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