\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}}{2 x^{6}-2 x^{4}+2 x^{2}+2}-\frac {\arctan \left (\frac {\sqrt {3}\, \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}}{-2 x +\left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{6}+\frac {\ln \left (x +\left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}\right )}{6}-\frac {\ln \left (x^{2}-x \left (x^{7}-x^{5}+x \right )^{\frac {1}{3}}+\left (x^{7}-x^{5}+x \right )^{\frac {2}{3}}\right )}{12} \]
command
Integrate[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x-x^5+x^7} \left (-\frac {6 x^{4/3}}{1+x^2-x^4+x^6}+\frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{1-x^4+x^6}}\right )}{\sqrt [3]{1-x^4+x^6}}+\frac {2 \log \left (x^{2/3}+\sqrt [3]{1-x^4+x^6}\right )}{\sqrt [3]{1-x^4+x^6}}-\frac {\log \left (x^{4/3}-x^{2/3} \sqrt [3]{1-x^4+x^6}+\left (1-x^4+x^6\right )^{2/3}\right )}{\sqrt [3]{1-x^4+x^6}}\right )}{12 \sqrt [3]{x}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \]________________________________________________________________________________________