\[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right )-\ln \left (x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )+\frac {\ln \left (x^{2}-x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[(-1 + x^2)/((1 + x + x^2)*(x^2 + x^4)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {x^{2/3} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2+x^4}} \]
Mathematica 12.3 output
\[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx \]________________________________________________________________________________________