\[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt {-1+x^4}} \, dx \]
Optimal antiderivative \[ -\frac {x \left (-x^{2}+1\right )}{2 \sqrt {x^{4}-1}}-\frac {\EllipticE \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {x^{4}-1}}+\frac {\EllipticF \left (\frac {x \sqrt {2}}{\sqrt {x^{2}-1}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2}-1}\, \sqrt {x^{2}+1}\, \sqrt {2}}{2 \sqrt {x^{4}-1}} \]
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
\[ \frac {\sqrt {x^{4} - 1} x}{2 \, {\left (x^{2} + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{4} - 1} x^{2}}{x^{6} + x^{4} - x^{2} - 1}, x\right ) \]