\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a x +1\right )^{2}}{3 a^{2} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}} x}-\frac {2 \left (-a x +1\right )^{2} \left (a x +1\right ) \left (2 a x +5\right )}{3 a^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}} x^{3}}-\frac {2 \left (-a x +1\right )^{\frac {3}{2}} \left (a x +1\right )^{\frac {3}{2}} \arcsin \! \left (a x \right )}{a^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}} x^{3}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {6 \, {\left (a x + 1\right )} a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} + \frac {24 \, a^{2} c \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {a^{2} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{2} + 15 \, a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} c^{3}}{c^{3}}}{6 \, a^{3} c^{2} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]