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