\[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (66913566720 x^{8}-2099249152 x^{7}+248171986944 x^{6}-11794907136 x^{5}+407581982720 x^{4}-1415707308 x^{3}+220397520304 x^{2}+5227043711 x +15903121112\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (1968046080 x^{7}+1130364928 x^{6}+10550149120 x^{5}+6568280064 x^{4}+96561463296 x^{3}+1984342244 x^{2}+66830366096 x -1176816782\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\sqrt {x^{2}+1}\, \left (\left (66913566720 x^{7}-2099249152 x^{6}+214715203584 x^{5}-10745282560 x^{4}+308588576768 x^{3}+3694527828 x^{2}+88760534448 x +2167822549\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (1968046080 x^{6}+1130364928 x^{5}+9566126080 x^{4}+6003097600 x^{3}+92024406016 x^{2}-875910940 x +21890925968\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{39729930240 \left (x +\sqrt {x^{2}+1}\right )^{\frac {7}{2}}}-\frac {545 \arctanh \left (\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{8192} \]
command
Integrate[(1 + x^2)^(3/2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]
Mathematica 13.1 output
\[ \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (15903121112+5227043711 x+220397520304 x^2-1415707308 x^3+407581982720 x^4-11794907136 x^5+248171986944 x^6-2099249152 x^7+66913566720 x^8+2 \left (-588408391+33415183048 x+992171122 x^2+48280731648 x^3+3284140032 x^4+5275074560 x^5+565182464 x^6+984023040 x^7\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (2167822549+88760534448 x+3694527828 x^2+308588576768 x^3-10745282560 x^4+214715203584 x^5-2099249152 x^6+66913566720 x^7+4 \left (5472731492-218977735 x+23006101504 x^2+1500774400 x^3+2391531520 x^4+282591232 x^5+492011520 x^6\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{39729930240 \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {545 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{8192} \]
Mathematica 12.3 output
\[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________