\[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (30720 x^{4}-4096 x^{3}-21570 x^{2}-2680 x -24993\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (3072 x^{3}+4096 x^{2}+1814 x +1712\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\sqrt {x^{2}+1}\, \left (\left (30720 x^{3}-4096 x^{2}-36930 x -632\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (3072 x^{2}+4096 x +278\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{26880 \left (x +\sqrt {x^{2}+1}\right )^{\frac {5}{2}}}-\frac {263 \arctanh \left (\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{256} \]
command
Integrate[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x + Sqrt[1 + x^2]],x]
Mathematica 13.1 output
\[ \frac {\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4+2 \left (856+907 x+2048 x^2+1536 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-632-36930 x-4096 x^2+30720 x^3+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{5/2}}-27615 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________