\[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx \]
Optimal antiderivative \[ \frac {2 x}{a^{2}}+\frac {x^{3}}{3}+\frac {\arctanh \left (\sqrt {1+\frac {1}{a^{2} x^{2}}}\right )}{a^{3}}+\frac {x^{2} \sqrt {1+\frac {1}{a^{2} x^{2}}}}{a} \]
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
\[ \frac {\sqrt {a^{2} x^{2} + 1} x {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {a^{2} x^{3} + 6 \, x}{3 \, a^{2}} - \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) \mathrm {sgn}\left (x\right )}{a^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________