\[ \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{2}+2\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ), \frac {\sqrt {2}}{4}\right ) \sqrt {\frac {x^{4}+3 x^{2}+4}{\left (x^{2}+2\right )^{2}}}\, \sqrt {2}}{4 \cos \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ) \sqrt {x^{4}+3 x^{2}+4}} \]
command
integrate(1/(x^4+3*x^2+4)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{16} \, \sqrt {2} {\left (\sqrt {-7} + 3\right )} \sqrt {\sqrt {-7} - 3} {\rm ellipticF}\left (\frac {1}{4} \, \sqrt {2} x \sqrt {\sqrt {-7} - 3}, \frac {3}{8} \, \sqrt {-7} + \frac {1}{8}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 4}}, x\right ) \]