20.1 Problem number 18

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^2} \, dx \]

Optimal antiderivative \[ -\frac {b \mathrm {arcsech}\! \left (b x +a \right )^{3}}{a}-\frac {\mathrm {arcsech}\! \left (b x +a \right )^{3}}{x}+\frac {3 b \mathrm {arcsech}\! \left (b x +a \right )^{2} \ln \! \left (1-\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1-\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}}-\frac {3 b \mathrm {arcsech}\! \left (b x +a \right )^{2} \ln \! \left (1-\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1+\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}}+\frac {6 b \,\mathrm {arcsech}\! \left (b x +a \right ) \polylog \! \left (2, \frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1-\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}}-\frac {6 b \,\mathrm {arcsech}\! \left (b x +a \right ) \polylog \! \left (2, \frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1+\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}}-\frac {6 b \polylog \! \left (3, \frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1-\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}}+\frac {6 b \polylog \! \left (3, \frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {1+\frac {1}{b x +a}}\right )}{1+\sqrt {-a^{2}+1}}\right )}{a \sqrt {-a^{2}+1}} \]

command

Integrate[ArcSech[a + b*x]^3/x^2,x]

Mathematica 13.1 output

\[ \text {\$Aborted} \]

Mathematica 12.3 output

\[ \text {output too large to display} \]