\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} \left (2 x^{4}-3 x^{3}-2 x -4\right )}{10 x^{5}}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}}\right )-\ln \left (-x +\left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}\right )+\frac {\ln \left (x^{2}+x \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}+\left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)),x]
Mathematica 13.1 output
\[ \frac {3 \left (2+x-x^3-x^4\right )^{2/3} \left (-4-2 x-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x-x^3-x^4}}\right )-\log \left (-x+\sqrt [3]{2+x-x^3-x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{2+x-x^3-x^4}+\left (2+x-x^3-x^4\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx \]________________________________________________________________________________________