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