\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (-x^{8}+2 x^{3}-2 x \right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{2}-\frac {\ln \left (-2 x +2^{\frac {2}{3}} \left (-x^{8}+2 x^{3}-2 x \right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{2}+\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (-x^{8}+2 x^{3}-2 x \right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (-x^{8}+2 x^{3}-2 x \right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{4} \]
command
Integrate[((-4 + 5*x^7)*(-2*x + 2*x^3 - x^8)^(1/3))/((2 + x^7)*(2 - 2*x^2 + x^7)),x]
Mathematica 13.1 output
\[ \frac {x^{2/3} \left (2-2 x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{2-2 x^2+x^7}}\right )-2 \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{2-2 x^2+x^7}\right )+\log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{2-2 x^2+x^7}-\sqrt [3]{2} \left (2-2 x^2+x^7\right )^{2/3}\right )\right )}{2\ 2^{2/3} \left (-x \left (2-2 x^2+x^7\right )\right )^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \]________________________________________________________________________________________