\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {2 x^{6}+1}}{\sqrt {2}-x^{2}+2 \sqrt {2}\, x^{6}}\right ) 2^{\frac {3}{4}}}{8}-\frac {\arctanh \left (\frac {2^{\frac {3}{4}} x \sqrt {2 x^{6}+1}}{\sqrt {2}+x^{2}+2 \sqrt {2}\, x^{6}}\right ) 2^{\frac {3}{4}}}{8} \]
command
Integrate[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )+\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \]________________________________________________________________________________________