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-2*ArcTan[(Sin[x]*Sqrt[g^2 - Sin[x]^2])/(-g^2 + Sin[x]^2)]

Integrate[(2*Cos[x])/Sqrt[g^2 - Sin[x]^2], x] ==
-2*ArcTan[(Sin[x]*Sqrt[g^2 - Sin[x]^2])/ (-g^2 + Sin[x]^2)] 

               2 Cos[x]
Integrate[------------------, x] == 
                2         2
          Sqrt[g  - Sin[x] ]

                       2         2
          Sin[x] Sqrt[g  - Sin[x] ]
-2 ArcTan[-------------------------]
                  2         2
                -g  + Sin[x]
Time to compute: 0.04 second
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