\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
Mathematica 13.1 output
\[ \frac {1}{64} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-1+2 x-x^2+2 x^3-\left (3+2 x-5 x^2+2 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (1-x+2 x^2+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+32 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {36 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+136 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________