\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[Sqrt[x - Sqrt[1 + x^2]]/(x^2 + Sqrt[1 + x^2]),x]
Mathematica 13.1 output
\[ \text {RootSum}\left [1-2 \text {$\#$1}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-2 \text {$\#$1}^2-3 \text {$\#$1}^4+2 \text {$\#$1}^6}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx \]________________________________________________________________________________________