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Sqrt[a - x^(-2)]*x + (Sqrt[a - x^(-2)]*x*ArcTan[1/Sqrt[-1 + a*x^2]])/Sqrt[-1 + a*x^2]

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

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

          -2
Sqrt[a - x  ] x + 
 
             -2                  1
   Sqrt[a - x  ] x ArcTan[---------------]
                                       2
                          Sqrt[-1 + a x ]
   ---------------------------------------
                            2
               Sqrt[-1 + a x ]
Time to compute: 0.05 second
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