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