\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{7}-1\right )^{\frac {1}{3}}}\right )+\ln \left (x +\left (x^{7}-1\right )^{\frac {1}{3}}\right )-\frac {\ln \left (x^{2}-x \left (x^{7}-1\right )^{\frac {1}{3}}+\left (x^{7}-1\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[((-1 + x^7)^(2/3)*(3 + 4*x^7))/(x^3*(-1 + x^3 + x^7)),x]
Mathematica 13.1 output
\[ \frac {3 \left (-1+x^7\right )^{2/3}}{2 x^2}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^7}}\right )+\log \left (x+\sqrt [3]{-1+x^7}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^7}+\left (-1+x^7\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx \]________________________________________________________________________________________