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