\[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx \]
Optimal antiderivative \[ \frac {d \,x^{2}}{6 a}+c x \,\mathrm {arccoth}\left (a x \right )+\frac {d \,x^{3} \mathrm {arccoth}\left (a x \right )}{3}+\frac {\left (3 a^{2} c +d \right ) \ln \left (-a^{2} x^{2}+1\right )}{6 a^{3}} \]
command
integrate((d*x^2+c)*arccoth(a*x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c + d\right )} \log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {{\left (3 \, a^{2} c + d\right )} \log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (a x + 1\right )} d}{{\left (a x - 1\right )} a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )}^{2} a^{2} c}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{2} c}{a x - 1} + 3 \, a^{2} c + \frac {3 \, {\left (a x + 1\right )}^{2} d}{{\left (a x - 1\right )}^{2}} + d\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (d x^{2} + c\right )} \operatorname {arcoth}\left (a x\right )\,{d x} \]________________________________________________________________________________________