\[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^2)^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
Mathematica 13.1 output
\[ \frac {1}{24} \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6+2 \left (1-24 x+x^2-8 x^3-2 x^4+32 x^5\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5+4 \left (-4+x-12 x^2-x^3+16 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{7/2}}-3 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+24 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]-6 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-24 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]-6 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________