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

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

             Sin[x]
Integrate[------------, x] == 
          a + b Sin[x]

                         x
               b + a Tan[-]
                         2
    2 a ArcTan[-------------]
                     2    2
               Sqrt[a  - b ]
x - -------------------------
                2    2
          Sqrt[a  - b ]
-----------------------------
              b
Time to compute: 0.15 second
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