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