# Maple integration test file: "4 Trig functions\4.7 Miscellaneous\4.7.5 x^m trig(a+b log(c x^n))^p.txt"

lst:=[

# Problems for integrands of the form (e x)^m Trig[d (a+b Log[c x^n])]^p

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

# Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p
[x^2*sin(a+b*log(c*x^n)),x,1,-b*n*x^3*cos(a+b*log(c*x^n))/(9+b^2*n^2)+3*x^3*sin(a+b*log(c*x^n))/(9+b^2*n^2)],
[x*sin(a+b*log(c*x^n)),x,1,-b*n*x^2*cos(a+b*log(c*x^n))/(4+b^2*n^2)+2*x^2*sin(a+b*log(c*x^n))/(4+b^2*n^2)],
[sin(a+b*log(c*x^n)),x,1,-b*n*x*cos(a+b*log(c*x^n))/(1+b^2*n^2)+x*sin(a+b*log(c*x^n))/(1+b^2*n^2)],
[sin(a+b*log(c*x^n))/x,x,2,-cos(a+b*log(c*x^n))/(b*n)],
[sin(a+b*log(c*x^n))/x^2,x,1,-b*n*cos(a+b*log(c*x^n))/((1+b^2*n^2)*x)-sin(a+b*log(c*x^n))/((1+b^2*n^2)*x)],
[sin(a+b*log(c*x^n))/x^3,x,1,-b*n*cos(a+b*log(c*x^n))/((4+b^2*n^2)*x^2)-2*sin(a+b*log(c*x^n))/((4+b^2*n^2)*x^2)],
[x^2*sin(a+b*log(c*x^n))^2,x,2,2/3*b^2*n^2*x^3/(9+4*b^2*n^2)-2*b*n*x^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(9+4*b^2*n^2)+3*x^3*sin(a+b*log(c*x^n))^2/(9+4*b^2*n^2)],
[x*sin(a+b*log(c*x^n))^2,x,2,1/4*b^2*n^2*x^2/(1+b^2*n^2)-1/2*b*n*x^2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+b^2*n^2)+1/2*x^2*sin(a+b*log(c*x^n))^2/(1+b^2*n^2)],
[sin(a+b*log(c*x^n))^2,x,2,2*b^2*n^2*x/(1+4*b^2*n^2)-2*b*n*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+4*b^2*n^2)+x*sin(a+b*log(c*x^n))^2/(1+4*b^2*n^2)],
[sin(a+b*log(c*x^n))^2/x,x,3,1/2*log(x)-1/2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(b*n)],
[sin(a+b*log(c*x^n))^2/x^2,x,2,-2*b^2*n^2/((1+4*b^2*n^2)*x)-2*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+4*b^2*n^2)*x)-sin(a+b*log(c*x^n))^2/((1+4*b^2*n^2)*x)],
[sin(a+b*log(c*x^n))^2/x^3,x,2,-1/4*b^2*n^2/((1+b^2*n^2)*x^2)-1/2*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+b^2*n^2)*x^2)-1/2*sin(a+b*log(c*x^n))^2/((1+b^2*n^2)*x^2)],
[x^2*sin(a+b*log(c*x^n))^3,x,2,-2/3*b^3*n^3*x^3*cos(a+b*log(c*x^n))/(9+10*b^2*n^2+b^4*n^4)+2*b^2*n^2*x^3*sin(a+b*log(c*x^n))/(9+10*b^2*n^2+b^4*n^4)-1/3*b*n*x^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^2/(1+b^2*n^2)+1/3*x^3*sin(a+b*log(c*x^n))^3/(1+b^2*n^2)],
[x*sin(a+b*log(c*x^n))^3,x,2,-6*b^3*n^3*x^2*cos(a+b*log(c*x^n))/(16+40*b^2*n^2+9*b^4*n^4)+12*b^2*n^2*x^2*sin(a+b*log(c*x^n))/(16+40*b^2*n^2+9*b^4*n^4)-3*b*n*x^2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^2/(4+9*b^2*n^2)+2*x^2*sin(a+b*log(c*x^n))^3/(4+9*b^2*n^2)],
[sin(a+b*log(c*x^n))^3,x,2,-6*b^3*n^3*x*cos(a+b*log(c*x^n))/(1+10*b^2*n^2+9*b^4*n^4)+6*b^2*n^2*x*sin(a+b*log(c*x^n))/(1+10*b^2*n^2+9*b^4*n^4)-3*b*n*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^2/(1+9*b^2*n^2)+x*sin(a+b*log(c*x^n))^3/(1+9*b^2*n^2)],
[sin(a+b*log(c*x^n))^3/x,x,3,-cos(a+b*log(c*x^n))/(b*n)+1/3*cos(a+b*log(c*x^n))^3/(b*n)],
[sin(a+b*log(c*x^n))^3/x^2,x,2,-6*b^3*n^3*cos(a+b*log(c*x^n))/((1+10*b^2*n^2+9*b^4*n^4)*x)-6*b^2*n^2*sin(a+b*log(c*x^n))/((1+10*b^2*n^2+9*b^4*n^4)*x)-3*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^2/((1+9*b^2*n^2)*x)-sin(a+b*log(c*x^n))^3/((1+9*b^2*n^2)*x)],
[sin(a+b*log(c*x^n))^3/x^3,x,2,-6*b^3*n^3*cos(a+b*log(c*x^n))/((16+40*b^2*n^2+9*b^4*n^4)*x^2)-12*b^2*n^2*sin(a+b*log(c*x^n))/((16+40*b^2*n^2+9*b^4*n^4)*x^2)-3*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^2/((4+9*b^2*n^2)*x^2)-2*sin(a+b*log(c*x^n))^3/((4+9*b^2*n^2)*x^2)],
[x^2*sin(a+b*log(c*x^n))^4,x,3,8*b^4*n^4*x^3/(81+180*b^2*n^2+64*b^4*n^4)-24*b^3*n^3*x^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(81+180*b^2*n^2+64*b^4*n^4)+36*b^2*n^2*x^3*sin(a+b*log(c*x^n))^2/(81+180*b^2*n^2+64*b^4*n^4)-4*b*n*x^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/(9+16*b^2*n^2)+3*x^3*sin(a+b*log(c*x^n))^4/(9+16*b^2*n^2)],
[x*sin(a+b*log(c*x^n))^4,x,3,3/4*b^4*n^4*x^2/(1+5*b^2*n^2+4*b^4*n^4)-3/2*b^3*n^3*x^2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+5*b^2*n^2+4*b^4*n^4)+3/2*b^2*n^2*x^2*sin(a+b*log(c*x^n))^2/(1+5*b^2*n^2+4*b^4*n^4)-b*n*x^2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/(1+4*b^2*n^2)+1/2*x^2*sin(a+b*log(c*x^n))^4/(1+4*b^2*n^2)],
[sin(a+b*log(c*x^n))^4,x,3,24*b^4*n^4*x/(1+20*b^2*n^2+64*b^4*n^4)-24*b^3*n^3*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+20*b^2*n^2+64*b^4*n^4)+12*b^2*n^2*x*sin(a+b*log(c*x^n))^2/(1+20*b^2*n^2+64*b^4*n^4)-4*b*n*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/(1+16*b^2*n^2)+x*sin(a+b*log(c*x^n))^4/(1+16*b^2*n^2)],
[sin(a+b*log(c*x^n))^4/x,x,4,3/8*log(x)-3/8*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(b*n)-1/4*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/(b*n)],
[sin(a+b*log(c*x^n))^4/x^2,x,3,-24*b^4*n^4/((1+20*b^2*n^2+64*b^4*n^4)*x)-24*b^3*n^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+20*b^2*n^2+64*b^4*n^4)*x)-12*b^2*n^2*sin(a+b*log(c*x^n))^2/((1+20*b^2*n^2+64*b^4*n^4)*x)-4*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/((1+16*b^2*n^2)*x)-sin(a+b*log(c*x^n))^4/((1+16*b^2*n^2)*x)],
[sin(a+b*log(c*x^n))^4/x^3,x,3,-3/4*b^4*n^4/((1+5*b^2*n^2+4*b^4*n^4)*x^2)-3/2*b^3*n^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+5*b^2*n^2+4*b^4*n^4)*x^2)-3/2*b^2*n^2*sin(a+b*log(c*x^n))^2/((1+5*b^2*n^2+4*b^4*n^4)*x^2)-b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))^3/((1+4*b^2*n^2)*x^2)-1/2*sin(a+b*log(c*x^n))^4/((1+4*b^2*n^2)*x^2)],
[sin(log(a+b*x)),x,2,-1/2*(a+b*x)*cos(log(a+b*x))/b+1/2*(a+b*x)*sin(log(a+b*x))/b],

# Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when b^2 n^2 p^2+(m+1)^2=0
[x^m*sin(a+log(c*x^n)*sqrt(-(1+m)^2/n^2)),x,3,-1/4*exp(a*(1+m)/(n*sqrt(-(1+m)^2/n^2)))*x^(1+m)*(c*x^n)^((1+m)/n)/(n*sqrt(-(1+m)^2/n^2))+1/2*exp(a*n*sqrt(-(1+m)^2/n^2)/(1+m))*(1+m)*x^(1+m)*log(x)/(n*(c*x^n)^((1+m)/n)*sqrt(-(1+m)^2/n^2))],
[x^2*sin(a+log(c*x^n)*sqrt((-9)/n^2)),x,3,1/12*n*x^3*(c*x^n)^(3/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/2*exp(a*n*sqrt((-1)/n^2))*n*x^3*log(x)*sqrt((-1)/n^2)/(c*x^n)^(3/n)],
[x*sin(a+log(c*x^n)*sqrt((-4)/n^2)),x,3,1/8*n*x^2*(c*x^n)^(2/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/2*exp(a*n*sqrt((-1)/n^2))*n*x^2*log(x)*sqrt((-1)/n^2)/(c*x^n)^(2/n)],
[sin(a+log(c*x^n)*sqrt((-1)/n^2)),x,3,1/4*n*x*(c*x^n)^(1/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/2*exp(a*n*sqrt((-1)/n^2))*n*x*log(x)*sqrt((-1)/n^2)/(c*x^n)^(1/n)],
[sin(a+log(c*x^n)*sqrt(0))/x,x,2,log(x)*sin(a)],
[sin(a+log(c*x^n)*sqrt((-1)/n^2))/x^2,x,3,1/4*exp(a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x*(c*x^n)^(1/n))+1/2*n*(c*x^n)^(1/n)*log(x)*sqrt((-1)/n^2)/(exp(a*n*sqrt((-1)/n^2))*x)],
[sin(a+log(c*x^n)*sqrt((-4)/n^2))/x^3,x,3,1/8*exp(a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x^2*(c*x^n)^(2/n))+1/2*n*(c*x^n)^(2/n)*log(x)*sqrt((-1)/n^2)/(exp(a*n*sqrt((-1)/n^2))*x^2)],
[x^m*sin(a+log(c*x^n)*sqrt(-1/4*(1+m)^2/n^2))^2,x,3,1/2*x^(1+m)/(1+m)-1/8*x^(1+m)*(c*x^n)^((1+m)/n)/(exp(2*a*n*sqrt(-(1+m)^2/n^2)/(1+m))*(1+m))-1/4*exp(2*a*n*sqrt(-(1+m)^2/n^2)/(1+m))*x^(1+m)*log(x)/(c*x^n)^((1+m)/n)],
[x^2*sin(a+log(c*x^n)*sqrt((-9/4)/n^2))^2,x,3,1/6*x^3-1/24*x^3*(c*x^n)^(3/n)/exp(2*a*n*sqrt((-1)/n^2))-1/4*exp(2*a*n*sqrt((-1)/n^2))*x^3*log(x)/(c*x^n)^(3/n)],
[x*sin(a+log(c*x^n)*sqrt((-1)/n^2))^2,x,3,1/4*x^2-1/16*x^2*(c*x^n)^(2/n)/exp(2*a*n*sqrt((-1)/n^2))-1/4*exp(2*a*n*sqrt((-1)/n^2))*x^2*log(x)/(c*x^n)^(2/n)],
[sin(a+log(c*x^n)*sqrt((-1/4)/n^2))^2,x,3,1/2*x-1/8*x*(c*x^n)^(1/n)/exp(2*a*n*sqrt((-1)/n^2))-1/4*exp(2*a*n*sqrt((-1)/n^2))*x*log(x)/(c*x^n)^(1/n)],
[sin(a+log(c*x^n)*sqrt(0))^2/x,x,2,log(x)*sin(a)^2],
[sin(a+log(c*x^n)*sqrt((-1/4)/n^2))^2/x^2,x,3,(-1/2)/x+1/8*exp(2*a*n*sqrt((-1)/n^2))/(x*(c*x^n)^(1/n))-1/4*(c*x^n)^(1/n)*log(x)/(exp(2*a*n*sqrt((-1)/n^2))*x)],
[sin(a+log(c*x^n)*sqrt((-1)/n^2))^2/x^3,x,3,(-1/4)/x^2+1/16*exp(2*a*n*sqrt((-1)/n^2))/(x^2*(c*x^n)^(2/n))-1/4*(c*x^n)^(2/n)*log(x)/(exp(2*a*n*sqrt((-1)/n^2))*x^2)],
[x^m*sin(a+log(c*x^n)*sqrt(-1/4*(1+m)^2/n^2))^3,x,2,8/5*x^(1+m)*sin(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))/(1+m)-4/5*x^(1+m)*sin(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))^3/(1+m)-4/5*n*x^(1+m)*cos(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))*sqrt(-(1+m)^2/n^2)/(1+m)^2+6/5*n*x^(1+m)*cos(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))*sin(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))^2*sqrt(-(1+m)^2/n^2)/(1+m)^2],
[x^2*sin(a+log(c*x^n)*sqrt((-1)/n^2))^3,x,3,-3/16*exp(a*n*sqrt((-1)/n^2))*n*x^3*sqrt((-1)/n^2)/(c*x^n)^(1/n)+3/32*n*x^3*(c*x^n)^(1/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/48*n*x^3*(c*x^n)^(3/n)*sqrt((-1)/n^2)/exp(3*a*n*sqrt((-1)/n^2))+1/8*exp(3*a*n*sqrt((-1)/n^2))*n*x^3*log(x)*sqrt((-1)/n^2)/(c*x^n)^(3/n)],
[x*sin(a+log(c*x^n)*sqrt((-4/9)/n^2))^3,x,3,-9/32*exp(a*n*sqrt((-1)/n^2))*n*x^2*sqrt((-1)/n^2)/(c*x^n)^(2/3/n)+9/64*n*x^2*(c*x^n)^(2/3/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/32*n*x^2*(c*x^n)^(2/n)*sqrt((-1)/n^2)/exp(3*a*n*sqrt((-1)/n^2))+1/8*exp(3*a*n*sqrt((-1)/n^2))*n*x^2*log(x)*sqrt((-1)/n^2)/(c*x^n)^(2/n)],
[sin(a+log(c*x^n)*sqrt((-1/9)/n^2))^3,x,3,-9/16*exp(a*n*sqrt((-1)/n^2))*n*x*sqrt((-1)/n^2)/(c*x^n)^(1/3/n)+9/32*n*x*(c*x^n)^(1/3/n)*sqrt((-1)/n^2)/exp(a*n*sqrt((-1)/n^2))-1/16*n*x*(c*x^n)^(1/n)*sqrt((-1)/n^2)/exp(3*a*n*sqrt((-1)/n^2))+1/8*exp(3*a*n*sqrt((-1)/n^2))*n*x*log(x)*sqrt((-1)/n^2)/(c*x^n)^(1/n)],
[sin(a+log(c*x^n)*sqrt(0))^3/x,x,2,log(x)*sin(a)^3],
[sin(a+log(c*x^n)*sqrt((-1/9)/n^2))^3/x^2,x,3,-1/16*exp(3*a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x*(c*x^n)^(1/n))+9/32*exp(a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x*(c*x^n)^(1/3/n))-9/16*n*(c*x^n)^(1/3/n)*sqrt((-1)/n^2)/(exp(a*n*sqrt((-1)/n^2))*x)-1/8*n*(c*x^n)^(1/n)*log(x)*sqrt((-1)/n^2)/(exp(3*a*n*sqrt((-1)/n^2))*x)],
[sin(a+log(c*x^n)*sqrt((-4/9)/n^2))^3/x^3,x,3,-1/32*exp(3*a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x^2*(c*x^n)^(2/n))+9/64*exp(a*n*sqrt((-1)/n^2))*n*sqrt((-1)/n^2)/(x^2*(c*x^n)^(2/3/n))-9/32*n*(c*x^n)^(2/3/n)*sqrt((-1)/n^2)/(exp(a*n*sqrt((-1)/n^2))*x^2)-1/8*n*(c*x^n)^(2/n)*log(x)*sqrt((-1)/n^2)/(exp(3*a*n*sqrt((-1)/n^2))*x^2)],
[x^m*sin(a+log(c*x^2)*sqrt(-1/4*(1+m)^2)),x,3,-1/4*exp(a*(1+m)/sqrt(-(1+m)^2))*x^(1+m)*(c*x^2)^(1/2*(1+m))/sqrt(-(1+m)^2)+1/2*exp(a*sqrt(-(1+m)^2)/(1+m))*(1+m)*x^(1+m)*(c*x^2)^(1/2*(-1-m))*log(x)/sqrt(-(1+m)^2)],
[sin(a+log(c*x^2)*sqrt(-1/4)),x,3,1/4*I*c*x^3/(exp(I*a)*sqrt(c*x^2))-1/2*I*exp(I*a)*x*log(x)/sqrt(c*x^2)],
[x^m*sin(a+log(c*x^2)*sqrt(-1/16*(1+m)^2))^2,x,3,1/2*x^(1+m)/(1+m)-1/8*exp(2*a*(1+m)/sqrt(-(1+m)^2))*x^(1+m)*(c*x^2)^(1/2*(1+m))/(1+m)-1/4*x^(1+m)*(c*x^2)^(1/2*(-1-m))*log(x)/exp(2*a*(1+m)/sqrt(-(1+m)^2))],
[sin(a+log(c*x^2)*sqrt(-1/16))^2,x,3,1/2*x-1/8*c*x^3/(exp(2*I*a)*sqrt(c*x^2))-1/4*exp(2*I*a)*x*log(x)/sqrt(c*x^2)],
[x^m*sin(a+log(c*x^2)*sqrt(-1/36*(1+m)^2))^3,x,3,9/16*exp(a*sqrt(-(1+m)^2)/(1+m))*x^(1+m)*(c*x^2)^(1/6*(-1-m))/sqrt(-(1+m)^2)-9/32*exp(a*(1+m)/sqrt(-(1+m)^2))*x^(1+m)*(c*x^2)^(1/6*(1+m))/sqrt(-(1+m)^2)+1/16*exp(3*a*(1+m)/sqrt(-(1+m)^2))*x^(1+m)*(c*x^2)^(1/2*(1+m))/sqrt(-(1+m)^2)-1/8*(1+m)*x^(1+m)*(c*x^2)^(1/2*(-1-m))*log(x)/(exp(3*a*(1+m)/sqrt(-(1+m)^2))*sqrt(-(1+m)^2))],
[sin(a+log(c*x^2)*sqrt(-1/36))^3,x,3,-9/16*I*exp(I*a)*x/(c*x^2)^(1/6)+9/32*I*x*(c*x^2)^(1/6)/exp(I*a)-1/16*I*c*x^3/(exp(3*I*a)*sqrt(c*x^2))+1/8*I*exp(3*I*a)*x*log(x)/sqrt(c*x^2)],

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

# p>0
[x*sqrt(sin(a+b*log(c*x^n))),x,3,2*x^2*hypergeom([-1/2,1/4*(-1+(-4*I)/(b*n))],[1/4*(3+(-4*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sin(a+b*log(c*x^n)))/((4-I*b*n)*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b)))],
[sqrt(sin(a+b*log(c*x^n))),x,3,2*x*hypergeom([-1/2,1/4*(-2*I-b*n)/(b*n)],[1/4*(3+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sin(a+b*log(c*x^n)))/((2-I*b*n)*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b)))],
[sqrt(sin(a+b*log(c*x^n)))/x,x,2,2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticE(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))/(b*n)],
[sqrt(sin(a+b*log(c*x^n)))/x^2,x,3,-2*hypergeom([-1/2,1/4*(-1+2*I/(b*n))],[1/4*(3+2*I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sin(a+b*log(c*x^n)))/((2+I*b*n)*x*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b)))],
[sqrt(sin(a+b*log(c*x^n)))/x^3,x,3,-2*hypergeom([-1/2,1/4*(-1+4*I/(b*n))],[1/4*(3+4*I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sin(a+b*log(c*x^n)))/((4+I*b*n)*x^2*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b)))],
[x*sin(a+b*log(c*x^n))^(3/2),x,3,2*x^2*hypergeom([-3/2,1/4*(-3+(-4*I)/(b*n))],[1/4*(1+(-4*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^(3/2)/((4-3*I*b*n)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],
[sin(a+b*log(c*x^n))^(3/2),x,3,2*x*hypergeom([-3/2,1/4*(-3+(-2*I)/(b*n))],[1/4*(1+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^(3/2)/((2-3*I*b*n)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],
[sin(a+b*log(c*x^n))^(3/2)/x,x,3,2/3*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))/(b*n)-2/3*cos(a+b*log(c*x^n))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],
[sin(a+b*log(c*x^n))^(3/2)/x^2,x,3,-2*hypergeom([-3/2,1/4*(-3+2*I/(b*n))],[1/4*(1+2*I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^(3/2)/((2+3*I*b*n)*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],
[sin(a+b*log(c*x^n))^(3/2)/x^3,x,3,-2*hypergeom([-3/2,1/4*(-3+4*I/(b*n))],[1/4*(1+4*I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^(3/2)/((4+3*I*b*n)*x^2*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],

# p<0
[1/sqrt(sin(a+b*log(c*x^n))),x,3,2*x*hypergeom([1/2,1/4*(1+(-2*I)/(b*n))],[1/4*(5+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+I*b*n)*sqrt(sin(a+b*log(c*x^n))))],
[1/(x*sqrt(sin(a+b*log(c*x^n)))),x,2,2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))/(b*n)],
[1/sin(a+b*log(c*x^n))^(3/2),x,3,2*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2,1/4*(3+(-2*I)/(b*n))],[1/4*(7+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((2+3*I*b*n)*sin(a+b*log(c*x^n))^(3/2))],
[1/(x*sin(a+b*log(c*x^n))^(3/2)),x,3,-2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticE(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))/(b*n)-2*cos(a+b*log(c*x^n))/(b*n*sqrt(sin(a+b*log(c*x^n))))],
[1/sin(a+b*log(c*x^n))^(5/2),x,3,2*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2,1/4*(5+(-2*I)/(b*n))],[1/4*(9+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((2+5*I*b*n)*sin(a+b*log(c*x^n))^(5/2))],
[1/(x*sin(a+b*log(c*x^n))^(5/2)),x,3,2/3*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))/(b*n)-2/3*cos(a+b*log(c*x^n))/(b*n*sin(a+b*log(c*x^n))^(3/2))],
[1/sin(a-2*I*log(c*x))^(3/2),x,3,1/2*(1-c^4*exp(2*I*a)*x^4)/(c^4*exp(2*I*a)*x^3*sin(a-2*I*log(c*x))^(3/2))],

# Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when m symbolic
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^4,x,3,24*b^4*d^4*n^4*(e*x)^(1+m)/(e*(1+m)*((1+m)^2+4*b^2*d^2*n^2)*((1+m)^2+16*b^2*d^2*n^2))-24*b^3*d^3*n^3*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))*sin(d*(a+b*log(c*x^n)))/(e*((1+m)^2+4*b^2*d^2*n^2)*((1+m)^2+16*b^2*d^2*n^2))+12*b^2*d^2*(1+m)*n^2*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))^2/(e*((1+m)^2+4*b^2*d^2*n^2)*((1+m)^2+16*b^2*d^2*n^2))-4*b*d*n*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))*sin(d*(a+b*log(c*x^n)))^3/(e*((1+m)^2+16*b^2*d^2*n^2))+(1+m)*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))^4/(e*((1+m)^2+16*b^2*d^2*n^2))],
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^3,x,2,-6*b^3*d^3*n^3*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))/(e*((1+m)^2+b^2*d^2*n^2)*((1+m)^2+9*b^2*d^2*n^2))+6*b^2*d^2*(1+m)*n^2*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))/(e*((1+m)^2+b^2*d^2*n^2)*((1+m)^2+9*b^2*d^2*n^2))-3*b*d*n*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))*sin(d*(a+b*log(c*x^n)))^2/(e*((1+m)^2+9*b^2*d^2*n^2))+(1+m)*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))^3/(e*((1+m)^2+9*b^2*d^2*n^2))],
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^2,x,2,2*b^2*d^2*n^2*(e*x)^(1+m)/(e*(1+m)*((1+m)^2+4*b^2*d^2*n^2))-2*b*d*n*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))*sin(d*(a+b*log(c*x^n)))/(e*((1+m)^2+4*b^2*d^2*n^2))+(1+m)*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))^2/(e*((1+m)^2+4*b^2*d^2*n^2))],
[(e*x)^m*sin(d*(a+b*log(c*x^n))),x,1,-b*d*n*(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))/(e*((1+m)^2+b^2*d^2*n^2))+(1+m)*(e*x)^(1+m)*sin(d*(a+b*log(c*x^n)))/(e*((1+m)^2+b^2*d^2*n^2))],
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^(3/2),x,3,2*(e*x)^(1+m)*hypergeom([-3/2,1/4*(-2*I-2*I*m-3*b*d*n)/(b*d*n)],[1/4*(-2*I-2*I*m+b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sin(d*(a+b*log(c*x^n)))^(3/2)/(e*(2+2*m-3*I*b*d*n)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2))],
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^(1/2),x,3,2*(e*x)^(1+m)*hypergeom([-1/2,1/4*(-2*I-2*I*m-b*d*n)/(b*d*n)],[1/4*(-2*I-2*I*m+3*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sqrt(sin(d*(a+b*log(c*x^n))))/(e*(2+2*m-I*b*d*n)*sqrt(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))],
[(e*x)^m/sin(d*(a+b*log(c*x^n)))^(1/2),x,3,2*(e*x)^(1+m)*hypergeom([1/2,1/4*(-2*I-2*I*m+b*d*n)/(b*d*n)],[1/4*(-2*I-2*I*m+5*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sqrt(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(2+2*m+I*b*d*n)*sqrt(sin(d*(a+b*log(c*x^n)))))],
[(e*x)^m/sin(d*(a+b*log(c*x^n)))^(3/2),x,3,2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2)*hypergeom([3/2,1/4*(-2*I-2*I*m+3*b*d*n)/(b*d*n)],[1/4*(-2*I-2*I*m+7*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(2+2*m+3*I*b*d*n)*sin(d*(a+b*log(c*x^n)))^(3/2))],
[(e*x)^m/sin(d*(a+b*log(c*x^n)))^(5/2),x,3,2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(5/2)*hypergeom([5/2,1/4*(-2*I-2*I*m+5*b*d*n)/(b*d*n)],[1/4*(-2*I-2*I*m+9*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(2+2*m+5*I*b*d*n)*sin(d*(a+b*log(c*x^n)))^(5/2))],

# Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when p symbolic
[(e*x)^m*sin(d*(a+b*log(c*x^n)))^p,x,3,(e*x)^(1+m)*hypergeom([-p,1/2*(-I-I*m-b*d*n*p)/(b*d*n)],[1/2*(2-I*(1+m)/(b*d*n)-p)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sin(d*(a+b*log(c*x^n)))^p/(e*(1+m-I*b*d*n*p)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)],
[x^2*sin(a+b*log(c*x^n))^p,x,3,x^3*hypergeom([-p,1/2*(-3*I-b*n*p)/(b*n)],[1/2*(2+(-3*I)/(b*n)-p)],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^p/((3-I*b*n*p)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)],
[x*sin(a+b*log(c*x^n))^p,x,3,x^2*hypergeom([1/2*((-2*I)/(b*n)-p),-p],[1/2*(2+(-2*I)/(b*n)-p)],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^p/((2-I*b*n*p)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)],
[sin(a+b*log(c*x^n))^p,x,3,x*hypergeom([-p,1/2*(-I-b*n*p)/(b*n)],[1/2*(2+(-I)/(b*n)-p)],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^p/((1-I*b*n*p)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)],
[sin(a+b*log(c*x^n))^p/x,x,2,cos(a+b*log(c*x^n))*hypergeom([1/2,1/2*(1+p)],[1/2*(3+p)],sin(a+b*log(c*x^n))^2)*sin(a+b*log(c*x^n))^(1+p)/(b*n*(1+p)*sqrt(cos(a+b*log(c*x^n))^2))],
[sin(a+b*log(c*x^n))^p/x^2,x,3,-hypergeom([1/2*(I/(b*n)-p),-p],[1/2*(2+I/(b*n)-p)],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^p/((1+I*b*n*p)*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)],
[sin(a+b*log(c*x^n))^p/x^3,x,3,-hypergeom([1/2*(2*I/(b*n)-p),-p],[1/2*(2+2*I/(b*n)-p)],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*log(c*x^n))^p/((2+I*b*n*p)*x^2*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)],

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

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

# p>0
[x^2*cos(a+b*log(c*x^n)),x,1,3*x^3*cos(a+b*log(c*x^n))/(9+b^2*n^2)+b*n*x^3*sin(a+b*log(c*x^n))/(9+b^2*n^2)],
[x*cos(a+b*log(c*x^n)),x,1,2*x^2*cos(a+b*log(c*x^n))/(4+b^2*n^2)+b*n*x^2*sin(a+b*log(c*x^n))/(4+b^2*n^2)],
[cos(a+b*log(c*x^n)),x,1,x*cos(a+b*log(c*x^n))/(1+b^2*n^2)+b*n*x*sin(a+b*log(c*x^n))/(1+b^2*n^2)],
[cos(a+b*log(c*x^n))/x,x,2,sin(a+b*log(c*x^n))/(b*n)],
[cos(a+b*log(c*x^n))/x^2,x,1,-cos(a+b*log(c*x^n))/((1+b^2*n^2)*x)+b*n*sin(a+b*log(c*x^n))/((1+b^2*n^2)*x)],
[x^2*cos(a+b*log(c*x^n))^2,x,2,2/3*b^2*n^2*x^3/(9+4*b^2*n^2)+3*x^3*cos(a+b*log(c*x^n))^2/(9+4*b^2*n^2)+2*b*n*x^3*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(9+4*b^2*n^2)],
[x*cos(a+b*log(c*x^n))^2,x,2,1/4*b^2*n^2*x^2/(1+b^2*n^2)+1/2*x^2*cos(a+b*log(c*x^n))^2/(1+b^2*n^2)+1/2*b*n*x^2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+b^2*n^2)],
[cos(a+b*log(c*x^n))^2,x,2,2*b^2*n^2*x/(1+4*b^2*n^2)+x*cos(a+b*log(c*x^n))^2/(1+4*b^2*n^2)+2*b*n*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+4*b^2*n^2)],
[cos(a+b*log(c*x^n))^2/x,x,3,1/2*log(x)+1/2*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(b*n)],
[cos(a+b*log(c*x^n))^2/x^2,x,2,-2*b^2*n^2/((1+4*b^2*n^2)*x)-cos(a+b*log(c*x^n))^2/((1+4*b^2*n^2)*x)+2*b*n*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+4*b^2*n^2)*x)],
[x^2*cos(a+b*log(c*x^n))^3,x,2,2*b^2*n^2*x^3*cos(a+b*log(c*x^n))/(9+10*b^2*n^2+b^4*n^4)+1/3*x^3*cos(a+b*log(c*x^n))^3/(1+b^2*n^2)+2/3*b^3*n^3*x^3*sin(a+b*log(c*x^n))/(9+10*b^2*n^2+b^4*n^4)+1/3*b*n*x^3*cos(a+b*log(c*x^n))^2*sin(a+b*log(c*x^n))/(1+b^2*n^2)],
[x*cos(a+b*log(c*x^n))^3,x,2,12*b^2*n^2*x^2*cos(a+b*log(c*x^n))/(16+40*b^2*n^2+9*b^4*n^4)+2*x^2*cos(a+b*log(c*x^n))^3/(4+9*b^2*n^2)+6*b^3*n^3*x^2*sin(a+b*log(c*x^n))/(16+40*b^2*n^2+9*b^4*n^4)+3*b*n*x^2*cos(a+b*log(c*x^n))^2*sin(a+b*log(c*x^n))/(4+9*b^2*n^2)],
[cos(a+b*log(c*x^n))^3,x,2,6*b^2*n^2*x*cos(a+b*log(c*x^n))/(1+10*b^2*n^2+9*b^4*n^4)+x*cos(a+b*log(c*x^n))^3/(1+9*b^2*n^2)+6*b^3*n^3*x*sin(a+b*log(c*x^n))/(1+10*b^2*n^2+9*b^4*n^4)+3*b*n*x*cos(a+b*log(c*x^n))^2*sin(a+b*log(c*x^n))/(1+9*b^2*n^2)],
[cos(a+b*log(c*x^n))^3/x,x,3,sin(a+b*log(c*x^n))/(b*n)-1/3*sin(a+b*log(c*x^n))^3/(b*n)],
[cos(a+b*log(c*x^n))^3/x^2,x,2,-6*b^2*n^2*cos(a+b*log(c*x^n))/((1+10*b^2*n^2+9*b^4*n^4)*x)-cos(a+b*log(c*x^n))^3/((1+9*b^2*n^2)*x)+6*b^3*n^3*sin(a+b*log(c*x^n))/((1+10*b^2*n^2+9*b^4*n^4)*x)+3*b*n*cos(a+b*log(c*x^n))^2*sin(a+b*log(c*x^n))/((1+9*b^2*n^2)*x)],
[cos(a+b*log(c*x^n))^4,x,3,24*b^4*n^4*x/(1+20*b^2*n^2+64*b^4*n^4)+12*b^2*n^2*x*cos(a+b*log(c*x^n))^2/(1+20*b^2*n^2+64*b^4*n^4)+x*cos(a+b*log(c*x^n))^4/(1+16*b^2*n^2)+24*b^3*n^3*x*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(1+20*b^2*n^2+64*b^4*n^4)+4*b*n*x*cos(a+b*log(c*x^n))^3*sin(a+b*log(c*x^n))/(1+16*b^2*n^2)],
[cos(a+b*log(c*x^n))^4/x,x,4,3/8*log(x)+3/8*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(b*n)+1/4*cos(a+b*log(c*x^n))^3*sin(a+b*log(c*x^n))/(b*n)],
[cos(log(6+3*x)),x,2,1/2*(2+x)*cos(log(3*(2+x)))+1/2*(2+x)*sin(log(3*(2+x)))],

# Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when b^2 n^2 p^2+(m+1)^2=0
[x^m*cos(a+log(c*x^n)*sqrt(-(1+m)^2/n^2)),x,3,1/4*exp(a*(1+m)/(n*sqrt(-(1+m)^2/n^2)))*x^(1+m)*(c*x^n)^((1+m)/n)/(1+m)+1/2*exp(a*n*sqrt(-(1+m)^2/n^2)/(1+m))*x^(1+m)*log(x)/(c*x^n)^((1+m)/n)],
[cos(a+log(c*x^n)*sqrt((-1)/n^2)),x,3,1/4*x*(c*x^n)^(1/n)/exp(a*n*sqrt((-1)/n^2))+1/2*exp(a*n*sqrt((-1)/n^2))*x*log(x)/(c*x^n)^(1/n)],
[x^m*cos(a+log(c*x^n)*sqrt(-1/4*(1+m)^2/n^2))^2,x,3,1/2*x^(1+m)/(1+m)+1/8*x^(1+m)*(c*x^n)^((1+m)/n)/(exp(2*a*n*sqrt(-(1+m)^2/n^2)/(1+m))*(1+m))+1/4*exp(2*a*n*sqrt(-(1+m)^2/n^2)/(1+m))*x^(1+m)*log(x)/(c*x^n)^((1+m)/n)],
[cos(a+log(c*x^n)*sqrt((-1/4)/n^2))^2,x,3,1/2*x+1/8*x*(c*x^n)^(1/n)/exp(2*a*n*sqrt((-1)/n^2))+1/4*exp(2*a*n*sqrt((-1)/n^2))*x*log(x)/(c*x^n)^(1/n)],
[x^m*cos(a+log(c*x^n)*sqrt(-1/4*(1+m)^2/n^2))^3,x,2,8/5*x^(1+m)*cos(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))/(1+m)-4/5*x^(1+m)*cos(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))^3/(1+m)+4/5*n*x^(1+m)*sin(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))*sqrt(-(1+m)^2/n^2)/(1+m)^2-6/5*n*x^(1+m)*cos(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))^2*sin(a+1/2*log(c*x^n)*sqrt(-(1+m)^2/n^2))*sqrt(-(1+m)^2/n^2)/(1+m)^2],
[cos(a+log(c*x^n)*sqrt((-1/9)/n^2))^3,x,3,9/16*exp(a*n*sqrt((-1)/n^2))*x/(c*x^n)^(1/3/n)+9/32*x*(c*x^n)^(1/3/n)/exp(a*n*sqrt((-1)/n^2))+1/16*x*(c*x^n)^(1/n)/exp(3*a*n*sqrt((-1)/n^2))+1/8*exp(3*a*n*sqrt((-1)/n^2))*x*log(x)/(c*x^n)^(1/n)],

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

# p>0
[sqrt(cos(a+b*log(c*x^n))),x,3,2*x*hypergeom([-1/2,1/4*(-2*I-b*n)/(b*n)],[1/4*(3+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(cos(a+b*log(c*x^n)))/((2-I*b*n)*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b)))],
[sqrt(cos(a+b*log(c*x^n)))/x,x,2,2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)],
[cos(a+b*log(c*x^n))^(3/2),x,3,2*x*cos(a+b*log(c*x^n))^(3/2)*hypergeom([-3/2,1/4*(-3+(-2*I)/(b*n))],[1/4*(1+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-3*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],
[cos(a+b*log(c*x^n))^(3/2)/x,x,3,2/3*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)+2/3*sin(a+b*log(c*x^n))*sqrt(cos(a+b*log(c*x^n)))/(b*n)],
[cos(a+b*log(c*x^n))^(5/2),x,3,2*x*cos(a+b*log(c*x^n))^(5/2)*hypergeom([-5/2,1/4*(-5+(-2*I)/(b*n))],[1/4*(-2*I-b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-5*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2))],
[cos(a+b*log(c*x^n))^(5/2)/x,x,3,6/5*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)+2/5*cos(a+b*log(c*x^n))^(3/2)*sin(a+b*log(c*x^n))/(b*n)],

# p<0
[1/sqrt(cos(a+b*log(c*x^n))),x,3,2*x*hypergeom([1/2,1/4*(1+(-2*I)/(b*n))],[1/4*(5+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))/((2+I*b*n)*sqrt(cos(a+b*log(c*x^n))))],
[1/(x*sqrt(cos(a+b*log(c*x^n)))),x,2,2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)],
[1/cos(a+b*log(c*x^n))^(3/2),x,3,2*x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2,1/4*(3+(-2*I)/(b*n))],[1/4*(7+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+3*I*b*n)*cos(a+b*log(c*x^n))^(3/2))],
[1/(x*cos(a+b*log(c*x^n))^(3/2)),x,3,-2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)+2*sin(a+b*log(c*x^n))/(b*n*sqrt(cos(a+b*log(c*x^n))))],
[1/cos(a+b*log(c*x^n))^(5/2),x,3,2*x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2,1/4*(5+(-2*I)/(b*n))],[1/4*(9+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+5*I*b*n)*cos(a+b*log(c*x^n))^(5/2))],
[1/(x*cos(a+b*log(c*x^n))^(5/2)),x,3,2/3*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))/(b*n)+2/3*sin(a+b*log(c*x^n))/(b*n*cos(a+b*log(c*x^n))^(3/2))],
[1/cos(a-2*I*log(c*x))^(3/2),x,3,1/2*(-1-c^4*exp(2*I*a)*x^4)/(c^4*exp(2*I*a)*x^3*cos(a-2*I*log(c*x))^(3/2))],

# Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when m symbolic
[x^m*cos(a+b*log(c*x^n))^4,x,3,24*b^4*n^4*x^(1+m)/((1+m)*((1+m)^2+4*b^2*n^2)*((1+m)^2+16*b^2*n^2))+12*b^2*(1+m)*n^2*x^(1+m)*cos(a+b*log(c*x^n))^2/(((1+m)^2+4*b^2*n^2)*((1+m)^2+16*b^2*n^2))+(1+m)*x^(1+m)*cos(a+b*log(c*x^n))^4/((1+m)^2+16*b^2*n^2)+24*b^3*n^3*x^(1+m)*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/(((1+m)^2+4*b^2*n^2)*((1+m)^2+16*b^2*n^2))+4*b*n*x^(1+m)*cos(a+b*log(c*x^n))^3*sin(a+b*log(c*x^n))/((1+m)^2+16*b^2*n^2)],
[x^m*cos(a+b*log(c*x^n))^3,x,2,6*b^2*(1+m)*n^2*x^(1+m)*cos(a+b*log(c*x^n))/(((1+m)^2+b^2*n^2)*((1+m)^2+9*b^2*n^2))+(1+m)*x^(1+m)*cos(a+b*log(c*x^n))^3/((1+m)^2+9*b^2*n^2)+6*b^3*n^3*x^(1+m)*sin(a+b*log(c*x^n))/(((1+m)^2+b^2*n^2)*((1+m)^2+9*b^2*n^2))+3*b*n*x^(1+m)*cos(a+b*log(c*x^n))^2*sin(a+b*log(c*x^n))/((1+m)^2+9*b^2*n^2)],
[x^m*cos(a+b*log(c*x^n))^2,x,2,2*b^2*n^2*x^(1+m)/((1+m)*((1+m)^2+4*b^2*n^2))+(1+m)*x^(1+m)*cos(a+b*log(c*x^n))^2/((1+m)^2+4*b^2*n^2)+2*b*n*x^(1+m)*cos(a+b*log(c*x^n))*sin(a+b*log(c*x^n))/((1+m)^2+4*b^2*n^2)],
[x^m*cos(a+b*log(c*x^n)),x,1,(1+m)*x^(1+m)*cos(a+b*log(c*x^n))/((1+m)^2+b^2*n^2)+b*n*x^(1+m)*sin(a+b*log(c*x^n))/((1+m)^2+b^2*n^2)],
[x^m*cos(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*cos(a+b*log(c*x^n))^(3/2)*hypergeom([-3/2,1/4*(-2*I-2*I*m-3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m-3*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2))],
[x^m*cos(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([-1/2,1/4*(-2*I-2*I*m-b*n)/(b*n)],[1/4*(-2*I-2*I*m+3*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(cos(a+b*log(c*x^n)))/((2+2*m-I*b*n)*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b)))],
[x^m/cos(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([1/2,1/4*(-2*I-2*I*m+b*n)/(b*n)],[1/4*(-2*I-2*I*m+5*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m+I*b*n)*sqrt(cos(a+b*log(c*x^n))))],
[x^m/cos(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2,1/4*(-2*I-2*I*m+3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+7*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m+3*I*b*n)*cos(a+b*log(c*x^n))^(3/2))],
[x^m/cos(a+b*log(c*x^n))^(5/2),x,3,2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2,1/4*(-2*I-2*I*m+5*b*n)/(b*n)],[1/4*(-2*I-2*I*m+9*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m+5*I*b*n)*cos(a+b*log(c*x^n))^(5/2))],

# Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when p symbolic
[(e*x)^m*cos(d*(a+b*log(c*x^n)))^p,x,3,(e*x)^(1+m)*cos(d*(a+b*log(c*x^n)))^p*hypergeom([-p,1/2*(-I-I*m-b*d*n*p)/(b*d*n)],[1/2*(2-I*(1+m)/(b*d*n)-p)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m-I*b*d*n*p)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)],
[x*cos(a+b*log(c*x^n))^p,x,3,x^2*cos(a+b*log(c*x^n))^p*hypergeom([1/2*((-2*I)/(b*n)-p),-p],[1/2*(2+(-2*I)/(b*n)-p)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-I*b*n*p)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^p)],
[cos(a+b*log(c*x^n))^p,x,3,x*cos(a+b*log(c*x^n))^p*hypergeom([-p,1/2*(-I-b*n*p)/(b*n)],[1/2*(2+(-I)/(b*n)-p)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-I*b*n*p)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^p)],

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

# Integrands of the form x^m Tan[a+b Log[x]]^p
[x^3*tan(a+I*log(x)),x,5,-I*exp(2*I*a)*x^2+1/4*I*x^4+I*exp(4*I*a)*log(exp(2*I*a)+x^2)],
[x^2*tan(a+I*log(x)),x,5,-2*I*exp(2*I*a)*x+1/3*I*x^3+2*I*exp(3*I*a)*arctan(x/exp(I*a))],
[x*tan(a+I*log(x)),x,5,1/2*I*x^2-I*exp(2*I*a)*log(exp(2*I*a)+x^2)],
[tan(a+I*log(x)),x,4,I*x-2*I*exp(I*a)*arctan(x/exp(I*a))],
[tan(a+I*log(x))/x,x,2,I*log(cos(a+I*log(x)))],
[tan(a+I*log(x))/x^2,x,4,I/x+2*I*arctan(x/exp(I*a))/exp(I*a)],
[tan(a+I*log(x))/x^3,x,4,1/2*I/x^2-I*log(1+exp(2*I*a)/x^2)/exp(2*I*a)],
[tan(a+I*log(x))/x^4,x,5,1/3*I/x^3+(-2*I)/(exp(2*I*a)*x)-2*I*arctan(x/exp(I*a))/exp(3*I*a)],
[x^3*tan(a+I*log(x))^2,x,5,2*exp(2*I*a)*x^2-1/4*x^4-2*exp(6*I*a)/(exp(2*I*a)+x^2)-4*exp(4*I*a)*log(exp(2*I*a)+x^2)],
[x^2*tan(a+I*log(x))^2,x,6,6*exp(2*I*a)*x-1/3*x^3-2*exp(2*I*a)*x^3/(exp(2*I*a)+x^2)-6*exp(3*I*a)*arctan(x/exp(I*a))],
[x*tan(a+I*log(x))^2,x,5,-1/2*x^2+2*exp(4*I*a)/(exp(2*I*a)+x^2)+2*exp(2*I*a)*log(exp(2*I*a)+x^2)],
[tan(a+I*log(x))^2,x,6,-x-2*exp(2*I*a)*x/(exp(2*I*a)+x^2)+2*exp(I*a)*arctan(x/exp(I*a))],
[tan(a+I*log(x))^2/x,x,3,-log(x)-I*tan(a+I*log(x))],
[tan(a+I*log(x))^2/x^2,x,5,exp(2*I*a)/(x*(exp(2*I*a)+x^2))+3*x/(exp(2*I*a)+x^2)+2*arctan(x/exp(I*a))/exp(I*a)],
[tan(a+I*log(x))^2/x^3,x,4,(-2)/(exp(2*I*a)*(1+exp(2*I*a)/x^2))+1/2/x^2-2*log(1+exp(2*I*a)/x^2)/exp(2*I*a)],

