38.43 Problem number 779

$$\int \frac{e^{{\coth }^{-1}(ax)}}{{(c-\frac{c}{a^2{x}^2})}^4}dx$$

Optimal antiderivative $$-\frac{8}{7a{c}^4{(1-\frac{1}{ax})}^{\frac{7}{2}}{(1+\frac{1}{ax})}^{\frac{5}{2}}}-\frac{11}{7a{c}^4{(1-\frac{1}{ax})}^{\frac{5}{2}}{(1+\frac{1}{ax})}^{\frac{5}{2}}}-\frac{62}{21a{c}^4{(1-\frac{1}{ax})}^{\frac{3}{2}}{(1+\frac{1}{ax})}^{\frac{5}{2}}}+\frac{x}{c^4{(1-\frac{1}{ax})}^{\frac{7}{2}}{(1+\frac{1}{ax})}^{\frac{5}{2}}}+\frac{\mathrm{arctanh}\left(\sqrt{1-\frac{1}{ax}}\sqrt{1+\frac{1}{ax}}\right)}{a{c}^4}-\frac{269}{21a{c}^4{(1+\frac{1}{ax})}^{\frac{5}{2}}\sqrt{1-\frac{1}{ax}}}+\frac{262\sqrt{1-\frac{1}{ax}}}{35a{c}^4{(1+\frac{1}{ax})}^{\frac{5}{2}}}+\frac{163\sqrt{1-\frac{1}{ax}}}{35a{c}^4{(1+\frac{1}{ax})}^{\frac{3}{2}}}+\frac{128\sqrt{1-\frac{1}{ax}}}{35a{c}^4\sqrt{1+\frac{1}{ax}}}$$

command

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,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

$$\frac{1}{6720}a(\frac{6720\log (\sqrt{\frac{ax-1}{ax+1}}+1)}{a^2{c}^4}-\frac{6720\log \left(|\sqrt{\frac{ax-1}{ax+1}}-1|\right)}{a^2{c}^4}-\frac{5{(ax+1)}^3(\frac{42(ax-1)}{ax+1}+\frac{329{(ax-1)}^2}{{(ax+1)}^2}+\frac{2940{(ax-1)}^3}{{(ax+1)}^3}+3)}{{(ax-1)}^3{a}^2{c}^4\sqrt{\frac{ax-1}{ax+1}}}-\frac{13440\sqrt{\frac{ax-1}{ax+1}}}{a^2{c}^4(\frac{ax-1}{ax+1}-1)}+\frac{7(\frac{50(ax-1)a^8{c}^{16}\sqrt{\frac{ax-1}{ax+1}}}{ax+1}+\frac{3{(ax-1)}^2{a}^8{c}^{16}\sqrt{\frac{ax-1}{ax+1}}}{{(ax+1)}^2}+705a^8{c}^{16}\sqrt{\frac{ax-1}{ax+1}})}{a^{10}{c}^{20}})$$