Relations between definite integrals not having a known closed form Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary relation connecting them (e.g. one integral is the natural logarithm of the other)?

$^\dagger$ non-trivial means we want to exclude non-interesing cases (especially, linear combinations) like one integral being twice larger then the other because its integrand is twice as large.

This is the most fitting nonlinear identity I found so far. There's still a closed form here somewhere, but without knowing the details, it's not at all obvious.
Here's the identity:

$$\int_{-\infty}^\infty \frac{e^{a x} \cos x}{(e^x+1)^{2a}}dx= \\ \frac{2 \sqrt{\pi}}{4^a} \frac{\Gamma(a)}{\Gamma(a+\frac12 )} \exp \left(-2 \int_0^\infty \frac{e^{(1-a) x} (1-\cos x)}{x (e^x-1)}dx \right), \quad a>0$$

If anyone is interested, I can provide the proof.
I'm sure some people will guess the closed form for the left hand side right away.
As a simple case, and also a hint, it follows for $a=1$ (Mathematica can't take even this integral):
$$\int_0^\infty \frac{1-\cos x}{x (e^x-1)}dx=\frac{1}{2} \log \frac{\sinh \pi}{\pi}$$
But the closed form is definitely not as simple for the general $a \in \mathbb{R}^+$.
