Difficult to Evaluate Numerical Integral

Question

Any tips for evaluating the integral

$$ I_{2n} = \frac{\pi}{2} \int_{-\infty}^\infty \int_{-\infty}^\infty \: \text{sech}(\pi x) \text{sech}(\pi y)\tanh^{2n}(\pi(x-y)) \: \text{d}x \text{d}y? $$

in a numerical but precise fashion? In particular, I'm interested in guessing the exact answer from the resultant decimal expression. Here's the current attempt ($2n = 8$):

exp = NIntegrate[1/2 π Sech[π x] Sech[π y] Tanh[π(x-y)]^8,       
    {x, -∞, ∞}, {y, -∞, ∞}, 
    PrecisionGoal -> 10];

Rationalize[exp, 10^-5]

Mathematica can easily do $2n =4,6$ with NIntegrate, but begins to run into problems for $2n = 8,10$ (values of interest). I've played around with various settings, but none seem to help.

Answer 1

There doesn't seem to be any problem. The integral seems to be well-behaved.

int[n_] :=
  NIntegrate[
    1/2 π Sech[π x] Sech[π y] Tanh[π (x - y)]^(2 n), {x, -∞, ∞}, {y, -∞, ∞}]]
vals = Table[int[n], {n, 6}]

{1., 0.833333, 0.741667, 0.680357, 0.635144, 0.599776}

vals = Table[int[n], {n, 6}]
ListPlot[vals]

These results obtained running V11.0.1 on OS X 10.10.2