\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a x +1\right )^{2}}{a^{2} x \sqrt {c -\frac {c}{a^{2} x^{2}}}}+\frac {2 \left (-a x +1\right ) \left (a x +1\right )}{a^{2} x \sqrt {c -\frac {c}{a^{2} x^{2}}}}+\frac {2 \arcsin \! \left (a x \right ) \sqrt {-a x +1}\, \sqrt {a x +1}}{a^{2} x \sqrt {c -\frac {c}{a^{2} x^{2}}}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(1/2),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
\[ -\frac {{\left (a x + 1\right )} a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} + \frac {4 \, a^{2} c \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}}}{a^{3} c \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]