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((-2 - 2*I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/4])/Sqrt[2]]*(Cos[x/2] + Sin[x/2]))/Sqrt[1 + Sin[x]]


Integrate[1/Sqrt[1 + Sin[x]], x] ==
((-2 - 2*I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/4])/ Sqrt[2]]*(Cos[x/2] + Sin[x/2]))/Sqrt[1 + Sin[x]]



                 1
Integrate[----------------, x] == 
          Sqrt[1 + Sin[x]]

                                     x
                            -1 + Tan[-]
                3/4                  4
((-2 - 2 I) (-1)    ArcTanh[-----------] 
                              Sqrt[2]
 
         x        x
    (Cos[-] + Sin[-])) / Sqrt[1 + Sin[x]]
         2        2

Time to compute: 0.08 second

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