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