\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {\left (c^{4}+\frac {1}{x^{4}}\right ) \sqrt {\mathrm {sech}\left (2 \ln \left (c x \right )\right )}}{3}+\frac {c^{3} \left (c^{2}+\frac {1}{x^{2}}\right ) x \sqrt {\frac {\cos \left (4 \,\mathrm {arccot}\left (c x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \,\mathrm {arccot}\left (c x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c^{4}+\frac {1}{x^{4}}}{\left (c^{2}+\frac {1}{x^{2}}\right )^{2}}}\, \sqrt {\mathrm {sech}\left (2 \ln \left (c x \right )\right )}}{6 \cos \left (2 \,\mathrm {arccot}\left (c x \right )\right )} \]
command
integrate(sech(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
\[ \frac {\sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} c x^{4} {\rm ellipticF}\left (\left (-c^{4}\right )^{\frac {1}{4}} x, -1\right ) - \sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{3 \, x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{5}}, x\right ) \]