\[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx \]
Optimal antiderivative \[ \frac {2 x^{3}}{3 a^{2}}+\frac {x^{5}}{5}-\frac {\arctanh \left (\sqrt {1+\frac {1}{a^{2} x^{2}}}\right )}{4 a^{5}}+\frac {x^{2} \sqrt {1+\frac {1}{a^{2} x^{2}}}}{4 a^{3}}+\frac {x^{4} \sqrt {1+\frac {1}{a^{2} x^{2}}}}{2 a} \]
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
\[ \frac {1}{4} \, \sqrt {a^{2} x^{2} + 1} x {\left (\frac {2 \, x^{2} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {{\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{5}}\right )} + \frac {3 \, a^{2} x^{5} + 10 \, x^{3}}{15 \, a^{2}} + \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) \mathrm {sgn}\left (x\right )}{4 \, a^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________