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


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



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

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

Time to compute: 0.25 second

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