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.

It seems natural to consider the gamma function and the log gamma function.  Recall that $$\Gamma(t) = \int_{0}^{\infty} x^{t-1} e^{-x} dx$$ for complex numbers with a positive real part.  Binet's second log gamma formula (see http://mathworld.wolfram.com/BinetsLogGammaFormulas.html) is $$\ln \Gamma(z) = \left( z - \frac{1}{2} \right) \ln z - z + \frac{1}{2} \ln (2 \pi) + 2 \int_{0}^{\infty} \frac{\tan^{-1}\left(\frac{t}{z}\right)}{e^{2 \pi t} - 1} dt$$ and we thus have that: $$\ln\left( \int_{0}^{\infty} x^{t-1} e^{-x} dx\right) = \left( z - \frac{1}{2} \right) \ln z - z + \frac{1}{2} \ln (2 \pi) + 2 \int_{0}^{\infty} \frac{\tan^{-1}\left(\frac{t}{z}\right)}{e^{2 \pi t} - 1} dt.$$ Certain values of the gamma function such as $\Gamma\left(\frac{1}{3}\right)$ have no closed form, so the above definite integrals have no known closed form for certain values of $t$.
