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