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


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



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

        x        x
2 (-Cos[-] + Sin[-]) Sqrt[1 + Sin[x]]
        2        2
-------------------------------------
               x        x
           Cos[-] + Sin[-]
               2        2

Time to compute: 0.04 second

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