\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (-x^{3}+1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{5 x^{5}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{12}+\frac {2^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {5}{6}} x}{3^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}}\right )}{4}+\frac {\ln \left (-x +2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{12}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (-3 x +2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{12}-\frac {\ln \left (x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{24}+\frac {\ln \left (3 x^{2}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} 3^{\frac {2}{3}}}{24} \]
command
Integrate[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(-4 + x^6)),x]
Mathematica 13.1 output
\[ \frac {1}{120} \left (-\frac {24 \left (-1+x^3\right )^{5/3}}{x^5}-10 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+30 \sqrt [3]{2} \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )-5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx \]________________________________________________________________________________________