\[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]
Mathematica 13.1 output
\[ \frac {1}{24} \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (8+32 x^2+3 \sqrt {1+x^2}-2 \sqrt {x+\sqrt {1+x^2}}+16 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+x \left (3+32 \sqrt {1+x^2}+16 \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{3/2}}+3 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-12 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ]+24 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]+12 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ]\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________