt(a+b*tan(e+f*x))/((c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*(b*c-a*d)*sqrt(a+b*tan(e+f*x))/((c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[sqrt(a+b*tan(e+f*x))/(c+d*tan(e+f*x))^(5/2),x,9,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/((c-I*d)^(5/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/((c+I*d)^(5/2)*f)+2/3*d*(6*a*c*d-b*(5*c^2-d^2))*sqrt(a+b*tan(e+f*x))/((b*c-a*d)*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*d*sqrt(a+b*tan(e+f*x))/((c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/(sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)),x,9,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((c-I*d)^(5/2)*f*sqrt(a-I*b))+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((c+I*d)^(5/2)*f*sqrt(a+I*b))-4/3*d^2*(3*a*c*d-b*(4*c^2+d^2))*sqrt(a+b*tan(e+f*x))/((b*c-a*d)^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*d^2*sqrt(a+b*tan(e+f*x))/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(5/2)),x,10,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*(c-I*d)^(5/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*(c+I*d)^(5/2)*f)+2/3*(6*a^3*c*d^4+6*a*b^2*c*d^4-a^2*b*d^3*(11*c^2+5*d^2)-b^3*(3*c^4*d+17*c^2*d^3+8*d^5))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)^3*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2*b^2/((a^2+b^2)*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))-2/3*d*(a^2*d^2+b^2*(3*c^2+4*d^2))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[1/((a+b*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(5/2)),x,11,-I*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*(c-I*d)^(5/2)*f)+I*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*(c+I*d)^(5/2)*f)-4/3*d*(3*a^5*c*d^4+6*a^3*b^2*c*d^4-a^4*b*d^3*(7*c^2+4*d^2)+3*a*b^4*c*(c^4+2*c^2*d^2+2*d^4)-b^5*d*(4*c^4+15*c^2*d^2+8*d^4)-a^2*b^3*d*(7*c^4+28*c^2*d^2+15*d^4))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^4*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-4*b^2*(a*b*c-2*a^2*d-b^2*d)/((a^2+b^2)^2*(b*c-a*d)^2*f*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2))+2/3*d*(a^4*d^3-6*a*b^3*c*(c^2+d^2)+b^4*d*(7*c^2+8*d^2)+a^2*b^2*d*(13*c^2+15*d^2))*sqrt(a+b*tan(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^3*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))-2/3*b^2/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2))],

# Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n with m and/or n symbolic
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n,x,7,1/2*AppellF1(1+m,-n,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^n/((I*a+b)*f*(1+m)*(b*(c+d*tan(e+f*x))/(b*c-a*d))^n)-1/2*AppellF1(1+m,-n,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^n/((I*a-b)*f*(1+m)*(b*(c+d*tan(e+f*x))/(b*c-a*d))^n)],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^3,x,7,d^2*(3*b*c-a*d)*(a+b*tan(e+f*x))^(1+m)/(b^2*f*(1+m))+1/2*(I*c+d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((a-I*b)*f*(1+m))-1/2*(I*c-d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*f*(1+m))+d^3*(a+b*tan(e+f*x))^(2+m)/(b^2*f*(2+m)),-d^2*(a*d-b*c*(5+2*m))*(a+b*tan(e+f*x))^(1+m)/(b^2*f*(1+m)*(2+m))+1/2*(c-I*d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))-1/2*(I*c-d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*f*(1+m))+d^2*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))/(b*f*(2+m))],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^2,x,6,d^2*(a+b*tan(e+f*x))^(1+m)/(b*f*(1+m))+1/2*(c-I*d)^2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))-1/2*(c+I*d)^2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*f*(1+m))],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x)),x,5,1/2*(c-I*d)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))+1/2*(I*c-d)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*f*(1+m))],
[(a+b*tan(e+f*x))^m,x,5,1/2*b*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-sqrt(-b^2)))*(a+b*tan(e+f*x))^(1+m)/(f*(1+m)*(a-sqrt(-b^2))*sqrt(-b^2))-1/2*b*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+sqrt(-b^2)))*(a+b*tan(e+f*x))^(1+m)/(f*(1+m)*sqrt(-b^2)*(a+sqrt(-b^2)))],
[(a+b*tan(e+f*x))^m/(c+d*tan(e+f*x)),x,8,1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(c-I*d)*f*(1+m))-1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*(c+I*d)*f*(1+m))+d^2*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(1+m))],
[(a+b*tan(e+f*x))^m/(c+d*tan(e+f*x))^2,x,9,1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(c-I*d)^2*f*(1+m))-1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*(c+I*d)^2*f*(1+m))-d^2*(2*a*c*d-b*(c^2*(2-m)-d^2*m))*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^2*(c^2+d^2)^2*f*(1+m))+d^2*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^m/(c+d*tan(e+f*x))^3,x,10,1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(c-I*d)^3*f*(1+m))+1/2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*(I*c-d)^3*f*(1+m))+1/2*d^2*(2*a^2*d^2*(3*c^2-d^2)-4*a*b*c*d*(c^2*(3-m)-d^2*(1+m))-b^2*(d^4*(1-m)*m+2*c^2*d^2*(1+3*m-m^2)-c^4*(6-5*m+m^2)))*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^3*(c^2+d^2)^3*f*(1+m))+1/2*d^2*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)-1/2*d^2*(4*a*c*d-b*(d^2*(1-m)+c^2*(5-m)))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^2*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^(3/2),x,7,1/2*(b*c-a*d)*AppellF1(1+m,-3/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(1+m)/(b*(I*a+b)*f*(1+m)*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d)))-1/2*(b*c-a*d)*AppellF1(1+m,-3/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*b*f*(1+m)*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d)))],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^m,x,7,1/2*AppellF1(1+m,-1/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m)*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d)))-1/2*AppellF1(1+m,-1/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*f*(1+m)*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d)))],
[(a+b*tan(e+f*x))^m/sqrt(c+d*tan(e+f*x)),x,7,1/2*AppellF1(1+m,1/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m)*sqrt(c+d*tan(e+f*x)))-1/2*AppellF1(1+m,1/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*f*(1+m)*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^m/(c+d*tan(e+f*x))^(3/2),x,7,1/2*b*AppellF1(1+m,3/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(b*c-a*d)*f*(1+m)*sqrt(c+d*tan(e+f*x)))-1/2*b*AppellF1(1+m,3/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*(b*c-a*d)*f*(1+m)*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^m/(c+d*tan(e+f*x))^(5/2),x,7,1/2*b^2*AppellF1(1+m,5/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(b*c-a*d)^2*f*(1+m)*sqrt(c+d*tan(e+f*x)))-1/2*b^2*AppellF1(1+m,5/2,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*sqrt(b*(c+d*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*(b*c-a*d)^2*f*(1+m)*sqrt(c+d*tan(e+f*x)))],

