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