\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a x +1\right )^{2}}{3 a^{2} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x}-\frac {10 \left (-a x +1\right )^{3}}{3 a^{3} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{2}}-\frac {12 \left (-a x +1\right )^{4}}{7 a^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{3}}-\frac {82 \left (-a x +1\right )^{4} \left (a x +1\right )}{105 a^{5} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{4}}-\frac {2 \left (-a x +1\right )^{4} \left (a x +1\right )^{2}}{35 a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{5}}-\frac {2 \left (-a x +1\right )^{4} \left (a x +1\right )^{3} \left (37 a x +72\right )}{35 a^{8} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7}}-\frac {2 \left (-a x +1\right )^{\frac {7}{2}} \left (a x +1\right )^{\frac {7}{2}} \arcsin \! \left (a x \right )}{a^{8} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(7/2),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
\[ -\frac {{\left (a x + 1\right )} \sqrt {c - \frac {2 \, c}{a x + 1}}}{a c^{4} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} - \frac {4 \, \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{a \sqrt {-c} c^{3} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} + \frac {14 \, c - \frac {27 \, c}{a x + 1}}{48 \, a {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{3} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} - \frac {15 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {7}{2}} c^{42} + 189 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {5}{2}} c^{43} + 1330 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{44} + 10710 \, a^{6} \sqrt {c - \frac {2 \, c}{a x + 1}} c^{45}}{3360 \, a^{7} c^{49} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]