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

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

# Integrands of the form (a+I a Tan[e+f x])^m (c (d Tan[e+f x])^p)^n with n and p symbolic
[(c*(d*tan(e+f*x))^p)^n*(a+I*a*tan(e+f*x))^m,x,4,AppellF1(1+n*p,1-m,1,2+n*p,-I*tan(e+f*x),I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n*(a+I*a*tan(e+f*x))^m/(f*(1+n*p)*(1+I*tan(e+f*x))^m)],
[(c*(d*tan(e+f*x))^p)^n*(a+I*a*tan(e+f*x))^3,x,8,-3*a^3*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+4*a^3*hypergeom([1,1+n*p],[2+n*p],I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))-I*a^3*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n*(a+I*a*tan(e+f*x))^2,x,5,-a^2*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+2*a^2*hypergeom([1,1+n*p],[2+n*p],I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))],
[(c*(d*tan(e+f*x))^p)^n*(a+I*a*tan(e+f*x)),x,4,a*hypergeom([1,1+n*p],[2+n*p],I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))],
[(c*(d*tan(e+f*x))^p)^n/(a+I*a*tan(e+f*x)),x,8,hypergeom([2,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a*f*(1+n*p))-I*hypergeom([2,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(a*f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n/(a+I*a*tan(e+f*x))^2,x,8,1/8*(1-4*n*p+2*n^2*p^2)*hypergeom([1,1+n*p],[2+n*p],-I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a^2*f*(1+n*p))+1/8*hypergeom([1,1+n*p],[2+n*p],I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a^2*f*(1+n*p))+1/4*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a^2*f*(1+I*tan(e+f*x))^2)+1/4*(2-n*p)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a^2*f*(1+I*tan(e+f*x)))],

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

# Integrands of the form (a+b Tan[e+f x])^m (c (d Tan[e+f x])^p)^n with n and p symbolic
[(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x))^m,x,8,1/2*AppellF1(1+n*p,-m,1,2+n*p,-b*tan(e+f*x)/a,-I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x))^m/(f*(1+n*p)*(1+b*tan(e+f*x)/a)^m)+1/2*AppellF1(1+n*p,-m,1,2+n*p,-b*tan(e+f*x)/a,I*tan(e+f*x))*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x))^m/(f*(1+n*p)*(1+b*tan(e+f*x)/a)^m)],
[(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x))^3,x,7,3*a*b^2*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+a*(a^2-3*b^2)*hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+b^3*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(f*(2+n*p))+b*(3*a^2-b^2)*hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x))^2,x,7,b^2*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+(a^2-b^2)*hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+2*a*b*hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n*(a+b*tan(e+f*x)),x,5,a*hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(f*(1+n*p))+b*hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/(f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n/(a+b*tan(e+f*x)),x,8,a*hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/((a^2+b^2)*f*(1+n*p))+b^2*hypergeom([1,1+n*p],[2+n*p],-b*tan(e+f*x)/a)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a*(a^2+b^2)*f*(1+n*p))-b*hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/((a^2+b^2)*f*(2+n*p))],
[(c*(d*tan(e+f*x))^p)^n/(a+b*tan(e+f*x))^2,x,9,(a^2-b^2)*hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/((a^2+b^2)^2*f*(1+n*p))+2*b^2*hypergeom([1,1+n*p],[2+n*p],-b*tan(e+f*x)/a)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/((a^2+b^2)^2*f*(1+n*p))+b^2*hypergeom([2,1+n*p],[2+n*p],-b*tan(e+f*x)/a)*tan(e+f*x)*(c*(d*tan(e+f*x))^p)^n/(a^2*(a^2+b^2)*f*(1+n*p))-2*a*b*hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(c*(d*tan(e+f*x))^p)^n/((a^2+b^2)^2*f*(2+n*p))]]:
