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((1 + k)^2*Log[-1 + k*(-1 + x)] + k*x*(1 + k + k*x*Log[(2*x)/(1 + k - k*x)]))/(2*k^2)

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

                     2 x
Integrate[x Log[-------------], x] == 
                1 - k (x - 1)

        2
((1 + k)  Log[-1 + k (-1 + x)] + 
 
                             2 x              2
    k x (1 + k + k x Log[-----------])) / (2 k )
                         1 + k - k x
Time to compute: 0.04 second
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