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