38.38 Problem number 579

\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx \]

Optimal antiderivative \[ -\frac {8}{35 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{3}}-\frac {-4 a x +3}{7 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{3} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {-\frac {24 a x}{35}+\frac {36}{35}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{3} \left (-a^{2} x^{2}+1\right )} \]

command

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^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

\[ -\frac {\frac {{\left (a x + 1\right )}^{3} {\left (\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}} + 35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a c^{3}} \]