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



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

                         x
                -1 + Tan[-]
                         4
((-2 Pi ArcTanh[-----------] + 
                  Sqrt[2]
 
                              1/4  I/2 x
      (Pi + 2 x) (Log[1 - (-1)    E     ] - 
 
                     1/4  I/2 x
         Log[1 + (-1)    E     ]) + 
 
                             1/4  I/2 x
      (4 I) PolyLog[2, -((-1)    E     )] - 
 
                           1/4  I/2 x
      (4 I) PolyLog[2, (-1)    E     ]) 
 
         x        x
    (Cos[-] + Sin[-])) / (Sqrt[2] Sqrt[1 + Sin[x]])
         2        2

Time to compute: 0.24 second

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