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