20.2 Problem number 58

$$\int e^{{\text{sech}}^{-1}(ax)}{x}^{m}dx$$

Optimal antiderivative $$\frac{x^m}{am(1+m)}+\frac{(\frac{1}{ax}+\sqrt{\frac{1}{ax}-1}\sqrt{1+\frac{1}{ax}})x^{1+m}}{1+m}+\frac{x^m\mathrm{hypergeom}([\frac{1}{2},\frac{m}{2}],[1+\frac{m}{2}],a^2{x}^2)\sqrt{\frac{1}{ax+1}}\sqrt{ax+1}}{am(1+m)}$$

command

Integrate[E^ArcSech[a*x]*x^m,x]

Mathematica 13.1 output

$$\int e^{{\text{sech}}^{-1}(ax)}{x}^{m}dx$$

Mathematica 12.3 output

$$-\frac{2^{m+1}{x}^m(ax{)}^{-m}{e}^{2{\text{sech}}^{-1}(ax)}{\left(\frac{e^{{\text{sech}}^{-1}(ax)}}{e^{2{\text{sech}}^{-1}(ax)}+1}\right)}^m{(e^{2{\text{sech}}^{-1}(ax)}+1)}^m((m+2)e^{2{\text{sech}}^{-1}(ax)}{}_2{F}_1(\frac{m}{2}+2,m+2;\frac{m}{2}+3;-e^{2{\text{sech}}^{-1}(ax)})-(m+4){}_2{F}_1(\frac{m}{2}+1,m+2;\frac{m}{2}+2;-e^{2{\text{sech}}^{-1}(ax)}))}{a(m+2)(m+4)}$$