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