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