# Integrands of the form (e x)^m Tan[a+b Log[x]]^p with m symbolic
[(e*x)^m*tan(a+I*log(x)),x,4,-I*(e*x)^(1+m)/(e*(1+m))+2*I*(e*x)^(1+m)*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],-exp(2*I*a)/x^2)/(e*(1+m))],
[(e*x)^m*tan(a+I*log(x))^2,x,5,-x*(e*x)^m/(1+m)+2*x*(e*x)^m/(1+exp(2*I*a)/x^2)-2*x*(e*x)^m*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],-exp(2*I*a)/x^2)],
[(e*x)^m*tan(a+I*log(x))^3,x,6,-1/2*I*(1-m)*m*x*(e*x)^m/(1+m)+1/2*I*(1-exp(2*I*a)/x^2)^2*x*(e*x)^m/(1+exp(2*I*a)/x^2)^2+1/2*I*(exp(2*I*a)*(3+m)+exp(4*I*a)*(1-m)/x^2)*x*(e*x)^m/(exp(2*I*a)*(1+exp(2*I*a)/x^2))-I*(3+2*m+m^2)*x*(e*x)^m*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],-exp(2*I*a)/x^2)/(1+m)],

# Integrands of the form (e x)^m Tan[a+b Log[x]]^p with p symbolic
[tan(a+b*log(x))^p,x,4,x*(I*(1-exp(2*I*a)*x^(2*I*b))/(1+exp(2*I*a)*x^(2*I*b)))^p*(1+exp(2*I*a)*x^(2*I*b))^p*AppellF1((-1/2*I)/b,-p,p,1+(-1/2*I)/b,exp(2*I*a)*x^(2*I*b),-exp(2*I*a)*x^(2*I*b))/(1-exp(2*I*a)*x^(2*I*b))^p],
[(e*x)^m*tan(a+b*log(x))^p,x,4,(e*x)^(1+m)*(I*(1-exp(2*I*a)*x^(2*I*b))/(1+exp(2*I*a)*x^(2*I*b)))^p*(1+exp(2*I*a)*x^(2*I*b))^p*AppellF1(-1/2*I*(1+m)/b,-p,p,1-1/2*I*(1+m)/b,exp(2*I*a)*x^(2*I*b),-exp(2*I*a)*x^(2*I*b))/(e*(1+m)*(1-exp(2*I*a)*x^(2*I*b))^p)],
[tan(a+log(x))^p,x,4,(I*(1-exp(2*I*a)*x^(2*I))/(1+exp(2*I*a)*x^(2*I)))^p*(1+exp(2*I*a)*x^(2*I))^p*x*AppellF1(-1/2*I,-p,p,1-1/2*I,exp(2*I*a)*x^(2*I),-exp(2*I*a)*x^(2*I))/(1-exp(2*I*a)*x^(2*I))^p],
[tan(a+2*log(x))^p,x,4,(I*(1-exp(2*I*a)*x^(4*I))/(1+exp(2*I*a)*x^(4*I)))^p*(1+exp(2*I*a)*x^(4*I))^p*x*AppellF1(-1/4*I,-p,p,1-1/4*I,exp(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I))/(1-exp(2*I*a)*x^(4*I))^p],
[tan(a+3*log(x))^p,x,4,(I*(1-exp(2*I*a)*x^(6*I))/(1+exp(2*I*a)*x^(6*I)))^p*(1+exp(2*I*a)*x^(6*I))^p*x*AppellF1(-1/6*I,-p,p,1-1/6*I,exp(2*I*a)*x^(6*I),-exp(2*I*a)*x^(6*I))/(1-exp(2*I*a)*x^(6*I))^p],

# Integrands of the form x^m Tan[a+b Log[c x^n]]^p
[x^3*tan(d*(a+b*log(c*x^n))),x,4,-1/4*I*x^4+1/2*I*x^4*hypergeom([1,(-2*I)/(b*d*n)],[1+(-2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[x^2*tan(d*(a+b*log(c*x^n))),x,4,-1/3*I*x^3+2/3*I*x^3*hypergeom([1,(-3/2*I)/(b*d*n)],[1+(-3/2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[x*tan(d*(a+b*log(c*x^n))),x,4,-1/2*I*x^2+I*x^2*hypergeom([1,(-I)/(b*d*n)],[1+(-I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[tan(d*(a+b*log(c*x^n))),x,4,-I*x+2*I*x*hypergeom([1,(-1/2*I)/(b*d*n)],[1+(-1/2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[tan(d*(a+b*log(c*x^n)))/x,x,2,-log(cos(a*d+b*d*log(c*x^n)))/(b*d*n)],
[tan(d*(a+b*log(c*x^n)))/x^2,x,4,I/x-2*I*hypergeom([1,1/2*I/(b*d*n)],[1+1/2*I/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/x],
[tan(d*(a+b*log(c*x^n)))/x^3,x,4,1/2*I/x^2-I*hypergeom([1,I/(b*d*n)],[1+I/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/x^2],
[x^3*tan(d*(a+b*log(c*x^n)))^2,x,5,1/4*(4*I-b*d*n)*x^4/(b*d*n)+I*x^4*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^4*hypergeom([1,(-2*I)/(b*d*n)],[1+(-2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[x^2*tan(d*(a+b*log(c*x^n)))^2,x,5,1/3*(3*I-b*d*n)*x^3/(b*d*n)+I*x^3*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^3*hypergeom([1,(-3/2*I)/(b*d*n)],[1+(-3/2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[x*tan(d*(a+b*log(c*x^n)))^2,x,5,1/2*(2*I-b*d*n)*x^2/(b*d*n)+I*x^2*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^2*hypergeom([1,(-I)/(b*d*n)],[1+(-I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[tan(d*(a+b*log(c*x^n)))^2,x,5,(I-b*d*n)*x/(b*d*n)+I*x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x*hypergeom([1,(-1/2*I)/(b*d*n)],[1+(-1/2*I)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[tan(d*(a+b*log(c*x^n)))^2/x,x,3,-log(x)+tan(a*d+b*d*log(c*x^n))/(b*d*n)],
[tan(d*(a+b*log(c*x^n)))^2/x^2,x,5,(1+I/(b*d*n))/x+I*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*hypergeom([1,1/2*I/(b*d*n)],[1+1/2*I/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x)],
[tan(d*(a+b*log(c*x^n)))^2/x^3,x,5,1/2*(1+2*I/(b*d*n))/x^2+I*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x^2*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*hypergeom([1,I/(b*d*n)],[1+I/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x^2)],
[tan(a+b*log(c*x^n))^3/x,x,3,log(cos(a+b*log(c*x^n)))/(b*n)+1/2*tan(a+b*log(c*x^n))^2/(b*n)],
[tan(a+b*log(c*x^n))^4/x,x,4,log(x)-tan(a+b*log(c*x^n))/(b*n)+1/3*tan(a+b*log(c*x^n))^3/(b*n)],
[tan(a+b*log(c*x^n))^5/x,x,4,-log(cos(a+b*log(c*x^n)))/(b*n)-1/2*tan(a+b*log(c*x^n))^2/(b*n)+1/4*tan(a+b*log(c*x^n))^4/(b*n)],

# Integrands of the form (e x)^m Tan[a+b Log[c x^n]]^p with m symbolic
[(e*x)^m*tan(d*(a+b*log(c*x^n))),x,4,-I*(e*x)^(1+m)/(e*(1+m))+2*I*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m))],
[(e*x)^m*tan(d*(a+b*log(c*x^n)))^2,x,5,(I*(1+m)-b*d*n)*(e*x)^(1+m)/(b*d*e*(1+m)*n)+I*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*e*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*e*n)],
[(e*x)^m*tan(d*(a+b*log(c*x^n)))^3,x,6,-1/2*(I*(1+m)-b*d*n)*(1+m+2*I*b*d*n)*(e*x)^(1+m)/(b^2*d^2*e*(1+m)*n^2)-1/2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^2/(b*d*e*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^2)-1/2*I*(e*x)^(1+m)*(exp(2*I*a*d)*(1+m-2*I*b*d*n)/n-exp(4*I*a*d)*(1+m+2*I*b*d*n)*(c*x^n)^(2*I*b*d)/n)/(b^2*d^2*e*exp(2*I*a*d)*n*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))+I*(1+2*m+m^2-2*b^2*d^2*n^2)*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b^2*d^2*e*(1+m)*n^2)],

# Integrands of the form (e x)^m Tan[a+b Log[c x^n]]^p with p symbolic
[tan(d*(a+b*log(c*x^n)))^p,x,5,x*(I*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*AppellF1((-1/2*I)/(b*d*n),-p,p,1+(-1/2*I)/(b*d*n),exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p],
[(e*x)^m*tan(d*(a+b*log(c*x^n)))^p,x,5,(e*x)^(1+m)*(I*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*AppellF1(-1/2*I*(1+m)/(b*d*n),-p,p,1-1/2*I*(1+m)/(b*d*n),exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)],

# Integrands of the form x^m Tan[a+b Log[c x^n]]^(p/2)
[tan(a+b*log(c*x^n))^(5/2)/x,x,13,arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+2/3*tan(a+b*log(c*x^n))^(3/2)/(b*n)],
[tan(a+b*log(c*x^n))^(3/2)/x,x,13,arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))-1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+2*sqrt(tan(a+b*log(c*x^n)))/(b*n)],
[tan(a+b*log(c*x^n))^(1/2)/x,x,12,-arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))-1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))],
[1/(x*tan(a+b*log(c*x^n))^(1/2)),x,12,-arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))],
[1/(x*tan(a+b*log(c*x^n))^(3/2)),x,13,arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+(-2)/(b*n*sqrt(tan(a+b*log(c*x^n))))],
[1/(x*tan(a+b*log(c*x^n))^(5/2)),x,13,arctan(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1-sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))-1/2*log(1+sqrt(2)*sqrt(tan(a+b*log(c*x^n)))+tan(a+b*log(c*x^n)))/(b*n*sqrt(2))+(-2/3)/(b*n*tan(a+b*log(c*x^n))^(3/2))],

