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