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(2*Cos[x/2]*(-Log[Cos[x/4] - Sin[x/4]] + Log[Cos[x/4] + Sin[x/4]]))/Sqrt[1 + Cos[x]]

Integrate[1/Sqrt[1 + Cos[x]], x] ==
(2*Cos[x/2]*(-Log[Cos[x/4] - Sin[x/4]] + Log[Cos[x/4] + Sin[x/4]]))/Sqrt[1 + Cos[x]] 

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

       x            x        x
(2 Cos[-] (-Log[Cos[-] - Sin[-]] + 
       2            4        4
 
              x        x
      Log[Cos[-] + Sin[-]])) / Sqrt[1 + Cos[x]]
              4        4
Time to compute: 0.04 second
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