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