\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx \]
Optimal antiderivative \[ 3 a \,\mathrm {arccsc}\! \left (a x \right )-\frac {2 \left (a +\frac {1}{x}\right )^{2}}{a \sqrt {1-\frac {1}{a^{2} x^{2}}}}-3 a \sqrt {1-\frac {1}{a^{2} x^{2}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^2,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
\[ -2 \, a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 2}{\frac {{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \sqrt {\frac {a x - 1}{a x + 1}}} + 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]