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