\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {a^{5} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {5}{2}}}{\left (a -\frac {1}{x}\right )^{3}}-\frac {3 a^{3} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{2 \left (a -\frac {1}{x}\right )}+\frac {9 a^{2} \mathrm {arccsc}\! \left (a x \right )}{2}-\frac {9 a^{2} \sqrt {1-\frac {1}{a^{2} x^{2}}}}{2} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^3,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
\[ -{\left (9 \, a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {4 \, a}{\sqrt {\frac {a x - 1}{a x + 1}}} + \frac {\frac {5 \, {\left (a x - 1\right )} a \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + 7 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]