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


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



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

      1 + x
(Sqrt[-----] ((-1 + x) Sqrt[1 + x] + 
      1 - x
 
                           Sqrt[1 + x]
      2 Sqrt[1 - x] ArcSin[-----------])) / 
                             Sqrt[2]
 
  Sqrt[1 + x]

Time to compute: 0.09 second

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