\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (-x^{7}-x^{4}+2 x^{3}+x \right )^{\frac {1}{3}}}\right )-\ln \left (-x +\left (-x^{7}-x^{4}+2 x^{3}+x \right )^{\frac {1}{3}}\right )+\frac {\ln \left (x^{2}+x \left (-x^{7}-x^{4}+2 x^{3}+x \right )^{\frac {1}{3}}+\left (-x^{7}-x^{4}+2 x^{3}+x \right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),x]
Mathematica 13.1 output
\[ -\frac {x^{2/3} \left (-1-2 x^2+x^3+x^6\right )^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-2 x^2+x^3+x^6}}\right )+2 \log \left (x^{2/3}+\sqrt [3]{-1-2 x^2+x^3+x^6}\right )-\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-2 x^2+x^3+x^6}+\left (-1-2 x^2+x^3+x^6\right )^{2/3}\right )\right )}{2 \left (-x \left (-1-2 x^2+x^3+x^6\right )\right )^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \]________________________________________________________________________________________