\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x} \, dx \]
Optimal antiderivative \[ b x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2}-\frac {\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}{2}+\frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}{3}-\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3} \ln \left (x \right ) \]
command
integrate(arccoth(tanh(b*x+a))^3/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{3} \, b^{3} x^{3} - \frac {3}{4} \, {\left (-i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} - \frac {3}{4} \, {\left (\pi ^{2} b - 4 i \, \pi a b - 4 \, a^{2} b\right )} x + \frac {1}{8} \, {\left (-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \left (x\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x}\,{d x} \]________________________________________________________________________________________