\[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^4} \, dx \]
Optimal antiderivative \[ \frac {2 a^{3} \left (1+\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3}-\frac {2 a^{3} \left (1+\frac {1}{a^{2} x^{2}}\right )^{\frac {5}{2}}}{5}-\frac {2}{5 a^{2} x^{5}}-\frac {1}{3 x^{3}} \]
command
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {8 \, {\left (15 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{6} a^{3} \mathrm {sgn}\left (x\right ) + 5 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a^{3} \mathrm {sgn}\left (x\right ) + 5 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} a^{3} \mathrm {sgn}\left (x\right ) - a^{3} \mathrm {sgn}\left (x\right )\right )}}{15 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{5}} - \frac {5 \, a^{2} x^{2} + 6}{15 \, a^{2} x^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________