\[ \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (x , \sqrt {2}\right ) \sqrt {-2 x^{2}+1}\, \sqrt {-x^{2}+1}}{\sqrt {-1+x}\, \sqrt {1+x}\, \sqrt {2 x^{2}-1}} \]
command
integrate(1/(-1+x)^(1/2)/(1+x)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ {\rm ellipticF}\left (x, 2\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}{2 \, x^{4} - 3 \, x^{2} + 1}, x\right ) \]