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Integrate[(x*Log[x]*Log[1 + x])/(1 + x), x] ==
x - x*(-1 + Log[x]) - Log[1 + x] + x*(-1 + Log[x])*Log[1 + x] + Log[x]*Log[1 + x] - ((-Log[-x] + Log[x])*Log[1 + x]^2)/2 + PolyLog[2, -x] + Log[1 + x]*PolyLog[2, 1 + x] - PolyLog[3, 1 + x] 

          x Log[x] Log[1 + x]
Integrate[-------------------, x] == 
                 1 + x

x - x (-1 + Log[x]) - Log[1 + x] + 
 
  x (-1 + Log[x]) Log[1 + x] + Log[x] Log[1 + x] - 
 
                                2
  (-Log[-x] + Log[x]) Log[1 + x]
  ------------------------------- + PolyLog[2, -x] + 
                 2
 
  Log[1 + x] PolyLog[2, 1 + x] - PolyLog[3, 1 + x]
Time to compute: 0.09 second
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