Integrate[-1/2*Log[1 + (\[Tau]*\[Nu])^2]/(\[Nu] - \[Omega]), {\[Nu], -Infinity, Infinity},
PrincipalValue -> True, Assumptions -> \[Tau] > 0, GenerateConditions -> False]/Pi
gives
(2 \[Pi] ArcCot[\[Tau] \[Omega]] + (Log[\[Tau]^2 + 1/\[Omega]^2] - Log[-(1/\[Omega])] -
Log[1/\[Omega]]) (Log[-(1/\[Omega])] - Log[1/\[Omega]]))/(2 \[Pi])
Always watch for the Mathematica way of spelling and capitalizing things. Using exact rationals instead of decimals often helps too. It is also gets cranky about leading plus signs and it is perhaps best to avoid those.
And if you tell it something about the domain of omega then you can avoid the default assumption that every variable takes on every value in the complex plane and you can do even better. Note: greater than or less than implies Real, as you probably expect.
Integrate[-1/2*Log[1 + (\[Tau]*\[Nu])^2]/(\[Nu] - \[Omega]), {\[Nu], -Infinity, Infinity},
PrincipalValue -> True, Assumptions -> \[Tau] >0 && \[Omega] >0, GenerateConditions -> False]/Pi
gives
-ArcTan[\[Tau] \[Omega]]
When posting here, separating your Mathematica expressions from your text with blank lines, then selecting the expression by scraping with the mouse and then tapping control-K on your keyboard will let it format and highlight your expressions, will do somewhat less damage to the contents of your expressions, like "eating" all your backslashes when it displays them, but it still won't translate your Greek characters.
You might experiment with all the different combinations of Assumptions you can think of. You should get back a rainbow of different kinds of result, depending on those options. For example, because Tau Nu is squared and Nu should be able to be inferred to be Real inside the integration, it would seem reasonable that knowing Tau is Real should be sufficient to get the same ArcTan result. But assuming only Tau Real gives a completely different result from Tau positive. You can also try using Simplify or FullSimplify on the result, also with assumptions, but stated in a slightly different way. That gives even more possible results. I can't see any clear pattern with the various reasonable assumptions I tried. Perhaps you will be better.