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

# Integrands of the form x^m Cot[a+b Log[x]]^p
[x^3*cot(a+I*log(x)),x,5,-I*exp(2*I*a)*x^2-1/4*I*x^4-I*exp(4*I*a)*log(exp(2*I*a)-x^2)],
[x^2*cot(a+I*log(x)),x,5,-2*I*exp(2*I*a)*x-1/3*I*x^3+2*I*exp(3*I*a)*arctanh(x/exp(I*a))],
[x*cot(a+I*log(x)),x,5,-1/2*I*x^2-I*exp(2*I*a)*log(exp(2*I*a)-x^2)],
[cot(a+I*log(x)),x,4,-I*x+2*I*exp(I*a)*arctanh(x/exp(I*a))],
[cot(a+I*log(x))/x,x,2,-I*log(sin(a+I*log(x)))],
[cot(a+I*log(x))/x^2,x,4,(-I)/x+2*I*arctanh(x/exp(I*a))/exp(I*a)],
[cot(a+I*log(x))/x^3,x,4,(-1/2*I)/x^2-I*log(1-exp(2*I*a)/x^2)/exp(2*I*a)],
[cot(a+I*log(x))/x^4,x,5,(-1/3*I)/x^3+(-2*I)/(exp(2*I*a)*x)+2*I*arctanh(x/exp(I*a))/exp(3*I*a)],
[x^3*cot(a+I*log(x))^2,x,5,-2*exp(2*I*a)*x^2-1/4*x^4-2*exp(6*I*a)/(exp(2*I*a)-x^2)-4*exp(4*I*a)*log(exp(2*I*a)-x^2)],
[x^2*cot(a+I*log(x))^2,x,6,-6*exp(2*I*a)*x-1/3*x^3-2*exp(2*I*a)*x^3/(exp(2*I*a)-x^2)+6*exp(3*I*a)*arctanh(x/exp(I*a))],
[x*cot(a+I*log(x))^2,x,5,-1/2*x^2-2*exp(4*I*a)/(exp(2*I*a)-x^2)-2*exp(2*I*a)*log(exp(2*I*a)-x^2)],
[cot(a+I*log(x))^2,x,6,-x-2*exp(2*I*a)*x/(exp(2*I*a)-x^2)+2*exp(I*a)*arctanh(x/exp(I*a))],
[cot(a+I*log(x))^2/x,x,3,I*cot(a+I*log(x))-log(x)],
[cot(a+I*log(x))^2/x^2,x,5,exp(2*I*a)/(x*(exp(2*I*a)-x^2))-3*x/(exp(2*I*a)-x^2)-2*arctanh(x/exp(I*a))/exp(I*a)],
[cot(a+I*log(x))^2/x^3,x,4,2/(exp(2*I*a)*(1-exp(2*I*a)/x^2))+1/2/x^2+2*log(1-exp(2*I*a)/x^2)/exp(2*I*a)],

# Integrands of the form (e x)^m Cot[a+b Log[x]]^p with m symbolic
[(e*x)^m*cot(a+I*log(x)),x,4,I*(e*x)^(1+m)/(e*(1+m))-2*I*(e*x)^(1+m)*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],exp(2*I*a)/x^2)/(e*(1+m))],
[(e*x)^m*cot(a+I*log(x))^2,x,5,-x*(e*x)^m/(1+m)+2*x*(e*x)^m/(1-exp(2*I*a)/x^2)-2*x*(e*x)^m*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],exp(2*I*a)/x^2)],
[(e*x)^m*cot(a+I*log(x))^3,x,6,1/2*I*(1-m)*m*x*(e*x)^m/(1+m)-1/2*I*(1+exp(2*I*a)/x^2)^2*x*(e*x)^m/(1-exp(2*I*a)/x^2)^2-1/2*I*(3+m-exp(2*I*a)*(1-m)/x^2)*x*(e*x)^m/(1-exp(2*I*a)/x^2)+I*(3+2*m+m^2)*x*(e*x)^m*hypergeom([1,1/2*(-1-m)],[1/2*(1-m)],exp(2*I*a)/x^2)/(1+m)],

# Integrands of the form (e x)^m Cot[a+b Log[x]]^p with p symbolic
[cot(a+b*log(x))^p,x,4,x*(1-exp(2*I*a)*x^(2*I*b))^p*(-I*(1+exp(2*I*a)*x^(2*I*b))/(1-exp(2*I*a)*x^(2*I*b)))^p*AppellF1((-1/2*I)/b,p,-p,1+(-1/2*I)/b,exp(2*I*a)*x^(2*I*b),-exp(2*I*a)*x^(2*I*b))/(1+exp(2*I*a)*x^(2*I*b))^p],
[(e*x)^m*cot(a+b*log(x))^p,x,4,(e*x)^(1+m)*(1-exp(2*I*a)*x^(2*I*b))^p*(-I*(1+exp(2*I*a)*x^(2*I*b))/(1-exp(2*I*a)*x^(2*I*b)))^p*AppellF1(-1/2*I*(1+m)/b,p,-p,1-1/2*I*(1+m)/b,exp(2*I*a)*x^(2*I*b),-exp(2*I*a)*x^(2*I*b))/(e*(1+m)*(1+exp(2*I*a)*x^(2*I*b))^p)],
[cot(a+log(x))^p,x,4,(1-exp(2*I*a)*x^(2*I))^p*(-I*(1+exp(2*I*a)*x^(2*I))/(1-exp(2*I*a)*x^(2*I)))^p*x*AppellF1(-1/2*I,p,-p,1-1/2*I,exp(2*I*a)*x^(2*I),-exp(2*I*a)*x^(2*I))/(1+exp(2*I*a)*x^(2*I))^p],
[cot(a+2*log(x))^p,x,4,(1-exp(2*I*a)*x^(4*I))^p*(-I*(1+exp(2*I*a)*x^(4*I))/(1-exp(2*I*a)*x^(4*I)))^p*x*AppellF1(-1/4*I,p,-p,1-1/4*I,exp(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I))/(1+exp(2*I*a)*x^(4*I))^p],
[cot(a+3*log(x))^p,x,4,(1-exp(2*I*a)*x^(6*I))^p*(-I*(1+exp(2*I*a)*x^(6*I))/(1-exp(2*I*a)*x^(6*I)))^p*x*AppellF1(-1/6*I,p,-p,1-1/6*I,exp(2*I*a)*x^(6*I),-exp(2*I*a)*x^(6*I))/(1+exp(2*I*a)*x^(6*I))^p],

# Integrands of the form x^m Cot[a+b Log[c x^n]]^p
[x^3*cot(d*(a+b*log(c*x^n))),x,4,1/4*I*x^4-1/2*I*x^4*hypergeom([1,(-2*I)/(b*d*n)],[1+(-2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[x^2*cot(d*(a+b*log(c*x^n))),x,4,1/3*I*x^3-2/3*I*x^3*hypergeom([1,(-3/2*I)/(b*d*n)],[1+(-3/2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[x*cot(d*(a+b*log(c*x^n))),x,4,1/2*I*x^2-I*x^2*hypergeom([1,(-I)/(b*d*n)],[1+(-I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[cot(d*(a+b*log(c*x^n))),x,4,I*x-2*I*x*hypergeom([1,(-1/2*I)/(b*d*n)],[1+(-1/2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))],
[cot(d*(a+b*log(c*x^n)))/x,x,2,log(sin(a*d+b*d*log(c*x^n)))/(b*d*n)],
[cot(d*(a+b*log(c*x^n)))/x^2,x,4,(-I)/x+2*I*hypergeom([1,1/2*I/(b*d*n)],[1+1/2*I/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/x],
[cot(d*(a+b*log(c*x^n)))/x^3,x,4,(-1/2*I)/x^2+I*hypergeom([1,I/(b*d*n)],[1+I/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/x^2],
[x^3*cot(d*(a+b*log(c*x^n)))^2,x,5,1/4*(4*I-b*d*n)*x^4/(b*d*n)+I*x^4*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^4*hypergeom([1,(-2*I)/(b*d*n)],[1+(-2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[x^2*cot(d*(a+b*log(c*x^n)))^2,x,5,1/3*(3*I-b*d*n)*x^3/(b*d*n)+I*x^3*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^3*hypergeom([1,(-3/2*I)/(b*d*n)],[1+(-3/2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[x*cot(d*(a+b*log(c*x^n)))^2,x,5,1/2*(2*I-b*d*n)*x^2/(b*d*n)+I*x^2*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x^2*hypergeom([1,(-I)/(b*d*n)],[1+(-I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[cot(d*(a+b*log(c*x^n)))^2,x,5,(I-b*d*n)*x/(b*d*n)+I*x*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*x*hypergeom([1,(-1/2*I)/(b*d*n)],[1+(-1/2*I)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n)],
[cot(d*(a+b*log(c*x^n)))^2/x,x,3,-cot(a*d+b*d*log(c*x^n))/(b*d*n)-log(x)],
[cot(d*(a+b*log(c*x^n)))^2/x^2,x,5,(1+I/(b*d*n))/x+I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*hypergeom([1,1/2*I/(b*d*n)],[1+1/2*I/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x)],
[cot(d*(a+b*log(c*x^n)))^2/x^3,x,5,1/2*(1+2*I/(b*d*n))/x^2+I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x^2*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*hypergeom([1,I/(b*d*n)],[1+I/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*n*x^2)],
[cot(a+b*log(c*x^n))^3/x,x,3,-1/2*cot(a+b*log(c*x^n))^2/(b*n)-log(sin(a+b*log(c*x^n)))/(b*n)],
[cot(a+b*log(c*x^n))^4/x,x,4,cot(a+b*log(c*x^n))/(b*n)-1/3*cot(a+b*log(c*x^n))^3/(b*n)+log(x)],
[cot(a+b*log(c*x^n))^5/x,x,4,1/2*cot(a+b*log(c*x^n))^2/(b*n)-1/4*cot(a+b*log(c*x^n))^4/(b*n)+log(sin(a+b*log(c*x^n)))/(b*n)],

# Integrands of the form (e x)^m Cot[a+b Log[c x^n]]^p with m symbolic
[(e*x)^m*cot(d*(a+b*log(c*x^n))),x,4,I*(e*x)^(1+m)/(e*(1+m))-2*I*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m))],
[(e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x,5,(I*(1+m)-b*d*n)*(e*x)^(1+m)/(b*d*e*(1+m)*n)+I*(e*x)^(1+m)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*e*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-2*I*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b*d*e*n)],
[(e*x)^m*cot(d*(a+b*log(c*x^n)))^3,x,6,1/2*(I*(1+m)-b*d*n)*(1+m+2*I*b*d*n)*(e*x)^(1+m)/(b^2*d^2*e*(1+m)*n^2)+1/2*(e*x)^(1+m)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^2/(b*d*e*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^2)+1/2*I*(e*x)^(1+m)*(exp(2*I*a*d)*(1+m-2*I*b*d*n)/n+exp(4*I*a*d)*(1+m+2*I*b*d*n)*(c*x^n)^(2*I*b*d)/n)/(b^2*d^2*e*exp(2*I*a*d)*n*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))-I*(1+2*m+m^2-2*b^2*d^2*n^2)*(e*x)^(1+m)*hypergeom([1,-1/2*I*(1+m)/(b*d*n)],[1-1/2*I*(1+m)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(b^2*d^2*e*(1+m)*n^2)],

# Integrands of the form (e x)^m Cot[a+b Log[c x^n]]^p with p symbolic
[cot(d*(a+b*log(c*x^n)))^p,x,5,x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*AppellF1((-1/2*I)/(b*d*n),p,-p,1+(-1/2*I)/(b*d*n),exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p],
[(e*x)^m*cot(d*(a+b*log(c*x^n)))^p,x,5,(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*AppellF1(-1/2*I*(1+m)/(b*d*n),p,-p,1-1/2*I*(1+m)/(b*d*n),exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)],

# Integrands of the form x^m Cot[a+b Log[c x^n]]^(p/2)
[cot(a+b*log(c*x^n))^(5/2)/x,x,13,-2/3*cot(a+b*log(c*x^n))^(3/2)/(b*n)-arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))],
[cot(a+b*log(c*x^n))^(3/2)/x,x,13,-arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-2*sqrt(cot(a+b*log(c*x^n)))/(b*n)],
[cot(a+b*log(c*x^n))^(1/2)/x,x,12,arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))],
[1/(x*cot(a+b*log(c*x^n))^(1/2)),x,12,arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))],
[1/(x*cot(a+b*log(c*x^n))^(3/2)),x,13,-arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+2/(b*n*sqrt(cot(a+b*log(c*x^n))))],
[1/(x*cot(a+b*log(c*x^n))^(5/2)),x,13,2/3/(b*n*cot(a+b*log(c*x^n))^(3/2))-arctan(1-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+arctan(1+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))-1/2*log(1+cot(a+b*log(c*x^n))-sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))+1/2*log(1+cot(a+b*log(c*x^n))+sqrt(2)*sqrt(cot(a+b*log(c*x^n))))/(b*n*sqrt(2))],

