\[ \int e^{3 \coth ^{-1}(a x)} x \, dx \]
Optimal antiderivative \[ \frac {9 \arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )}{2 a^{2}}-\frac {4 \sqrt {1-\frac {1}{a^{2} x^{2}}}}{a \left (a -\frac {1}{x}\right )}+\frac {3 x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{a}+\frac {x^{2} \sqrt {1-\frac {1}{a^{2} x^{2}}}}{2} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{2} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{3}} - \frac {8}{a^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {2 \, {\left (\frac {5 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 7 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]