\[ \int \frac {1}{\sqrt {2+4 x^2-3 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (\frac {x \sqrt {-4+2 \sqrt {10}}}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right ) \sqrt {12+6 \sqrt {10}}}{6} \]
command
integrate(1/(-3*x^4+4*x^2+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{6} \, {\left (\sqrt {10} + 2\right )} \sqrt {\sqrt {10} - 2} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {10} - 2}, -\frac {2}{3} \, \sqrt {10} - \frac {7}{3}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} + 4 \, x^{2} + 2}}{3 \, x^{4} - 4 \, x^{2} - 2}, x\right ) \]