\[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}}\right )-\ln \left (x +\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}\right )+\frac {\ln \left (x^{2}-x \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[(-3 + 5*x^8)/((1 + x^8)*(1 - x^3 + x^8)^(1/3)),x]
Mathematica 13.1 output
\[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx \]________________________________________________________________________________________