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-(x*Sqrt[-a^2 + x^2])/4 + (x^2*Log[x + Sqrt[-a^2 + x^2]])/2 - (a^2*Log[2*(x + Sqrt[-a^2 + x^2])])/4

Integrate[x*Log[x + Sqrt[-a^2 + x^2]], x] ==
-(x*Sqrt[-a^2 + x^2])/4 + (x^2*Log[x + Sqrt[-a^2 + x^2]])/2 - (a^2*Log[2*(x + Sqrt[-a^2 + x^2])])/4 

                           2    2
Integrate[x Log[x + Sqrt[-a  + x ]], x] == 

           2    2      2                2    2
-(x Sqrt[-a  + x ])   x  Log[x + Sqrt[-a  + x ]]
------------------- + -------------------------- - 
         4                        2
 
   2                   2    2
  a  Log[2 (x + Sqrt[-a  + x ])]
  ------------------------------
                4
Time to compute: 0.06 second
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