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