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(x*(1 - x^2/c^2)^n*Hypergeometric2F1[1/2, n, 3/2, x^2/c^2])/(-c^2 + x^2)^n


Integrate[(-c^2 + x^2)^(-n), x] ==
(x*(1 - x^2/c^2)^n*Hypergeometric2F1[1/2, n, 3/2, x^2/c^2])/(-c^2 + x^2)^n



             2    2 -n
Integrate[(-c  + x )  , x] == 

        2                               2
       x  n                   1     3  x
x (1 - --)  Hypergeometric2F1[-, n, -, --]
        2                     2     2   2
       c                               c
------------------------------------------
                  2    2 n
               (-c  + x )

Time to compute: 0.02 second

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