\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx \]
Optimal antiderivative \[ \frac {\left (c^{4}+\frac {1}{x^{4}}\right ) x^{2} \mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{2}-\frac {\left (c^{4}+\frac {1}{x^{4}}\right ) \left (c^{2}+\frac {1}{x^{2}}\right ) x^{3} \sqrt {\frac {\cos \left (4 \,\mathrm {arccot}\left (c x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \,\mathrm {arccot}\left (c x \right )\right ), \frac {\sqrt {2}}{2}\right ) \mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}} \sqrt {\frac {c^{4}+\frac {1}{x^{4}}}{\left (c^{2}+\frac {1}{x^{2}}\right )^{2}}}}{4 \cos \left (2 \,\mathrm {arccot}\left (c x \right )\right ) c} \]
command
integrate(sech(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
\[ \frac {\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{3} - \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\left (-c^{4}\right )^{\frac {1}{4}} x, -1\right )}{c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{3}}, x\right ) \]