\[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x} \, dx \]
Optimal antiderivative \[ -\frac {1}{a^{2} x^{2}}-\mathrm {arccsch}\left (a x \right )+\ln \left (x \right )-\frac {\sqrt {1+\frac {1}{a^{2} x^{2}}}}{a x} \]
command
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) - a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - {\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )}}{{\left (\sqrt {a^{2} x^{2} + 1} + 1\right )} {\left (\sqrt {a^{2} x^{2} + 1} - 1\right )}}}{2 \, a^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________