\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx \]
Optimal antiderivative \[ \mathrm {arccsc}\! \left (a x \right )+\arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )-\frac {4 a \sqrt {1-\frac {1}{a^{2} x^{2}}}}{a -\frac {1}{x}} \]
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
\[ -a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} + \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a} + \frac {4}{a \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]