\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (4 \left (\text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )+\text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx \]________________________________________________________________________________________