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Integrate[x*Log[x]*Log[1 + x]^2, x] ==
(28*x - 3*x^2 - 12*x*Log[x] + 2*x^2*Log[x] - 16*Log[1 + x] - 12*x*Log[1 + x] + 4*x^2*Log[1 + x] + 12*Log[x]*Log[1 + x] + 8*x*Log[x]*Log[1 + x] - 4*x^2*Log[x]*Log[1 + x] + 2*Log[1 + x]^2 - 2*x^2*Log[1 + x]^2 + 4*Log[-x]*Log[1 + x]^2 - 4*Log[x]*Log[1 + x]^2 + 4*x^2*Log[x]*Log[1 + x]^2 + 12*PolyLog[2, -x] + 8*Log[1 + x]*PolyLog[2, 1 + x] - 8*PolyLog[3, 1 + x])/8



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

           2                    2
(28 x - 3 x  - 12 x Log[x] + 2 x  Log[x] - 
 
    16 Log[1 + x] - 12 x Log[1 + x] + 
 
       2
    4 x  Log[1 + x] + 12 Log[x] Log[1 + x] + 
 
                               2
    8 x Log[x] Log[1 + x] - 4 x  Log[x] Log[1 + x] + 
 
                2      2           2
    2 Log[1 + x]  - 2 x  Log[1 + x]  + 
 
                        2                      2
    4 Log[-x] Log[1 + x]  - 4 Log[x] Log[1 + x]  + 
 
       2                  2
    4 x  Log[x] Log[1 + x]  + 12 PolyLog[2, -x] + 
 
    8 Log[1 + x] PolyLog[2, 1 + x] - 
 
    8 PolyLog[3, 1 + x]) / 8

Time to compute: 0.17 second

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