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x - x*(-1 + Log[x]) - Log[1 + x] + x*(-1 + Log[x])*Log[1 + x] + Log[x]*Log[1 + x] + PolyLog[2, -x]

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

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

x - x (-1 + Log[x]) - Log[1 + x] + 
 
  x (-1 + Log[x]) Log[1 + x] + Log[x] Log[1 + x] + 
 
  PolyLog[2, -x]
Time to compute: 0.04 second
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