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