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-(x*(-1 + Log[x])) + (p*(Log[x]*Log[1 + x/p] + PolyLog[2, -(x/p)]))/2 - (p*(Log[x]*Log[1 - x/p] + PolyLog[2, x/p]))/2

Integrate[(x^2*Log[x])/(p^2 - x^2), x] ==
-(x*(-1 + Log[x])) + (p*(Log[x]*Log[1 + x/p] + PolyLog[2, -(x/p)]))/2 - (p*(Log[x]*Log[1 - x/p] + PolyLog[2, x/p]))/2 

           2
          x  Log[x]
Integrate[---------, x] == 
            2    2
           p  - x

-(x (-1 + Log[x])) + 
 
                     x                 x
   p (Log[x] Log[1 + -] + PolyLog[2, -(-)])
                     p                 p
   ---------------------------------------- - 
                      2
 
                    x               x
  p (Log[x] Log[1 - -] + PolyLog[2, -])
                    p               p
  -------------------------------------
                    2
Time to compute: 0.05 second
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