\[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (-x^{7}+1\right )^{\frac {1}{3}}}{2 x}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {2}{3}} \left (-x^{7}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{4}-\frac {\ln \left (-x +2^{\frac {2}{3}} \left (-x^{7}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{4}+\frac {\ln \left (x^{2}+2^{\frac {2}{3}} x \left (-x^{7}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}} \left (-x^{7}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{8} \]
command
Integrate[((1 - x^7)^(1/3)*(-2 + x^3 + 2*x^7)*(3 + 4*x^7))/(x^2*(-1 + x^7)*(-4 + x^3 + 4*x^7)),x]
Mathematica 13.1 output
\[ \frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \]________________________________________________________________________________________