\[ \int \frac {1}{\sqrt {-3+2 x^2-2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {x 54^{\frac {1}{4}}}{3}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} x}{3}\right )\right ), \frac {\sqrt {18+3 \sqrt {6}}}{6}\right ) \left (3+x^{2} \sqrt {6}\right ) \sqrt {\frac {2 x^{4}-2 x^{2}+3}{\left (3+x^{2} \sqrt {6}\right )^{2}}}\, 6^{\frac {3}{4}}}{12 \cos \left (2 \arctan \left (\frac {2^{\frac {1}{4}} 3^{\frac {3}{4}} x}{3}\right )\right ) \sqrt {-2 x^{4}+2 x^{2}-3}} \]
command
integrate(1/(-2*x^4+2*x^2-3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{18} \, \sqrt {3} \sqrt {-3} \sqrt {\sqrt {-5} + 1} {\left (\sqrt {-5} - 1\right )} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {-5} + 1}, -\frac {1}{3} \, \sqrt {-5} - \frac {2}{3}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 2 \, x^{2} - 3}}{2 \, x^{4} - 2 \, x^{2} + 3}, x\right ) \]