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