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