How does Mathematica understand branchcuts of the complex logarithm?

The standard built-in logarithm function is defined for complex variables as follows:

Log[z] = Log[Abs[z]] + I Arg[z]

The location of the branch cut is simply caused by the convention that polar angles of z are assumed to be in the range $-\pi$ to $\pi$. This same branch cut is also part of the definition of the built-in Arg function.

Here is a different logarithm function called simply (lower-case) log to which you can supply a polar angle $\sigma$ at which the branch cut is going to be:

Clear[arg, log];
arg[z_, σ_: - Pi] := Arg[z Exp[-I (σ + Pi)]] + σ + Pi;
log[z_, σ_: - Pi] := Log[Abs[z]] + I arg[z, σ]

The first argument is the same as in the usual Log function. The second argument is optional, and if it's omitted then the new function agrees with its built-in counterpart. The default branch cut of the phase angle arg is $\sigma = -\pi$.

To illustrate these branch cut locations, I defined a "convenience function" that plots a function of the complex variable z in a square of side length 6 around the origin:

plot[f_] := Module[
  {fn = f /. z -> x + I y},
  Show[
   ContourPlot[
    Im[fn],
    {x, -3, 3}, {y, -3, 3},
    ContourShading -> Automatic,
    ExclusionsStyle -> Red
    ],
   ContourPlot[
    Re[fn],
    {x, -3, 3}, {y, -3, 3},
    ContourShading -> False,
    ContourStyle -> Blue,
    ExclusionsStyle -> Red
    ],
   FrameLabel -> {"Re(z)", "Im(z)"},
   Background -> Lighter[Orange],
   PlotRangePadding -> 0,
   PlotLabel -> 
    Framed[Grid[{{Style["-", Bold, Blue], 
        "Real part"}, {Style["-", Bold, Gray], "Imaginary part"}}, 
      Alignment -> Left], FrameStyle -> None, Background -> White, 
     RoundingRadius -> 5],
   ImageSize -> 300
   ]
  ]

For example, see the location of the branch cuts (red) rotate from the default to a non-standard direction:

GraphicsRow[{plot[Log[z]],plot[log[z,π/4]]}]

But now look at your function, with a = 1 because that constant is irrelevant:

plot[z  Tanh[π z] Log[z^2 + 1]]

Here, the default choice of branch cut of the Log function leads to vertical branch cuts in the complex plane. This means the upper half plane is not a safe place to be integrating in...

Depending on your needs, you could try to change the branch cut location to something else by using the log function with an appropriate choice for $\sigma$.

Here is where the freedom of choice in the function log comes in. Try this:

plot[z  Tanh[Pi z] log[z^2 + 1, 0]]

And as you can see, the branch cuts are now in a different position, no longer cutting through the middle of your integration domain.