38.34 Problem number 560

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

Optimal antiderivative \[ \frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c} \]

command

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

\[ \frac {1}{a c \sqrt {\frac {a x - 1}{a x + 1}}} \]