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(-(x*Sqrt[1 + x^2]) + ArcSinh[x] + 2*x^2*Log[x + Sqrt[1 + x^2]])/4

Integrate[x*Log[x + Sqrt[1 + x^2]], x] ==
(-(x*Sqrt[1 + x^2]) + ArcSinh[x] + 2*x^2*Log[x + Sqrt[1 + x^2]])/4 

                              2
Integrate[x Log[x + Sqrt[1 + x ]], x] == 

               2
(-(x Sqrt[1 + x ]) + ArcSinh[x] + 
 
       2                   2
    2 x  Log[x + Sqrt[1 + x ]]) / 4
Time to compute: 0.04 second
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