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