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Integrate[Sqrt[Tan[x]], x] ==
(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[x]] + Tan[x]] - Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]])/(2*Sqrt[2])



Integrate[Sqrt[Tan[x]], x] == 

(-2 ArcTan[1 - Sqrt[2] Sqrt[Tan[x]]] + 
 
    2 ArcTan[1 + Sqrt[2] Sqrt[Tan[x]]] + 
 
    Log[1 - Sqrt[2] Sqrt[Tan[x]] + Tan[x]] - 
 
    Log[1 + Sqrt[2] Sqrt[Tan[x]] + Tan[x]]) / 
 
  (2 Sqrt[2])

Time to compute: 0.17 second

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