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