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