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-ArcTan[x/Sqrt[a^2 - x^2]] + (Sqrt[a^2 + b^2]*ArcTan[(Sqrt[a^2 + b^2]*x)/(b*Sqrt[a^2 - x^2])])/b

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

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

              x
-ArcTan[-------------] + 
              2    2
        Sqrt[a  - x ]
 
                             2    2
        2    2         Sqrt[a  + b ] x
  Sqrt[a  + b ] ArcTan[---------------]
                               2    2
                       b Sqrt[a  - x ]
  -------------------------------------
                    b
Time to compute: 0.07 second
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