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

Integrate[1/(Sqrt[a^2 - x^2]*(b^2 + x^2)), x] ==
ArcTan[(Sqrt[a^2 + b^2]*x)/(b*Sqrt[a^2 - x^2])]/ (b*Sqrt[a^2 + b^2]) 

                     1
Integrate[-----------------------, x] == 
                2    2    2    2
          Sqrt[a  - x ] (b  + x )

             2    2
       Sqrt[a  + b ] x
ArcTan[---------------]
               2    2
       b Sqrt[a  - x ]
-----------------------
            2    2
    b Sqrt[a  + b ]
Time to compute: 0.05 second
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