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