I've come across an integral that when numerically evaluated is quite close to $1$. The integral is in fact loosely related to a recent question that similarly was suspiciously close to a simple result.
$$I = \int^{\infty}_{0} \int^{\infty}_{0} \frac{dx}{1+x^2} \frac{dy}{1+y^2} \frac{\sinh(\pi x) \sinh(\pi y)}{\sinh(\pi (x+y))} \approx 1.0088$$
While I'd be interested in an analytical answer to the integral, I suspect it might be tough to evaluate exactly. Instead, I'd like to see if there might be some appropriate series expansion or other approximation technique that could show that this integral is in fact close to $1$.
Here is a rough estimate that places $I$ in the right ballpark.
By rewriting $I$, we can note that:
$$I = \frac{1}{2}\int^{\infty}_{0} \int^{\infty}_{0} \frac{dx}{1+x^2} \frac{dy}{1+y^2} \frac{(1-e^{-2 \pi x}) (1-e^{-2 \pi y})}{(1-e^{-2 \pi (x+y)})} <\frac{1}{2}\int^{\infty}_{0} \int^{\infty}_{0} \frac{dx}{1+x^2} \frac{dy}{1+y^2} $$
This means
$$I < \frac{\pi^2}{8} \approx 1.2337$$
However, I'm less confident about systematic ways to tighten this upper bound or to introduce lower bounds.
