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