\[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^4)^(1/4)*(2 + x^4))/(x^2*(2 + 2*x^4 + x^8)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{8} \text {RootSum}\left [5-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-5 \log (x)+5 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx \]________________________________________________________________________________________