\[ \int \frac {x^2 \left (-2+x^8\right ) \sqrt [4]{2-2 x^4+x^8}}{\left (2+x^8\right ) \left (4-x^4+2 x^8\right )} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{8}-2 x^{4}+2\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{8}-2 x^{4}+2}}\right ) 2^{\frac {3}{4}}}{4}+\frac {3^{\frac {1}{4}} \arctan \left (\frac {6^{\frac {3}{4}} x \left (x^{8}-2 x^{4}+2\right )^{\frac {1}{4}}}{-3 x^{2}+\sqrt {6}\, \sqrt {x^{8}-2 x^{4}+2}}\right ) 2^{\frac {1}{4}}}{4}+\frac {\arctanh \left (\frac {2 \,2^{\frac {1}{4}} x \left (x^{8}-2 x^{4}+2\right )^{\frac {1}{4}}}{2 x^{2}+\sqrt {2}\, \sqrt {x^{8}-2 x^{4}+2}}\right ) 2^{\frac {3}{4}}}{4}-\frac {3^{\frac {1}{4}} \arctanh \left (\frac {6^{\frac {3}{4}} x \left (x^{8}-2 x^{4}+2\right )^{\frac {1}{4}}}{3 x^{2}+\sqrt {6}\, \sqrt {x^{8}-2 x^{4}+2}}\right ) 2^{\frac {1}{4}}}{4} \]
command
Integrate[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)),x]
Mathematica 13.1 output
\[ \frac {1}{4} \left (\sqrt [4]{6} \text {ArcTan}\left (\frac {\frac {3 \sqrt [4]{6} x^2}{\sqrt [4]{2-2 x^4+x^8}}-6^{3/4} \sqrt [4]{2-2 x^4+x^8}}{6 x}\right )+2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{2-2 x^4+x^8}}{\sqrt {2} x^2-\sqrt {2-2 x^4+x^8}}\right )+2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{2-2 x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {2-2 x^4+x^8}}\right )-\sqrt [4]{6} \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{2-2 x^4+x^8}}{3 x^2+\sqrt {6} \sqrt {2-2 x^4+x^8}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {x^2 \left (-2+x^8\right ) \sqrt [4]{2-2 x^4+x^8}}{\left (2+x^8\right ) \left (4-x^4+2 x^8\right )} \, dx \]________________________________________________________________________________________