\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^5} \, dx \]
Optimal antiderivative \[ \frac {\left (c^{4}-\frac {1}{x^{4}}\right ) \sqrt {\mathrm {csch}\left (2 \ln \left (c x \right )\right )}}{3}-\frac {c^{5} x \EllipticF \left (\frac {1}{c x}, i\right ) \sqrt {1-\frac {1}{c^{4} x^{4}}}\, \sqrt {\mathrm {csch}\left (2 \ln \left (c x \right )\right )}}{3} \]
command
integrate(csch(2*log(c*x))^(1/2)/x^5,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{3 \, x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{5}}, x\right ) \]