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-((x^(1 - 2*m)*Sqrt[1 - x^2/a^2]*Hypergeometric2F1[1/2, 1/2 - m, 3/2 - m, x^2/a^2])/((-1 + 2*m)*Sqrt[-a^2 + x^2]))


Integrate[1/(x^(2*m)*Sqrt[-a^2 + x^2]), x] ==
-((x^(1 - 2*m)*Sqrt[1 - x^2/a^2]*Hypergeometric2F1[ 1/2, 1/2 - m, 3/2 - m, x^2/a^2])/ ((-1 + 2*m)*Sqrt[-a^2 + x^2]))



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

                      2
    1 - 2 m          x
-((x        Sqrt[1 - --] 
                      2
                     a
 
                                          2
                        1  1      3      x
      Hypergeometric2F1[-, - - m, - - m, --]) / 
                        2  2      2       2
                                         a
 
                       2    2
    ((-1 + 2 m) Sqrt[-a  + x ]))

Time to compute: 0.07 second

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