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Integrate[Sqrt[x^2 + 1 + 1/x^2], x] ==
(x*Sqrt[1 + x^(-2) + x^2]* (ArcSinh[(1 + 2*x^2)/Sqrt[3]] + 2*(Sqrt[1 + x^2 + x^4] + Log[2*x^2] - Log[2 + x^2 + 2*Sqrt[1 + x^2 + x^4]])))/ (4*Sqrt[1 + x^2 + x^4])



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

             -2    2
(x Sqrt[1 + x   + x ] 
 
                    2
             1 + 2 x
    (ArcSinh[--------] + 
             Sqrt[3]
 
                   2    4           2
      2 (Sqrt[1 + x  + x ] + Log[2 x ] - 
 
                  2               2    4
         Log[2 + x  + 2 Sqrt[1 + x  + x ]]))) / 
 
               2    4
  (4 Sqrt[1 + x  + x ])

Time to compute: 0.15 second

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