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Integrate[Log[x]*Log[1 + x]^3, x] ==
6 + 24*x - 6*x*Log[x] - 18*Log[1 + x] - 18*x*Log[1 + x] + 6*Log[x]*Log[1 + x] + 6*x*Log[x]*Log[1 + x] + 6*Log[1 + x]^2 + 6*x*Log[1 + x]^2 + 3*Log[-x]*Log[1 + x]^2 - 3*Log[x]*Log[1 + x]^2 - 3*x*Log[x]*Log[1 + x]^2 - Log[1 + x]^3 - x*Log[1 + x]^3 - Log[-x]*Log[1 + x]^3 + Log[x]*Log[1 + x]^3 + x*Log[x]*Log[1 + x]^3 + 6*PolyLog[2, -x] - 3*(-2 + Log[1 + x])*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
Integrate[Log[x] Log[1 + x] , x] == 

6 + 24 x - 6 x Log[x] - 18 Log[1 + x] - 
 
  18 x Log[1 + x] + 6 Log[x] Log[1 + x] + 
 
                                      2
  6 x Log[x] Log[1 + x] + 6 Log[1 + x]  + 
 
                2                       2
  6 x Log[1 + x]  + 3 Log[-x] Log[1 + x]  - 
 
                     2                        2
  3 Log[x] Log[1 + x]  - 3 x Log[x] Log[1 + x]  - 
 
            3               3                     3
  Log[1 + x]  - x Log[1 + x]  - Log[-x] Log[1 + x]  + 
 
                   3                      3
  Log[x] Log[1 + x]  + x Log[x] Log[1 + x]  + 
 
  6 PolyLog[2, -x] - 3 (-2 + Log[1 + x]) 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]
Time to compute: 0.20 second
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