\[ \int \frac {\sqrt {-1+x^2+x^4+x^6} \left (1+x^4+2 x^6\right )}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {1+i}\, \arctan \left (\frac {\sqrt {-1-i}\, x}{\sqrt {x^{6}+x^{4}+x^{2}-1}}\right )}{2}-\frac {\sqrt {1-i}\, \arctan \left (\frac {\sqrt {-1+i}\, x}{\sqrt {x^{6}+x^{4}+x^{2}-1}}\right )}{2} \]
command
Integrate[(Sqrt[-1 + x^2 + x^4 + x^6]*(1 + x^4 + 2*x^6))/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12),x]
Mathematica 13.1 output
\[ -\frac {1}{2} \sqrt {1+i} \text {ArcTan}\left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^4+x^6}}\right )-\frac {1}{2} \sqrt {1-i} \text {ArcTan}\left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^4+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-1+x^2+x^4+x^6} \left (1+x^4+2 x^6\right )}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx \]________________________________________________________________________________________