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