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