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Integrate[x*Log[x]*Log[1 + x]^3, x] ==
(-45 - 270*x + 12*x^2 + 84*x*Log[x] - 6*x^2*Log[x] + 186*Log[1 + x] + 168*x*Log[1 + x] - 18*x^2*Log[1 + x] - 84*Log[x]*Log[1 + x] - 72*x*Log[x]*Log[1 + x] + 12*x^2*Log[x]*Log[1 + x] - 48*Log[1 + x]^2 - 36*x*Log[1 + x]^2 + 12*x^2*Log[1 + x]^2 - 36*Log[-x]*Log[1 + x]^2 + 36*Log[x]*Log[1 + x]^2 + 24*x*Log[x]*Log[1 + x]^2 - 12*x^2*Log[x]*Log[1 + x]^2 + 4*Log[1 + x]^3 - 4*x^2*Log[1 + x]^3 + 8*Log[-x]*Log[1 + x]^3 - 8*Log[x]*Log[1 + x]^3 + 8*x^2*Log[x]*Log[1 + x]^3 - 84*PolyLog[2, -x] + 24*(-3 + Log[1 + x])*Log[1 + x]* PolyLog[2, 1 + x] + 72*PolyLog[3, 1 + x] - 48*Log[1 + x]*PolyLog[3, 1 + x] + 48*PolyLog[4, 1 + x])/16



                             3
Integrate[x Log[x] Log[1 + x] , x] == 

                   2                    2
(-45 - 270 x + 12 x  + 84 x Log[x] - 6 x  Log[x] + 
 
    186 Log[1 + x] + 168 x Log[1 + x] - 
 
        2
    18 x  Log[1 + x] - 84 Log[x] Log[1 + x] - 
 
    72 x Log[x] Log[1 + x] + 
 
        2                                  2
    12 x  Log[x] Log[1 + x] - 48 Log[1 + x]  - 
 
                   2       2           2
    36 x Log[1 + x]  + 12 x  Log[1 + x]  - 
 
                         2                       2
    36 Log[-x] Log[1 + x]  + 36 Log[x] Log[1 + x]  + 
 
                          2
    24 x Log[x] Log[1 + x]  - 
 
        2                  2               3
    12 x  Log[x] Log[1 + x]  + 4 Log[1 + x]  - 
 
       2           3                       3
    4 x  Log[1 + x]  + 8 Log[-x] Log[1 + x]  - 
 
                       3      2                  3
    8 Log[x] Log[1 + x]  + 8 x  Log[x] Log[1 + x]  - 
 
    84 PolyLog[2, -x] + 
 
    24 (-3 + Log[1 + x]) Log[1 + x] 
 
     PolyLog[2, 1 + x] + 72 PolyLog[3, 1 + x] - 
 
    48 Log[1 + x] PolyLog[3, 1 + x] + 
 
    48 PolyLog[4, 1 + x]) / 16

Time to compute: 0.36 second

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