\[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {1}{9+x^{2} \left (9+3 \sqrt {3}\right )}}\, \sqrt {9+x^{2} \left (9+3 \sqrt {3}\right )}\, \EllipticF \left (\frac {x \sqrt {9+3 \sqrt {3}}}{\sqrt {9+x^{2} \left (9+3 \sqrt {3}\right )}}, \sqrt {\sqrt {3}-1}\right ) \left (3+x^{2} \left (3+\sqrt {3}\right )\right ) \sqrt {\frac {3+x^{2} \left (3-\sqrt {3}\right )}{3+x^{2} \left (3+\sqrt {3}\right )}}}{\sqrt {2 x^{4}+6 x^{2}+3}\, \sqrt {9+3 \sqrt {3}}} \]
command
integrate(1/(2*x^4+6*x^2+3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{6} \, {\left (\sqrt {3} + 3\right )} \sqrt {\sqrt {3} - 3} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {3} - 3}, \sqrt {3} + 2\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}}, x\right ) \]