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

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

           2 Log[x]
2 Log[x] - -------- - 2 Log[1 + x] - 
            1 + x
 
  2 Log[x] Log[1 + x]             2
  ------------------- - Log[1 + x]  + 
         1 + x
 
                                         2
                    2   Log[x] Log[1 + x]
  Log[-x] Log[1 + x]  - ------------------ - 
                              1 + x
 
            3
  Log[1 + x]
  ----------- - 2 PolyLog[2, -x] + 
       3
 
  2 Log[1 + x] PolyLog[2, 1 + x] - 2 PolyLog[3, 1 + x]
Time to compute: 0.12 second
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