\[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 a^{2} x^{4}}-\frac {1}{2 x^{2}}+\frac {a^{2} \mathrm {arccsch}\left (a x \right )}{4}-\frac {\sqrt {1+\frac {1}{a^{2} x^{2}}}}{2 x^{3} a}-\frac {a \sqrt {1+\frac {1}{a^{2} x^{2}}}}{4 x} \]
command
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left ({\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + \sqrt {a^{2} x^{2} + 1} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2 \, {\left (a^{2} x^{2} + 1\right )} a^{7}\right )}}{a^{4} x^{4}}}{8 \, a^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________