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Integrate[x^3*Log[x]*Log[1 + x]^3, x] ==
(-5845 - 53540*x + 4270*x^2 - 700*x^3 + 108*x^4 + 19920*x*Log[x] - 2760*x^2*Log[x] + 592*x^3*Log[x] - 108*x^4*Log[x] + 33620*Log[1 + x] + 27840*x*Log[1 + x] - 4272*x^2*Log[1 + x] + 1184*x^3*Log[1 + x] - 324*x^4*Log[1 + x] - 19920*Log[x]*Log[1 + x] - 14400*x*Log[x]* Log[1 + x] + 3744*x^2*Log[x]*Log[1 + x] - 1344*x^3*Log[x]*Log[1 + x] + 432*x^4*Log[x]* Log[1 + x] - 6720*Log[1 + x]^2 - 4320*x*Log[1 + x]^2 + 1296*x^2*Log[1 + x]^2 - 672*x^3*Log[1 + x]^2 + 432*x^4*Log[1 + x]^2 - 7200*Log[-x]*Log[1 + x]^2 + 7200*Log[x]*Log[1 + x]^2 + 3456*x*Log[x]* Log[1 + x]^2 - 1728*x^2*Log[x]*Log[1 + x]^2 + 1152*x^3*Log[x]*Log[1 + x]^2 - 864*x^4*Log[x]*Log[1 + x]^2 + 288*Log[1 + x]^3 - 288*x^4*Log[1 + x]^3 + 1152*Log[-x]*Log[1 + x]^3 - 1152*Log[x]*Log[1 + x]^3 + 1152*x^4*Log[x]* Log[1 + x]^3 - 19920*PolyLog[2, -x] + 576*Log[1 + x]*(-25 + 6*Log[1 + x])* PolyLog[2, 1 + x] + 14400*PolyLog[3, 1 + x] - 6912*Log[1 + x]*PolyLog[3, 1 + x] + 6912*PolyLog[4, 1 + x])/4608



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

                         2        3        4
(-5845 - 53540 x + 4270 x  - 700 x  + 108 x  + 
 
                           2               3
    19920 x Log[x] - 2760 x  Log[x] + 592 x  Log[x] - 
 
         4
    108 x  Log[x] + 33620 Log[1 + x] + 
 
                               2
    27840 x Log[1 + x] - 4272 x  Log[1 + x] + 
 
          3                   4
    1184 x  Log[1 + x] - 324 x  Log[1 + x] - 
 
    19920 Log[x] Log[1 + x] - 
 
    14400 x Log[x] Log[1 + x] + 
 
          2
    3744 x  Log[x] Log[1 + x] - 
 
          3
    1344 x  Log[x] Log[1 + x] + 
 
         4                                    2
    432 x  Log[x] Log[1 + x] - 6720 Log[1 + x]  - 
 
                     2         2           2
    4320 x Log[1 + x]  + 1296 x  Log[1 + x]  - 
 
         3           2        4           2
    672 x  Log[1 + x]  + 432 x  Log[1 + x]  - 
 
                           2
    7200 Log[-x] Log[1 + x]  + 
 
                          2
    7200 Log[x] Log[1 + x]  + 
 
                            2
    3456 x Log[x] Log[1 + x]  - 
 
          2                  2
    1728 x  Log[x] Log[1 + x]  + 
 
          3                  2
    1152 x  Log[x] Log[1 + x]  - 
 
         4                  2                 3
    864 x  Log[x] Log[1 + x]  + 288 Log[1 + x]  - 
 
         4           3                          3
    288 x  Log[1 + x]  + 1152 Log[-x] Log[1 + x]  - 
 
                          3
    1152 Log[x] Log[1 + x]  + 
 
          4                  3
    1152 x  Log[x] Log[1 + x]  - 
 
    19920 PolyLog[2, -x] + 
 
    576 Log[1 + x] (-25 + 6 Log[1 + x]) 
 
     PolyLog[2, 1 + x] + 14400 PolyLog[3, 1 + x] - 
 
    6912 Log[1 + x] PolyLog[3, 1 + x] + 
 
    6912 PolyLog[4, 1 + x]) / 4608

Time to compute: 0.72 second

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