\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {a}{35 c \left (a^{2} c +d \right ) \left (d \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {a \left (11 a^{2} c +6 d \right )}{105 c^{2} \left (a^{2} c +d \right )^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {x \,\mathrm {arccoth}\left (a x \right )}{7 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {6 x \,\mathrm {arccoth}\left (a x \right )}{35 c^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {8 x \,\mathrm {arccoth}\left (a x \right )}{35 c^{3} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {\left (35 a^{6} c^{3}+70 a^{4} c^{2} d +56 a^{2} c \,d^{2}+16 d^{3}\right ) \arctanh \left (\frac {a \sqrt {d \,x^{2}+c}}{\sqrt {a^{2} c +d}}\right )}{35 c^{4} \left (a^{2} c +d \right )^{\frac {7}{2}}}+\frac {a \left (19 a^{4} c^{2}+22 a^{2} c d +8 d^{2}\right )}{35 c^{3} \left (a^{2} c +d \right )^{3} \sqrt {d \,x^{2}+c}}+\frac {16 x \,\mathrm {arccoth}\left (a x \right )}{35 c^{4} \sqrt {d \,x^{2}+c}} \]
command
integrate(arccoth(a*x)/(d*x^2+c)^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} + 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} + 16 \, d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{6} c^{7} + 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} + c^{4} d^{3}\right )} \sqrt {-a^{2} c - d} a} + \frac {57 \, {\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \, {\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} + 66 \, {\left (d x^{2} + c\right )}^{2} a^{2} c d + 17 \, {\left (d x^{2} + c\right )} a^{2} c^{2} d + 6 \, a^{2} c^{3} d + 24 \, {\left (d x^{2} + c\right )}^{2} d^{2} + 6 \, {\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} + 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{70 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]________________________________________________________________________________________