TITLE: 'Almost rational' integrals with no known closed form?
QUESTION [6 upvotes]: I recently stumbled upon an 'almost rational' integral, namely:
$$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= \frac{77}{333}$$
The error here is only:
$$\frac{77}{333}-I \approx 9 \cdot 10^{-9}$$
I don't think this integral has any known closed form (although I could be mistaken).

Do you know any integrals without a known closed form which are 'almost rational', meaning their value can be approximated by a rational number with the conditions:

$$|I-\frac{p}{q}|<10^{-7}$$
$$|p|,|q|<500$$
I hope there are some non-trivial integrals like the example I provided.
I tried to be as clear as possible, because the MathWorld page about almost integers is filled with 'questionable' entries.

REPLY [4 votes]: At least your integral turns out to have a closed form:
$$ \int_{0}^{\pi/2} x \, \frac{\sqrt{\sin x} - \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, \mathrm{d}x = G + \pi \left( \frac{1+2\sqrt{2}}{4} \log 2 - \log (1+\sqrt{2}) \right), $$
where $G$ is the Catalan's constant. So it seems to me that the 'almost rationality' of this integral is just a coincidence.