\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{5 b n \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}-\frac {6 i \sqrt {\frac {1}{2}+\frac {\sin \left (i a +i b \ln \left (c \,x^{n}\right )\right )}{2}}\, \EllipticE \left (\cos \left (\frac {i a}{2}+\frac {\pi }{4}+\frac {i b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {i a}{2}+\frac {\pi }{4}+\frac {i b \ln \left (c \,x^{n}\right )}{2}\right ) b n \sqrt {\mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}} \]
command
integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{6} + 6 \, \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 )^{5} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{6} + {\left (15 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 11\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 11 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 11 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (15 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 66 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 13\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 13 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 22 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 13 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right )} \sqrt {\frac {\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 )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \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 ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1}} + 24 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, \sqrt {2} \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 )^{2} + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )} {\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 )}{20 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, b n \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 )^{2} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}, x\right ) \]