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

# Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p
[x^2*sec(a+b*log(c*x^n)),x,3,2*exp(I*a)*x^3*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-3*I)/(b*n))],[3/2*(1+(-I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(3+I*b*n)],
[x*sec(a+b*log(c*x^n)),x,3,2*exp(I*a)*x^2*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-2*I)/(b*n))],[1/2*(3+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(2+I*b*n)],
[sec(a+b*log(c*x^n)),x,3,2*exp(I*a)*x*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-I)/(b*n))],[1/2*(3+(-I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+I*b*n)],
[sec(a+b*log(c*x^n))/x,x,2,arctanh(sin(a+b*log(c*x^n)))/(b*n)],
[sec(a+b*log(c*x^n))/x^2,x,3,-2*exp(I*a)*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+I/(b*n))],[1/2*(3+I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-I*b*n)*x)],
[sec(a+b*log(c*x^n))/x^3,x,3,-2*exp(I*a)*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+2*I/(b*n))],[1/2*(3+2*I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-I*b*n)*x^2)],
[x^2*sec(a+b*log(c*x^n))^2,x,3,4*exp(2*I*a)*x^3*(c*x^n)^(2*I*b)*hypergeom([2,1/2*(2+(-3*I)/(b*n))],[1/2*(4+(-3*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(3+2*I*b*n)],
[x*sec(a+b*log(c*x^n))^2,x,3,2*exp(2*I*a)*x^2*(c*x^n)^(2*I*b)*hypergeom([2,1+(-I)/(b*n)],[2+(-I)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+I*b*n)],
[sec(a+b*log(c*x^n))^2,x,3,4*exp(2*I*a)*x*(c*x^n)^(2*I*b)*hypergeom([2,1/2*(2+(-I)/(b*n))],[1/2*(4+(-I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+2*I*b*n)],
[sec(a+b*log(c*x^n))^2/x,x,3,tan(a+b*log(c*x^n))/(b*n)],
[sec(a+b*log(c*x^n))^2/x^2,x,3,-4*exp(2*I*a)*(c*x^n)^(2*I*b)*hypergeom([2,1/2*(2+I/(b*n))],[1/2*(4+I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-2*I*b*n)*x)],
[sec(a+b*log(c*x^n))^2/x^3,x,3,-2*exp(2*I*a)*(c*x^n)^(2*I*b)*hypergeom([2,1+I/(b*n)],[2+I/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-I*b*n)*x^2)],
[x*sec(a+b*log(c*x^n))^3,x,3,8*exp(3*I*a)*x^2*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(3+(-2*I)/(b*n))],[1/2*(5+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(2+3*I*b*n)],
[sec(a+b*log(c*x^n))^3,x,3,8*exp(3*I*a)*x*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(3+(-I)/(b*n))],[1/2*(5+(-I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+3*I*b*n)],
[sec(a+b*log(c*x^n))^3/x,x,3,1/2*arctanh(sin(a+b*log(c*x^n)))/(b*n)+1/2*sec(a+b*log(c*x^n))*tan(a+b*log(c*x^n))/(b*n)],
[sec(a+b*log(c*x^n))^3/x^2,x,3,-8*exp(3*I*a)*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(3+I/(b*n))],[1/2*(5+I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-3*I*b*n)*x)],
[sec(a+b*log(c*x^n))^3/x^3,x,3,-8*exp(3*I*a)*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(3+2*I/(b*n))],[1/2*(5+2*I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-3*I*b*n)*x^2)],
[x*sec(a+b*log(c*x^n))^4,x,3,8*exp(4*I*a)*x^2*(c*x^n)^(4*I*b)*hypergeom([4,2+(-I)/(b*n)],[3+(-I)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+2*I*b*n)],
[sec(a+b*log(c*x^n))^4,x,3,16*exp(4*I*a)*x*(c*x^n)^(4*I*b)*hypergeom([4,1/2*(4+(-I)/(b*n))],[1/2*(6+(-I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+4*I*b*n)],
[sec(a+b*log(c*x^n))^4/x,x,3,tan(a+b*log(c*x^n))/(b*n)+1/3*tan(a+b*log(c*x^n))^3/(b*n)],
[sec(a+b*log(c*x^n))^4/x^2,x,3,-16*exp(4*I*a)*(c*x^n)^(4*I*b)*hypergeom([4,1/2*(4+I/(b*n))],[1/2*(6+I/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-4*I*b*n)*x)],
[sec(a+b*log(c*x^n))^4/x^3,x,3,-8*exp(4*I*a)*(c*x^n)^(4*I*b)*hypergeom([4,2+I/(b*n)],[3+I/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((1-2*I*b*n)*x^2)],
[-(1+b^2*n^2)*sec(a+b*log(c*x^n))+2*b^2*n^2*sec(a+b*log(c*x^n))^3,x,-7,-x*sec(a+b*log(c*x^n))+b*n*x*sec(a+b*log(c*x^n))*tan(a+b*log(c*x^n))],

# Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when b^2 n^2 (p-2)^2+(m+1)^2=0
[x^m*sec(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))^3,x,-3,1/2*x^(1+m)*sec(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))/(1+m)+1/2*x^(1+m)*sec(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))*tan(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))/sqrt(-(1+m)^2)],
[x*sec(a+2*log(c*x^I))^3,x,3,exp(I*a)*(c*x^I)^(2*I)*x^2/(1+exp(2*I*a)*(c*x^I)^(4*I))^2],
[sec(a+2*log(c*x^(1/2*I)))^3,x,3,1/2*x*sec(a+2*log(c*x^(1/2*I)))-1/2*I*x*sec(a+2*log(c*x^(1/2*I)))*tan(a+2*log(c*x^(1/2*I))),2*exp(I*a)*(c*x^(1/2*I))^(2*I)*x/(1+exp(2*I*a)*(c*x^(1/2*I))^(4*I))^2],
[sec(a+2*log(c/x^(1/2*I)))^3,x,3,2*exp(3*I*a)*(c/x^(1/2*I))^(6*I)*x/(1+exp(2*I*a)*(c/x^(1/2*I))^(4*I))^2],
[sec(a+I*log(c*x^n)/(n*(-2+p)))^p,x,3,1/2*(2-p)*x*(1+exp(2*I*a)*(c*x^n)^(2/(n*(2-p))))*sec(a-I*log(c*x^n)/(n*(2-p)))^p/(exp(2*I*a)*(1-p)*(c*x^n)^(2/(n*(2-p))))],
[sec(a-I*log(c*x^n)/(n*(-2+p)))^p,x,3,1/2*(2-p)*x*(1+exp(2*I*a)/(c*x^n)^(2/(n*(2-p))))*sec(a+I*log(c*x^n)/(n*(2-p)))^p/(1-p)],

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

# p>0
[sqrt(sec(a+b*log(c*x^n))),x,3,2*x*hypergeom([1/2,1/4*(1+(-2*I)/(b*n))],[1/4*(5+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sec(a+b*log(c*x^n)))/(2+I*b*n)],
[sqrt(sec(a+b*log(c*x^n)))/x,x,3,2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],
[sec(a+b*log(c*x^n))^(3/2),x,3,2*x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2,1/4*(3+(-2*I)/(b*n))],[1/4*(7+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^(3/2)/(2+3*I*b*n)],
[sec(a+b*log(c*x^n))^(3/2)/x,x,4,2*sin(a+b*log(c*x^n))*sqrt(sec(a+b*log(c*x^n)))/(b*n)-2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],
[sec(a+b*log(c*x^n))^(5/2),x,3,2*x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2,1/4*(5+(-2*I)/(b*n))],[1/4*(9+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^(5/2)/(2+5*I*b*n)],
[sec(a+b*log(c*x^n))^(5/2)/x,x,4,2/3*sec(a+b*log(c*x^n))^(3/2)*sin(a+b*log(c*x^n))/(b*n)+2/3*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],

# p<0
[1/sqrt(sec(a+b*log(c*x^n))),x,3,2*x*hypergeom([-1/2,1/4*(-2*I-b*n)/(b*n)],[1/4*(3+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-I*b*n)*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sec(a+b*log(c*x^n))))],
[1/(x*sqrt(sec(a+b*log(c*x^n)))),x,3,2*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],
[1/sec(a+b*log(c*x^n))^(3/2),x,3,2*x*hypergeom([-3/2,1/4*(-3+(-2*I)/(b*n))],[1/4*(1+(-2*I)/(b*n))],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-3*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*sec(a+b*log(c*x^n))^(3/2))],
[1/(x*sec(a+b*log(c*x^n))^(3/2)),x,4,2/3*sin(a+b*log(c*x^n))/(b*n*sqrt(sec(a+b*log(c*x^n))))+2/3*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticF(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],
[1/sec(a+b*log(c*x^n))^(5/2),x,3,2*x*hypergeom([-5/2,1/4*(-5+(-2*I)/(b*n))],[1/4*(-2*I-b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2-5*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*sec(a+b*log(c*x^n))^(5/2))],
[1/(x*sec(a+b*log(c*x^n))^(5/2)),x,4,2/5*sin(a+b*log(c*x^n))/(b*n*sec(a+b*log(c*x^n))^(3/2))+6/5*sqrt(cos(1/2*(a+b*log(c*x^n)))^2)/cos(1/2*(a+b*log(c*x^n)))*EllipticE(sin(1/2*(a+b*log(c*x^n))),sqrt(2))*sqrt(cos(a+b*log(c*x^n)))*sqrt(sec(a+b*log(c*x^n)))/(b*n)],

# Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when m symbolic
[x^m*sec(a+b*log(c*x^n))^3,x,3,8*exp(3*I*a)*x^(1+m)*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(-I*(1+m)+3*b*n)/(b*n)],[1/2*(-I*(1+m)+5*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+m+3*I*b*n)],
[x^m*sec(a+b*log(c*x^n))^2,x,3,4*exp(2*I*a)*x^(1+m)*(c*x^n)^(2*I*b)*hypergeom([2,1/2*(-I*(1+m)+2*b*n)/(b*n)],[1/2*(-I*(1+m)+4*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+m+2*I*b*n)],
[x^m*sec(a+b*log(c*x^n)),x,3,2*exp(I*a)*x^(1+m)*(c*x^n)^(I*b)*hypergeom([1,1/2*(-I-I*m+b*n)/(b*n)],[1/2*(-I*(1+m)+3*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1+m+I*b*n)],
[x^m*sec(a+b*log(c*x^n))^(5/2),x,3,2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2,1/4*(-2*I-2*I*m+5*b*n)/(b*n)],[1/4*(-2*I-2*I*m+9*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^(5/2)/(2+2*m+5*I*b*n)],
[x^m*sec(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2,1/4*(-2*I-2*I*m+3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+7*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^(3/2)/(2+2*m+3*I*b*n)],
[x^m*sec(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([1/2,1/4*(-2*I-2*I*m+b*n)/(b*n)],[1/4*(-2*I-2*I*m+5*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sec(a+b*log(c*x^n)))/(2+2*m+I*b*n)],
[x^m/sec(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([-1/2,1/4*(-2*I-2*I*m-b*n)/(b*n)],[1/4*(-2*I-2*I*m+3*b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m-I*b*n)*sqrt(1+exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(sec(a+b*log(c*x^n))))],
[x^m/sec(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*hypergeom([-3/2,1/4*(-2*I-2*I*m-3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+b*n)/(b*n)],-exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m-3*I*b*n)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*sec(a+b*log(c*x^n))^(3/2))],

# Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when p symbolic
[(e*x)^m*sec(d*(a+b*log(c*x^n)))^p,x,3,(e*x)^(1+m)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*hypergeom([p,1/2*(-I-I*m+b*d*n*p)/(b*d*n)],[1/2*(2-I*(1+m)/(b*d*n)+p)],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sec(d*(a+b*log(c*x^n)))^p/(e*(1+m+I*b*d*n*p))],
[x*sec(a+b*log(c*x^n))^p,x,3,x^2*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^p*hypergeom([p,1/2*((-2*I)/(b*n)+p)],[1/2*(2+(-2*I)/(b*n)+p)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^p/(2+I*b*n*p)],
[sec(a+b*log(c*x^n))^p,x,3,x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^p*hypergeom([p,1/2*(-I+b*n*p)/(b*n)],[1/2*(2+(-I)/(b*n)+p)],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*log(c*x^n))^p/(1+I*b*n*p)],

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

# Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p
[x^2*csc(a+b*log(c*x^n)),x,3,2*exp(I*a)*x^3*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-3*I)/(b*n))],[3/2*(1+(-I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(3*I-b*n)],
[x*csc(a+b*log(c*x^n)),x,3,2*exp(I*a)*x^2*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-2*I)/(b*n))],[1/2*(3+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(2*I-b*n)],
[csc(a+b*log(c*x^n)),x,3,2*exp(I*a)*x*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+(-I)/(b*n))],[1/2*(3+(-I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(I-b*n)],
[csc(a+b*log(c*x^n))/x,x,2,-arctanh(cos(a+b*log(c*x^n)))/(b*n)],
[csc(a+b*log(c*x^n))/x^2,x,3,-2*exp(I*a)*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+I/(b*n))],[1/2*(3+I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((I+b*n)*x)],
[csc(a+b*log(c*x^n))/x^3,x,3,-2*exp(I*a)*(c*x^n)^(I*b)*hypergeom([1,1/2*(1+2*I/(b*n))],[1/2*(3+2*I/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((2*I+b*n)*x^2)],
[csc(a+b*log(c*x^n))^2,x,3,-4*exp(2*I*a)*x*(c*x^n)^(2*I*b)*hypergeom([2,1/2*(2+(-I)/(b*n))],[1/2*(4+(-I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(1+2*I*b*n)],
[csc(a+b*log(c*x^n))^2/x,x,3,-cot(a+b*log(c*x^n))/(b*n)],
[csc(a+b*log(c*x^n))^3,x,3,-8*exp(3*I*a)*x*(c*x^n)^(3*I*b)*hypergeom([3,1/2*(3+(-I)/(b*n))],[1/2*(5+(-I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(I-3*b*n)],
[csc(a+b*log(c*x^n))^3/x,x,3,-1/2*arctanh(cos(a+b*log(c*x^n)))/(b*n)-1/2*cot(a+b*log(c*x^n))*csc(a+b*log(c*x^n))/(b*n)],
[csc(a+b*log(c*x^n))^4,x,3,16*exp(4*I*a)*x*(c*x^n)^(4*I*b)*hypergeom([4,1/2*(4+(-I)/(b*n))],[1/2*(6+(-I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(1+4*I*b*n)],
[csc(a+b*log(c*x^n))^4/x,x,3,-cot(a+b*log(c*x^n))/(b*n)-1/3*cot(a+b*log(c*x^n))^3/(b*n)],
[-(1+b^2*n^2)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3,x,-7,-x*csc(a+b*log(c*x^n))-b*n*x*cot(a+b*log(c*x^n))*csc(a+b*log(c*x^n))],

# Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when b^2 n^2 (p-2)^2+(m+1)^2=0
[x^m*csc(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))^3,x,-3,1/2*x^(1+m)*csc(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))/(1+m)-1/2*x^(1+m)*cot(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))*csc(a+2*log(c*x^(1/2*sqrt(-(1+m)^2))))/sqrt(-(1+m)^2)],
[x*csc(a+2*log(c*x^I))^3,x,3,-I*exp(I*a)*(c*x^I)^(2*I)*x^2/(1-exp(2*I*a)*(c*x^I)^(4*I))^2],
[csc(a+2*log(c*x^(1/2*I)))^3,x,3,1/2*x*csc(a+2*log(c*x^(1/2*I)))+1/2*I*x*cot(a+2*log(c*x^(1/2*I)))*csc(a+2*log(c*x^(1/2*I))),-2*I*exp(I*a)*(c*x^(1/2*I))^(2*I)*x/(1-exp(2*I*a)*(c*x^(1/2*I))^(4*I))^2],
[csc(a+2*log(c/x^(1/2*I)))^3,x,3,2*I*exp(3*I*a)*(c/x^(1/2*I))^(6*I)*x/(1-exp(2*I*a)*(c/x^(1/2*I))^(4*I))^2],
[csc(a+I*log(c*x^n)/(n*(-2+p)))^p,x,3,-1/2*(2-p)*x*(1-exp(2*I*a)*(c*x^n)^(2/(n*(2-p))))*csc(a-I*log(c*x^n)/(n*(2-p)))^p/(exp(2*I*a)*(1-p)*(c*x^n)^(2/(n*(2-p))))],
[csc(a-I*log(c*x^n)/(n*(-2+p)))^p,x,3,1/2*(2-p)*x*(1-exp(2*I*a)/(c*x^n)^(2/(n*(2-p))))*csc(a+I*log(c*x^n)/(n*(2-p)))^p/(1-p)],

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

# p>0
[sqrt(csc(a+b*log(c*x^n))),x,3,2*x*hypergeom([1/2,1/4*(1+(-2*I)/(b*n))],[1/4*(5+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(csc(a+b*log(c*x^n)))/(2+I*b*n)],
[sqrt(csc(a+b*log(c*x^n)))/x,x,3,2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],
[csc(a+b*log(c*x^n))^(3/2),x,3,2*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*csc(a+b*log(c*x^n))^(3/2)*hypergeom([3/2,1/4*(3+(-2*I)/(b*n))],[1/4*(7+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+3*I*b*n)],
[csc(a+b*log(c*x^n))^(3/2)/x,x,4,-2*cos(a+b*log(c*x^n))*sqrt(csc(a+b*log(c*x^n)))/(b*n)-2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticE(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],
[csc(a+b*log(c*x^n))^(5/2),x,3,2*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*csc(a+b*log(c*x^n))^(5/2)*hypergeom([5/2,1/4*(5+(-2*I)/(b*n))],[1/4*(9+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+5*I*b*n)],
[csc(a+b*log(c*x^n))^(5/2)/x,x,4,-2/3*cos(a+b*log(c*x^n))*csc(a+b*log(c*x^n))^(3/2)/(b*n)+2/3*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],

# p<0
[1/sqrt(csc(a+b*log(c*x^n))),x,3,2*x*hypergeom([-1/2,1/4*(-2*I-b*n)/(b*n)],[1/4*(3+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((2-I*b*n)*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(csc(a+b*log(c*x^n))))],
[1/(x*sqrt(csc(a+b*log(c*x^n)))),x,3,2*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticE(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],
[1/csc(a+b*log(c*x^n))^(3/2),x,3,2*x*hypergeom([-3/2,1/4*(-3+(-2*I)/(b*n))],[1/4*(1+(-2*I)/(b*n))],exp(2*I*a)*(c*x^n)^(2*I*b))/((2-3*I*b*n)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*csc(a+b*log(c*x^n))^(3/2))],
[1/(x*csc(a+b*log(c*x^n))^(3/2)),x,4,-2/3*cos(a+b*log(c*x^n))/(b*n*sqrt(csc(a+b*log(c*x^n))))+2/3*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticF(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],
[1/csc(a+b*log(c*x^n))^(5/2),x,3,2*x*hypergeom([-5/2,1/4*(-5+(-2*I)/(b*n))],[1/4*(-2*I-b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))/((2-5*I*b*n)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*csc(a+b*log(c*x^n))^(5/2))],
[1/(x*csc(a+b*log(c*x^n))^(5/2)),x,4,-2/5*cos(a+b*log(c*x^n))/(b*n*csc(a+b*log(c*x^n))^(3/2))+6/5*sqrt(cos(1/2*(a-1/2*Pi+b*log(c*x^n)))^2)/cos(1/2*(a-1/2*Pi+b*log(c*x^n)))*EllipticE(sin(1/2*(a-1/2*Pi+b*log(c*x^n))),sqrt(2))*sqrt(csc(a+b*log(c*x^n)))*sqrt(sin(a+b*log(c*x^n)))/(b*n)],

# Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when m symbolic
[(e*x)^m*csc(d*(a+b*log(c*x^n)))^3,x,3,-8*exp(3*I*a*d)*(e*x)^(1+m)*(c*x^n)^(3*I*b*d)*hypergeom([3,1/2*(-I*(1+m)+3*b*d*n)/(b*d*n)],[1/2*(-I*(1+m)+5*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(I*(1+m)-3*b*d*n))],
[(e*x)^m*csc(d*(a+b*log(c*x^n)))^2,x,3,-4*exp(2*I*a*d)*(e*x)^(1+m)*(c*x^n)^(2*I*b*d)*hypergeom([2,1/2*(-I*(1+m)+2*b*d*n)/(b*d*n)],[1/2*(-I*(1+m)+4*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m+2*I*b*d*n))],
[(e*x)^m*csc(d*(a+b*log(c*x^n))),x,3,2*exp(I*a*d)*(e*x)^(1+m)*(c*x^n)^(I*b*d)*hypergeom([1,1/2*(-I-I*m+b*d*n)/(b*d*n)],[1/2*(-I*(1+m)+3*b*d*n)/(b*d*n)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(I*(1+m)-b*d*n))],
[x^m*csc(a+b*log(c*x^n))^(5/2),x,3,2*x^(1+m)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*csc(a+b*log(c*x^n))^(5/2)*hypergeom([5/2,1/4*(-2*I-2*I*m+5*b*n)/(b*n)],[1/4*(-2*I-2*I*m+9*b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+2*m+5*I*b*n)],
[x^m*csc(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*csc(a+b*log(c*x^n))^(3/2)*hypergeom([3/2,1/4*(-2*I-2*I*m+3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+7*b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+2*m+3*I*b*n)],
[x^m*csc(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([1/2,1/4*(-2*I-2*I*m+b*n)/(b*n)],[1/4*(-2*I-2*I*m+5*b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(csc(a+b*log(c*x^n)))/(2+2*m+I*b*n)],
[x^m/csc(a+b*log(c*x^n))^(1/2),x,3,2*x^(1+m)*hypergeom([-1/2,1/4*(-2*I-2*I*m-b*n)/(b*n)],[1/4*(-2*I-2*I*m+3*b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m-I*b*n)*sqrt(1-exp(2*I*a)*(c*x^n)^(2*I*b))*sqrt(csc(a+b*log(c*x^n))))],
[x^m/csc(a+b*log(c*x^n))^(3/2),x,3,2*x^(1+m)*hypergeom([-3/2,1/4*(-2*I-2*I*m-3*b*n)/(b*n)],[1/4*(-2*I-2*I*m+b*n)/(b*n)],exp(2*I*a)*(c*x^n)^(2*I*b))/((2+2*m-3*I*b*n)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*csc(a+b*log(c*x^n))^(3/2))],

# Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when p symbolic
[(e*x)^m*csc(d*(a+b*log(c*x^n)))^p,x,3,(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*csc(d*(a+b*log(c*x^n)))^p*hypergeom([p,1/2*(-I-I*m+b*d*n*p)/(b*d*n)],[1/2*(2-I*(1+m)/(b*d*n)+p)],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/(e*(1+m+I*b*d*n*p))],
[x*csc(a+b*log(c*x^n))^p,x,3,x^2*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p*csc(a+b*log(c*x^n))^p*hypergeom([p,1/2*((-2*I)/(b*n)+p)],[1/2*(2+(-2*I)/(b*n)+p)],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+I*b*n*p)],
[csc(a+b*log(c*x^n))^p,x,3,x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p*csc(a+b*log(c*x^n))^p*hypergeom([p,1/2*(-I+b*n*p)/(b*n)],[1/2*(2+(-I)/(b*n)+p)],exp(2*I*a)*(c*x^n)^(2*I*b))/(1+I*b*n*p)]]:
