\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x \left (x^{2}-1\right )}-\frac {2 \arctan \left (\frac {3^{\frac {1}{6}} x}{\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right ) 3^{\frac {5}{6}}}{9}-\frac {\arctan \left (\frac {3^{\frac {5}{6}} x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{-3 x^{2}+3^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}\right ) 3^{\frac {5}{6}}}{9}-\frac {\arctanh \left (\frac {\frac {3^{\frac {2}{3}} x^{2}}{3}+\frac {\left (x^{4}-x^{2}\right )^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}}{x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right ) 3^{\frac {1}{3}}}{3} \]
command
Integrate[(1 + x^6)/((-x^2 + x^4)^(1/3)*(-1 + x^6)),x]
Mathematica 13.1 output
\[ -\frac {x^{2/3} \left (9 \sqrt [3]{x}+2\ 3^{5/6} \sqrt [3]{-1+x^2} \text {ArcTan}\left (\frac {\sqrt [6]{3} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+3^{5/6} \sqrt [3]{-1+x^2} \text {ArcTan}\left (\frac {3^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )+3 \sqrt [3]{3} \sqrt [3]{-1+x^2} \tanh ^{-1}\left (\frac {3 \sqrt [3]{3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )\right )}{9 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx \]________________________________________________________________________________________