Probably this will all get deleted, but I will point out that there has not been a release of version 12.1, ergo no rationale for the claims that nothing has been done to address the issue.

In the meantime, are you aware that you can use FourierCoefficient[] to evaluate this integral?

Ff[n_Integer] := (-1)^n FourierCoefficient[Sinc[x + ?] Log[x + ?], x,
                                           n, FourierParameters -> {-1, -1}]

N[Ff[1]]
   -1.0706908375028643 + 0.6492569703685498*I

So Mr. Lichtblau, you're saying Wolfram Research may NOT fix this? In the case that it doesn't "...survive stress-testing on our integration suite." Really ?

It is difficult to tell what exactly caused the degraded performance. Some debugging evidence suggests it might be from a change to an internal time constraint, causing an attempt to give up too soon, but that is far from certainty.

In this particular instance the mix of approximate numbers and symbolic methods is somewhat toxic. I will experiment with a couple of changes intended to improve the behavior of this example, but it is by no means a given that they will survive stress-testing on our integration suite.

Integrate is a symbolic solver,then use a exact numbers not numeric one.

For me calculating time is almost the same.

Mathematica 10.2

 F[n_] := Integrate[Exp[I n x] Sin[x] x^-1 Log[x], {x, 0, 2 Pi}]; 
 F[1] // N // AbsoluteTiming
 (* {16.2086,-1.07069+0.649257 \[ImaginaryI]}*)

Mathematica 12.0

 F[n_] := Integrate[Exp[I n x] Sin[x] x^-1 Log[x], {x, 0, 2 Pi}]; 
 F[1] // N // AbsoluteTiming
 (* {17.1023,-1.07069+0.649257 \[ImaginaryI]}*)

Regards M.I.