\[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (x \sqrt {2}, \frac {i \sqrt {2}}{2}\right ) \sqrt {x^{2}+1}}{2 \sqrt {-x^{2}-1}} \]
command
integrate(1/(-4*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
\[ -\frac {1}{4} \, \sqrt {2} \sqrt {-2} {\rm ellipticF}\left (\sqrt {2} x, -\frac {1}{2}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}}{2 \, {\left (2 \, x^{4} + x^{2} - 1\right )}}, x\right ) \]