\[ \int \frac {1}{\sqrt {2-2 x^2+3 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {x 24^{\frac {1}{4}}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {3^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2}\right )\right ), \frac {\sqrt {18+3 \sqrt {6}}}{6}\right ) \left (2+x^{2} \sqrt {6}\right ) \sqrt {\frac {3 x^{4}-2 x^{2}+2}{\left (2+x^{2} \sqrt {6}\right )^{2}}}\, 6^{\frac {3}{4}}}{12 \cos \left (2 \arctan \left (\frac {3^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2}\right )\right ) \sqrt {3 x^{4}-2 x^{2}+2}} \]
command
integrate(1/(3*x^4-2*x^2+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{6} \, \sqrt {\sqrt {-5} + 1} {\left (\sqrt {-5} - 1\right )} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-5} + 1}, -\frac {1}{3} \, \sqrt {-5} - \frac {2}{3}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - 2 \, x^{2} + 2}}, x\right ) \]