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