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Integrate[x^3*Log[x]*Log[1 + x]^4, x] ==
360005/20736 + (40999*x)/576 - (851*x^2)/288 + (781*x^3)/2592 - (15*x^4)/512 - (5845*x*Log[x])/288 + (865*x^2*Log[x])/576 - (175*x^3*Log[x])/864 + (3*x^4*Log[x])/128 - (29309*Log[1 + x])/576 - (13385*x*Log[1 + x])/288 + (2135*x^2*Log[1 + x])/576 - (175*x^3*Log[1 + x])/ 288 + (3*x^4*Log[1 + x])/32 + (5845*Log[x]*Log[1 + x])/288 + (415*x*Log[x]*Log[1 + x])/24 - (115*x^2*Log[x]*Log[1 + x])/48 + (37*x^3*Log[x]*Log[1 + x])/72 - (3*x^4*Log[x]*Log[1 + x])/32 + (8405*Log[1 + x]^2)/576 + (145*x*Log[1 + x]^2)/12 - (89*x^2*Log[1 + x]^2)/48 + (37*x^3*Log[1 + x]^2)/ 72 - (9*x^4*Log[1 + x]^2)/64 + (415*Log[-x]*Log[1 + x]^2)/48 - (415*Log[x]*Log[1 + x]^2)/48 - (25*x*Log[x]*Log[1 + x]^2)/4 + (13*x^2*Log[x]*Log[1 + x]^2)/8 - (7*x^3*Log[x]*Log[1 + x]^2)/12 + (3*x^4*Log[x]*Log[1 + x]^2)/16 - (35*Log[1 + x]^3)/18 - (5*x*Log[1 + x]^3)/4 + (3*x^2*Log[1 + x]^3)/8 - (7*x^3*Log[1 + x]^3)/36 + (x^4*Log[1 + x]^3)/8 - (25*Log[-x]*Log[1 + x]^3)/ 12 + (25*Log[x]*Log[1 + x]^3)/12 + x*Log[x]*Log[1 + x]^3 - (x^2*Log[x]*Log[1 + x]^3)/ 2 + (x^3*Log[x]*Log[1 + x]^3)/3 - (x^4*Log[x]*Log[1 + x]^3)/4 + Log[1 + x]^4/16 - (x^4*Log[1 + x]^4)/16 + (Log[-x]*Log[1 + x]^4)/4 - (Log[x]*Log[1 + x]^4)/4 + (x^4*Log[x]*Log[1 + x]^4)/ 4 + (5845*PolyLog[2, -x])/288 + ((415*Log[1 + x])/24 - (25*Log[1 + x]^2)/4 + Log[1 + x]^3)*PolyLog[2, 1 + x] - (415*PolyLog[3, 1 + x])/24 + (25*Log[1 + x]*PolyLog[3, 1 + x])/2 - 3*Log[1 + x]^2*PolyLog[3, 1 + x] - (25*PolyLog[4, 1 + x])/2 + 6*Log[1 + x]* PolyLog[4, 1 + x] - 6*PolyLog[5, 1 + x]



           3                  4
Integrate[x  Log[x] Log[1 + x] , x] == 

                        2        3       4
360005   40999 x   851 x    781 x    15 x
------ + ------- - ------ + ------ - ----- - 
20736      576      288      2592     512
 
                       2               3
  5845 x Log[x]   865 x  Log[x]   175 x  Log[x]
  ------------- + ------------- - ------------- + 
       288             576             864
 
     4
  3 x  Log[x]   29309 Log[1 + x]
  ----------- - ---------------- - 
      128             576
 
                             2
  13385 x Log[1 + x]   2135 x  Log[1 + x]
  ------------------ + ------------------ - 
         288                  576
 
       3                 4
  175 x  Log[1 + x]   3 x  Log[1 + x]
  ----------------- + --------------- + 
         288                32
 
  5845 Log[x] Log[1 + x]   415 x Log[x] Log[1 + x]
  ---------------------- + ----------------------- - 
           288                       24
 
       2
  115 x  Log[x] Log[1 + x]
  ------------------------ + 
             48
 
      3                        4
  37 x  Log[x] Log[1 + x]   3 x  Log[x] Log[1 + x]
  ----------------------- - ---------------------- + 
            72                        32
 
                 2                   2
  8405 Log[1 + x]    145 x Log[1 + x]
  ---------------- + ----------------- - 
        576                 12
 
      2           2       3           2
  89 x  Log[1 + x]    37 x  Log[1 + x]
  ----------------- + ----------------- - 
         48                  72
 
     4           2                         2
  9 x  Log[1 + x]    415 Log[-x] Log[1 + x]
  ---------------- + ----------------------- - 
         64                    48
 
                       2                         2
  415 Log[x] Log[1 + x]    25 x Log[x] Log[1 + x]
  ---------------------- - ----------------------- + 
            48                        4
 
      2                  2
  13 x  Log[x] Log[1 + x]
  ------------------------ - 
             8
 
     3                  2      4                  2
  7 x  Log[x] Log[1 + x]    3 x  Log[x] Log[1 + x]
  ----------------------- + ----------------------- - 
            12                        16
 
               3                 3
  35 Log[1 + x]    5 x Log[1 + x]
  -------------- - --------------- + 
        18                4
 
     2           3      3           3
  3 x  Log[1 + x]    7 x  Log[1 + x]
  ---------------- - ---------------- + 
         8                  36
 
   4           3                        3
  x  Log[1 + x]    25 Log[-x] Log[1 + x]
  -------------- - ---------------------- + 
        8                    12
 
                      3
  25 Log[x] Log[1 + x]                       3
  --------------------- + x Log[x] Log[1 + x]  - 
           12
 
   2                  3    3                  3
  x  Log[x] Log[1 + x]    x  Log[x] Log[1 + x]
  --------------------- + --------------------- - 
            2                       3
 
   4                  3             4
  x  Log[x] Log[1 + x]    Log[1 + x]
  --------------------- + ----------- - 
            4                 16
 
   4           4                     4
  x  Log[1 + x]    Log[-x] Log[1 + x]
  -------------- + ------------------- - 
        16                  4
 
                   4    4                  4
  Log[x] Log[1 + x]    x  Log[x] Log[1 + x]
  ------------------ + --------------------- + 
          4                      4
 
  5845 PolyLog[2, -x]
  ------------------- + 
          288
 
                                 2
   415 Log[1 + x]   25 Log[1 + x]              3
  (-------------- - -------------- + Log[1 + x] ) 
         24               4
 
                       415 PolyLog[3, 1 + x]
   PolyLog[2, 1 + x] - --------------------- + 
                                24
 
  25 Log[1 + x] PolyLog[3, 1 + x]
  ------------------------------- - 
                 2
 
              2
  3 Log[1 + x]  PolyLog[3, 1 + x] - 
 
  25 PolyLog[4, 1 + x]
  -------------------- + 
           2
 
  6 Log[1 + x] PolyLog[4, 1 + x] - 6 PolyLog[5, 1 + x]

Time to compute: 1.52 second

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