\[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {1}{3 a^{2} x^{6}}-\frac {1}{4 x^{4}}-\frac {a^{4} \mathrm {arccsch}\left (a x \right )}{8}-\frac {\sqrt {1+\frac {1}{a^{2} x^{2}}}}{3 a \,x^{5}}-\frac {a \sqrt {1+\frac {1}{a^{2} x^{2}}}}{12 x^{3}}+\frac {a^{3} \sqrt {1+\frac {1}{a^{2} x^{2}}}}{8 x} \]
command
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, a^{8} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - 3 \, a^{8} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {a^{2} x^{2} + 1} a^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 6 \, {\left (a^{2} x^{2} + 1\right )} a^{9} - 2 \, a^{9}\right )}}{a^{6} x^{6}}}{48 \, a^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________