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Integrate[Log[x]*Log[1 + x]^2, x] ==
-6*x + 2*x*Log[x] + 4*Log[1 + x] + 4*x*Log[1 + x] - 2*Log[x]*Log[1 + x] - 2*x*Log[x]*Log[1 + x] - Log[1 + x]^2 - x*Log[1 + x]^2 - Log[-x]*Log[1 + x]^2 + Log[x]*Log[1 + x]^2 + x*Log[x]*Log[1 + x]^2 - 2*PolyLog[2, -x] - 2*Log[1 + x]*PolyLog[2, 1 + x] + 2*PolyLog[3, 1 + x] 

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

-6 x + 2 x Log[x] + 4 Log[1 + x] + 4 x Log[1 + x] - 
 
  2 Log[x] Log[1 + x] - 2 x Log[x] Log[1 + x] - 
 
            2               2                     2
  Log[1 + x]  - x Log[1 + x]  - Log[-x] Log[1 + x]  + 
 
                   2                      2
  Log[x] Log[1 + x]  + x Log[x] Log[1 + x]  - 
 
  2 PolyLog[2, -x] - 2 Log[1 + x] PolyLog[2, 1 + x] + 
 
  2 PolyLog[3, 1 + x]
Time to compute: 0.10 second
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