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