\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right )}{6}-\frac {\arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{24}+\frac {\ln \left (-x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{2}-\frac {\ln \left (x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (-2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{24}-\frac {\ln \left (2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{8}+\frac {\ln \left (x^{2}-x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{12}-\frac {\ln \left (x^{2}+x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{4}+\frac {\ln \left (-2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{16}-\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{48} \]
command
Integrate[(1 - x^3 + x^6)/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]
Mathematica 13.1 output
\[ \frac {x^{2/3} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-24 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x^2}}\right )+6\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2^{2/3} \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+24 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )-8 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+2\ 2^{2/3} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-6\ 2^{2/3} \log \left (2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )+4 \log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-12 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+3\ 2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}-\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )-2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{x^2+x^4}} \]
Mathematica 12.3 output
\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \]________________________________________________________________________________________