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(a*Sqrt[a^2*(1 - x^2/a^2)]*(Cos[2*ArcSin[x/a]] + 2*ArcSin[x/a]*(ArcSin[x/a] + Sin[2*ArcSin[x/a]])))/(8*Sqrt[1 - x^2/a^2])


Integrate[Sqrt[a^2 - x^2]*ArcSin[x/a], x] ==
(a*Sqrt[a^2*(1 - x^2/a^2)]*(Cos[2*ArcSin[x/a]] + 2*ArcSin[x/a]*(ArcSin[x/a] + Sin[2*ArcSin[x/a]])))/ (8*Sqrt[1 - x^2/a^2])



                2    2         x
Integrate[Sqrt[a  - x ] ArcSin[-], x] == 
                               a

                 2
         2      x                  x
(a Sqrt[a  (1 - --)] (Cos[2 ArcSin[-]] + 
                 2                 a
                a
 
               x          x                 x
      2 ArcSin[-] (ArcSin[-] + Sin[2 ArcSin[-]]))) / 
               a          a                 a
 
               2
              x
  (8 Sqrt[1 - --])
               2
              a

Time to compute: 0.23 second

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