\[ \int \frac {\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x \left (-x^{2}+1\right )}{3 \sqrt {x^{4}+x^{2}+1}}+\frac {2 x \sqrt {x^{4}+x^{2}+1}}{3 \left (x^{2}+1\right )}-\frac {2 \left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {1}{2}\right ) \sqrt {\frac {x^{4}+x^{2}+1}{\left (x^{2}+1\right )^{2}}}}{3 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+x^{2}+1}}+\frac {\left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {1}{2}\right ) \sqrt {\frac {x^{4}+x^{2}+1}{\left (x^{2}+1\right )^{2}}}}{\cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+x^{2}+1}} \]
command
Integrate[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {-x+x^3+2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 (-1)^{5/6} \sqrt {3+3 \sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{3 \sqrt {1+x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________