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(2^(x/2)*ExpIntegralEi[-(x*Log[2])/2])/Sqrt[2^x]

Integrate[1/(Sqrt[2^x]*x), x] ==
(2^(x/2)*ExpIntegralEi[-(x*Log[2])/2])/Sqrt[2^x] 

              1
Integrate[----------, x] == 
                x
          Sqrt[2 ] x

 x/2               -(x Log[2])
2    ExpIntegralEi[-----------]
                        2
-------------------------------
                 x
           Sqrt[2 ]
Time to compute: < 0.01 second
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