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