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

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

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

(Sqrt[-((-1 + x) x)] (Sqrt[-1 + x] Sqrt[x] 
 
       (-1 + 2 x) - Log[2 (Sqrt[-1 + x] + Sqrt[x])]))\
 
   / (4 Sqrt[-1 + x] Sqrt[x])
Time to compute: 0.06 second
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