\[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]
Optimal antiderivative \[ \frac {x \left (x^{2}-5\right )}{2 \left (x^{2}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}-4 \arctan \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {2}+\sqrt {14+10 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )+\frac {\arctanh \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {2}}{2}-\sqrt {-14+10 \sqrt {2}}\, \arctanh \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right ) \]
command
Integrate[(1 + x^2)^2/((-1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Mathematica 13.1 output
\[ \frac {x \left (-5+x^2\right )}{2 \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-4 \sqrt {2} \text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]________________________________________________________________________________________