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