\[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctan \left (\frac {x}{\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right )}{3}-\frac {\arctan \left (\frac {2^{\frac {1}{3}} x}{\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{6}-\frac {\arctan \left (\frac {x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{-x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}\right )}{3}-\frac {\arctan \left (\frac {2^{\frac {2}{3}} x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{-2 x^{2}+2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}\right ) 2^{\frac {2}{3}}}{12}-\frac {\arctanh \left (\frac {\frac {\sqrt {3}\, x^{2}}{3}+\frac {\left (x^{4}-x^{2}\right )^{\frac {2}{3}} \sqrt {3}}{3}}{x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}-\frac {\sqrt {3}\, \arctanh \left (\frac {\frac {2^{\frac {1}{3}} x^{2} \sqrt {3}}{3}+\frac {\left (x^{4}-x^{2}\right )^{\frac {2}{3}} 2^{\frac {2}{3}} \sqrt {3}}{6}}{x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{12} \]
command
Integrate[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
Mathematica 13.1 output
\[ -\frac {x^{2/3} \sqrt [3]{-1+x^2} \left (8 \text {ArcTan}\left (\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+2\ 2^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+4 \text {ArcTan}\left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 i \sqrt {3} \text {ArcTan}\left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 \text {ArcTan}\left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )-4 i \sqrt {3} \text {ArcTan}\left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+2^{2/3} \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )+2^{2/3} \sqrt {3} \tanh ^{-1}\left (\frac {2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )\right )}{12 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx \]________________________________________________________________________________________