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


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



Integrate[Sqrt[2 + 2 Sin[x]], x] == 

                x        x
2 Sqrt[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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