\[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {1}{25 a \left (-a \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {4}{45 a^{2} \left (-a \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {x \,\mathrm {arccoth}\left (x \right )}{5 a \left (-a \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {4 x \,\mathrm {arccoth}\left (x \right )}{15 a^{2} \left (-a \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {8}{15 a^{3} \sqrt {-a \,x^{2}+a}}+\frac {8 x \,\mathrm {arccoth}\left (x \right )}{15 a^{3} \sqrt {-a \,x^{2}+a}} \]
command
integrate(arccoth(x)/(-a*x^2+a)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {-a x^{2} + a} {\left (4 \, x^{2} {\left (\frac {2 \, x^{2}}{a} - \frac {5}{a}\right )} + \frac {15}{a}\right )} x \log \left (-\frac {\frac {1}{x} + 1}{\frac {1}{x} - 1}\right )}{30 \, {\left (a x^{2} - a\right )}^{3}} - \frac {120 \, {\left (a x^{2} - a\right )}^{2} - 20 \, {\left (a x^{2} - a\right )} a + 9 \, a^{2}}{225 \, {\left (a x^{2} - a\right )}^{2} \sqrt {-a x^{2} + a} a^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (x\right )}{{\left (-a x^{2} + a\right )}^{\frac {7}{2}}}\,{d x} \]________________________________________________________________________________________