\[ \int x^m \tanh ^{-1}(\tanh (a+b x))^3 \, dx \]
Optimal antiderivative \[ -\frac {6 b^{3} x^{4+m}}{\left (1+m \right ) \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {6 b^{2} x^{3+m} \arctanh \left (\tanh \left (b x +a \right )\right )}{m^{3}+6 m^{2}+11 m +6}-\frac {3 b \,x^{2+m} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{m^{2}+3 m +2}+\frac {x^{1+m} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{1+m} \]
command
integrate(x^m*arctanh(tanh(b*x+a))^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {3 \, b x^{2} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{{\left (m + 2\right )} {\left (m + 1\right )}} + \frac {x^{m + 1} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{m + 1} - \frac {6 \, {\left (\frac {b^{2} x^{4} x^{m}}{{\left (m + 4\right )} {\left (m + 3\right )} {\left (m + 2\right )}} - \frac {b x^{3} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{{\left (m + 3\right )} {\left (m + 2\right )}}\right )} b}{m + 1} \]
Maxima 5.44 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________