Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ are the Bessel function of the first and second kind. It is supported by numerical calculation with hundreds of digits of precision for many different values of $\mu, \nu$. The question is open for several days with +500 bounty on it and is not resolved yet. But my question here is not about if this conjecture true or false.

Obviously, several possible closed forms matched my numeric calculations, for example: $$\left(\frac{\pi}{2}+7^{-7^{7^{7^{7^{7^{\sqrt{5}+\sin \mu\nu}}}}}}\right)(\mu^2-\nu^2).$$ But for some reason that I cannot clearly explain (or even understand) I selected the simpler one, and I am strongly inclined to search for its proof rather than a disproof. I believe most people would feel and behave exactly the same way.

(1) Are there any mathematical or philosophical reasons supporting this position?

Why when we calculate some sum or integral (which do not contain explicit tiny quantities like $10^{-10^{10^{.^{.^{10}}}}}$) with thousands of digits of precision and it matches some simple closed-form expression, we inclined to believe this is the exact equality rather than an accidental very close value?

(2) Are there known cases when such intuition turned out to be wrong?

And one more question:

(3) Do you believe there can be exact closed forms for some infinite sums or integrals, that cannot be proved in $ZFC$ or any its reasonable extension (like adding some large cardinal axioms) - so to speak, equalities that hold without any reason.