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