\[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (1+\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3}-\frac {2}{3 a^{2} x^{3}}-\frac {1}{x} \]
command
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a \mathrm {sgn}\left (x\right ) + a \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {3 \, a^{2} x^{2} + 2}{3 \, a^{2} x^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________