\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx \]
Optimal antiderivative \[ \frac {11 a^{3} \mathrm {arccsc}\! \left (a x \right )}{2}-\frac {\left (a +\frac {1}{x}\right )^{3}}{\sqrt {1-\frac {1}{a^{2} x^{2}}}}-\frac {a \left (3 a +\frac {1}{x}\right )^{2} \sqrt {1-\frac {1}{a^{2} x^{2}}}}{3}-\frac {a^{2} \left (28 a +\frac {3}{x}\right ) \sqrt {1-\frac {1}{a^{2} x^{2}}}}{6} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^4,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
\[ -\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {12 \, a^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} + \frac {\frac {52 \, {\left (a x - 1\right )} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {21 \, {\left (a x - 1\right )}^{2} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 39 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]