This particular integral is related to what is known as the 'dilogarithm'.
In our case,
It seems to me there is also a unit disk reference with the dilog. It says for n=2 and z in the unit disk, then
i.e, if z=1, then we have
There is also a trilogarithm.
Do a google to find out more. It is a cool function with lotsa implications.
For instance, it has the series:
In the event z = -1, then we have what PH posted.
Notice how the Basel sum is involved here.
I noticed the similarity between this and the posters integral and thought there was a connection. I am sure Wiki or Mathworld has something about it.
The dilog is the polylog with n=2. There are others. As I mentioned, a trilog.
It has implications to gamma and other things. Research it. You'll find it cool.