38.45 Problem number 795

$$\int \frac{e^{3{\coth }^{-1}(ax)}}{{(c-\frac{c}{a^2{x}^2})}^3}dx$$

Optimal antiderivative $$\frac{3\mathrm{arctanh}\left(\sqrt{1-\frac{1}{ax}}\sqrt{1+\frac{1}{ax}}\right)}{a{c}^3}-\frac{8}{7a{c}^3{(1-\frac{1}{ax})}^{\frac{7}{2}}\sqrt{1+\frac{1}{ax}}}-\frac{53}{35a{c}^3{(1-\frac{1}{ax})}^{\frac{5}{2}}\sqrt{1+\frac{1}{ax}}}-\frac{88}{35a{c}^3{(1-\frac{1}{ax})}^{\frac{3}{2}}\sqrt{1+\frac{1}{ax}}}+\frac{x}{c^3{(1-\frac{1}{ax})}^{\frac{7}{2}}\sqrt{1+\frac{1}{ax}}}-\frac{281}{35a{c}^3\sqrt{1-\frac{1}{ax}}\sqrt{1+\frac{1}{ax}}}+\frac{176\sqrt{1-\frac{1}{ax}}}{35a{c}^3\sqrt{1+\frac{1}{ax}}}$$

command

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

$$\text{could not integrate}$$

Giac 1.7.0 via sagemath 9.3 output

$$\frac{1}{560}a(\frac{1680\log (\sqrt{\frac{ax-1}{ax+1}}+1)}{a^2{c}^3}-\frac{1680\log \left(|\sqrt{\frac{ax-1}{ax+1}}-1|\right)}{a^2{c}^3}-\frac{{(ax+1)}^3(\frac{56(ax-1)}{ax+1}+\frac{350{(ax-1)}^2}{{(ax+1)}^2}+\frac{2520{(ax-1)}^3}{{(ax+1)}^3}+5)}{{(ax-1)}^3{a}^2{c}^3\sqrt{\frac{ax-1}{ax+1}}}+\frac{35\sqrt{\frac{ax-1}{ax+1}}}{a^2{c}^3}-\frac{1120\sqrt{\frac{ax-1}{ax+1}}}{a^2{c}^3(\frac{ax-1}{ax+1}-1)})$$