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Integrate[Sin[x]/(a + b*Sin[x])^2, x] ==
-((a*Sqrt[a^2 - b^2]*Cos[x] + 2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]]* (a + b*Sin[x]))/((a^2 - b^2)^(3/2)* (a + b*Sin[x])))



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

           2    2
-((a Sqrt[a  - b ] Cos[x] + 
 
                           x
                 b + a Tan[-]
                           2
      2 b ArcTan[-------------] (a + b Sin[x])) / 
                       2    2
                 Sqrt[a  - b ]
 
       2    2 3/2
    ((a  - b )    (a + b Sin[x])))

Time to compute: 0.27 second

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