What is going on here (Mathematica version 8.x):
NIntegrate[Log[1/2 + Sqrt[1/4 - 1/(4 x^2)]]/x, {x, 1, Infinity}]
--> -0.171007
Integrate[Log[1/2 + Sqrt[1/4 - 1/(4 x^2)]]/x, {x, 1, Infinity}] // N
--> 0.171007
The NIntegrate[] value is correct. I have run into problems with PrincipalValue selections before but a) those have been fixed in mma8 and b) this integral doesn't, or at least shouldn't, have poles in the integration region.
EDIT: Thanks to people suggesting solutions to this problem, a general solution would be, e.g., using exclusively NIntegrate. However, I am interested in finding out why specifically this happens and whether thus this bug is predictable.
This is a bug in Integrate, I am afraid. As a workaround, do the change of variables x->u^(-1/2):
In[12]:= Log[1/2 + Sqrt[1/4 - 1/(4*x^2)]]/x Dt[x]/Dt[u] /.
x -> 1/Sqrt[u]
Out[12]= Log[1/2 + Sqrt[1/4 - u/4]]/(2 u)
Then
In[14]:= Integrate[%, {u, 1, 0}]
Out[14]= 1/24 (-\[Pi]^2 + Log[8] Log[16])
In[15]:= N[%]
Out[15]= -0.171007
This agrees with NIntegrate.