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Integrate[x^2*Log[x]*Log[1 + x]^3, x] ==
(1150 + 8820*x - 585*x^2 + 64*x^3 - 3060*x*Log[x] + 342*x^2*Log[x] - 48*x^3*Log[x] - 5760*Log[1 + x] - 4932*x*Log[1 + x] + 684*x^2*Log[1 + x] - 144*x^3*Log[1 + x] + 3060*Log[x]*Log[1 + x] + 2376*x*Log[x]*Log[1 + x] - 540*x^2*Log[x]* Log[1 + x] + 144*x^3*Log[x]*Log[1 + x] + 1278*Log[1 + x]^2 + 864*x*Log[1 + x]^2 - 270*x^2*Log[1 + x]^2 + 144*x^3*Log[1 + x]^2 + 1188*Log[-x]*Log[1 + x]^2 - 1188*Log[x]*Log[1 + x]^2 - 648*x*Log[x]* Log[1 + x]^2 + 324*x^2*Log[x]*Log[1 + x]^2 - 216*x^3*Log[x]*Log[1 + x]^2 - 72*Log[1 + x]^3 - 72*x^3*Log[1 + x]^3 - 216*Log[-x]*Log[1 + x]^3 + 216*Log[x]*Log[1 + x]^3 + 216*x^3*Log[x]* Log[1 + x]^3 + 3060*PolyLog[2, -x] - 216*Log[1 + x]*(-11 + 3*Log[1 + x])* PolyLog[2, 1 + x] - 2376*PolyLog[3, 1 + x] + 1296*Log[1 + x]*PolyLog[3, 1 + x] - 1296*PolyLog[4, 1 + x])/648



           2                  3
Integrate[x  Log[x] Log[1 + x] , x] == 

                      2       3
(1150 + 8820 x - 585 x  + 64 x  - 3060 x Log[x] + 
 
         2              3
    342 x  Log[x] - 48 x  Log[x] - 5760 Log[1 + x] - 
 
                             2
    4932 x Log[1 + x] + 684 x  Log[1 + x] - 
 
         3
    144 x  Log[1 + x] + 3060 Log[x] Log[1 + x] + 
 
    2376 x Log[x] Log[1 + x] - 
 
         2
    540 x  Log[x] Log[1 + x] + 
 
         3                                    2
    144 x  Log[x] Log[1 + x] + 1278 Log[1 + x]  + 
 
                    2        2           2
    864 x Log[1 + x]  - 270 x  Log[1 + x]  + 
 
         3           2                          2
    144 x  Log[1 + x]  + 1188 Log[-x] Log[1 + x]  - 
 
                          2
    1188 Log[x] Log[1 + x]  - 
 
                           2
    648 x Log[x] Log[1 + x]  + 
 
         2                  2
    324 x  Log[x] Log[1 + x]  - 
 
         3                  2                3
    216 x  Log[x] Log[1 + x]  - 72 Log[1 + x]  - 
 
        3           3                         3
    72 x  Log[1 + x]  - 216 Log[-x] Log[1 + x]  + 
 
                         3
    216 Log[x] Log[1 + x]  + 
 
         3                  3
    216 x  Log[x] Log[1 + x]  + 3060 PolyLog[2, -x] - 
 
    216 Log[1 + x] (-11 + 3 Log[1 + x]) 
 
     PolyLog[2, 1 + x] - 2376 PolyLog[3, 1 + x] + 
 
    1296 Log[1 + x] PolyLog[3, 1 + x] - 
 
    1296 PolyLog[4, 1 + x]) / 648

Time to compute: 0.51 second

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