\[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]
Optimal antiderivative \[ -\frac {2 \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \left (\csc ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 b n}-\frac {2 \sqrt {\frac {1}{2}+\frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \EllipticF \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \left (\sqrt {\sin }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b n} \]
command
integrate(csc(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-i \, \sqrt {2 i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {-2 i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) - \frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{3 \, b n \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]