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