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Please explain this weird Integrate result


Expand[Integrate[(2*x)/(2*x - 1), x]]

gives

-1/2 + x + Log[-1 + 2*x]/2

But,

Expand[Integrate[1 + 1/(-1 + 2*x), x]]

gives

x + Log[-1 + 2*x]/2

even though

(2*x)/(2*x - 1) == 1 + 1/(-1 + 2*x)

Where did the extra -1/2 in the first result come from?


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