\[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx \]
Optimal antiderivative \[ -a \arctanh \! \left (\sqrt {\frac {-a x +1}{a x +1}}\right )-\frac {a}{\left (1+\sqrt {\frac {-a x +1}{a x +1}}\right )^{2}}+\frac {a}{1+\sqrt {\frac {-a x +1}{a x +1}}} \]
command
integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/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
\[ -\frac {1}{2} \, {\left (\sqrt {a^{2} + \frac {a}{x}} \sqrt {-a^{2} + \frac {a}{x}} {\left (\frac {1}{a^{2}} - \frac {a^{2} + \frac {a}{x}}{a^{4}}\right )} - \frac {2 \, {\left (a^{2} + \frac {a}{x}\right )} a^{2} - {\left (a^{2} + \frac {a}{x}\right )}^{2}}{a^{4}} - 2 \, \log \left (\sqrt {a^{2} + \frac {a}{x}} - \sqrt {-a^{2} + \frac {a}{x}}\right )\right )} a \]