\[ \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (40320 x^{4}-2560 x^{3}+112192 x^{2}+1545 x +31736\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (2240 x^{3}+1536 x^{2}+40688 x -1542\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\sqrt {x^{2}+1}\, \left (\left (40320 x^{3}-2560 x^{2}+92032 x +2825\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (2240 x^{2}+1536 x +39568\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{55440 \left (x +\sqrt {x^{2}+1}\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{16} \]
command
Integrate[Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]
Mathematica 13.1 output
\[ \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (31736+1545 x+112192 x^2-2560 x^3+40320 x^4+2 \left (-771+20344 x+768 x^2+1120 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (2825+92032 x-2560 x^2+40320 x^3+16 \left (2473+96 x+140 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{55440 \left (x+\sqrt {1+x^2}\right )^{3/2}}-\frac {1}{16} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]
Mathematica 12.3 output
\[ \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________