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