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


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



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

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

Time to compute: 0.21 second

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