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