\[ \int \frac {1}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(x + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]
Mathematica 13.1 output
\[ -\log \left (x+\sqrt {1+x^2}\right )+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {1}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________