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


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



           m     a + x
Integrate[x  Log[------], x] == 
                 -a + x

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

Time to compute: 0.10 second

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