\[ \int \frac {-1+x+x^3}{\left (1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-1+x} x^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )+4 \text {RootSum}\left [-1+\text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2}{-1+3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{(-1+x) x^2}} \]
Mathematica 12.3 output
\[ \int \frac {-1+x+x^3}{\left (1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx \]________________________________________________________________________________________