\[ \int e^{-\text {sech}^{-1}(a x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (a x +1\right )}{a}+\frac {2 \ln \left (1+\sqrt {\frac {-a x +1}{a x +1}}\right )}{a}-\frac {\left (a x +1\right ) \sqrt {\frac {-a x +1}{a x +1}}}{a} \]
command
integrate(1/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - 2 a^{2} \left (\begin {cases} \frac {2 \left (\frac {\sqrt {-1 + \frac {1}{a x}}}{2 \sqrt {1 + \frac {1}{a x}} \left (\frac {-1 + \frac {1}{a x}}{1 + \frac {1}{a x}} + 1\right )} - \frac {\log {\left (\frac {\sqrt {-1 + \frac {1}{a x}}}{\sqrt {1 + \frac {1}{a x}}} + 1 \right )}}{2} + \frac {\log {\left (\frac {-1 + \frac {1}{a x}}{1 + \frac {1}{a x}} + 1 \right )}}{4}\right )}{a^{3}} & \text {for}\: \sqrt {1 + \frac {1}{a x}} > - \sqrt {2} \wedge \sqrt {1 + \frac {1}{a x}} < \sqrt {2} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ a \int \frac {x}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]________________________________________________________________________________________