38.44 Problem number 793

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

Optimal antiderivative $$\frac{3\mathrm{arctanh}\left(\sqrt{1-\frac{1}{ax}}\sqrt{1+\frac{1}{ax}}\right)}{ac}-\frac{5\sqrt{1+\frac{1}{ax}}}{3ac{(1-\frac{1}{ax})}^{\frac{3}{2}}}+\frac{x\sqrt{1+\frac{1}{ax}}}{c{(1-\frac{1}{ax})}^{\frac{3}{2}}}-\frac{14\sqrt{1+\frac{1}{ax}}}{3ac\sqrt{1-\frac{1}{ax}}}$$

command

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

Giac 1.9.0-11 via sagemath 9.6 output

$$\text{Exception raised: TypeError}$$

Giac 1.7.0 via sagemath 9.3 output

$$\frac{1}{3}a(\frac{9\log (\sqrt{\frac{ax-1}{ax+1}}+1)}{a^{2}c}-\frac{9\log \left(|\sqrt{\frac{ax-1}{ax+1}}-1|\right)}{a^{2}c}-\frac{(ax+1)(\frac{12(ax-1)}{ax+1}+1)}{(ax-1)a^{2}c\sqrt{\frac{ax-1}{ax+1}}}-\frac{6\sqrt{\frac{ax-1}{ax+1}}}{a^{2}c(\frac{ax-1}{ax+1}-1)})$$