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