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



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

             6 Log[x]
(-6 Log[x] + -------- + 6 Log[1 + x] + 
              1 + x
 
    6 Log[x] Log[1 + x]               2
    ------------------- + 3 Log[1 + x]  - 
           1 + x
 
                                               2
                        2   3 Log[x] Log[1 + x]
    3 Log[-x] Log[1 + x]  + -------------------- + 
                                   1 + x
 
              3                     3
    Log[1 + x]  - Log[-x] Log[1 + x]  + 
 
                     3
    Log[x] Log[1 + x]  + 6 PolyLog[2, -x] - 
 
    3 Log[1 + x] (2 + Log[1 + x]) PolyLog[2, 1 + x] + 
 
    6 PolyLog[3, 1 + x] + 
 
    6 Log[1 + x] PolyLog[3, 1 + x] - 
 
    6 PolyLog[4, 1 + x]) / 3

Time to compute: 0.29 second

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