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



                       2    2       2
Integrate[Csc[x] Sqrt[a  + b  Sin[x] ], x] == 

                                    2
        2             Sqrt[2] Sqrt[a ] Cos[x]
-(Sqrt[a ] ArcTanh[-----------------------------]) + 
                           2    2    2
                   Sqrt[2 a  + b  - b  Cos[2 x]]
 
         2                     2
  Sqrt[-b ] Log[Sqrt[2] Sqrt[-b ] Cos[x] + 
 
             2    2    2
     Sqrt[2 a  + b  - b  Cos[2 x]]]

Time to compute: 5.79 second

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