\[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(1 + 2*x^6)/((x + x^3)^(1/3)*(-1 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}}\right )-8 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+2\ 2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}\right )+4 \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-2^{2/3} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )+4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{8 \sqrt [3]{x+x^3}} \]
Mathematica 12.3 output
\[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx \]________________________________________________________________________________________