Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

Question

In this question, Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$ , robjohn evaluates the integral to a nice summation with an approximate value. When plugged into W|A, it gives a possible closed form as $\dfrac{1512835691 \pi}{1983703776}$, correct to at least 20 decimal digits. When subtracting the two in W|A, it gives a nice result of $0$. (1) Can we prove that it equals the conjectured closed form?

Answer 1

This is not the correct value. The value of the integral with $70$ digits precision is $$\underline{2.395878633914562092}453189586490058501300873812447750729519628041973530$$

And the conjectured expression is $$\underline{2.395878633914562092}589756143323601049570678313084832875204827253396682$$

The two values are not equal.

Answer 2

Another expression for it, BTW, is $$ 4 \pi \sum_{j=0}^\infty \sigma_0(2j+1) (-1)^j \exp(-(j+1/2)\pi) $$ where $\sigma_0$ is the number of divisors function. If your number is $\pi x$, then the continued fraction of $x$ starts

$$[0; 1, 3, 4, 1, 2, 3, 4, 3, 1, 4, 12, 2, 4, 3, 104, 1, 2, 5, 1, 1, 21, 4, 1, 2, 1, 3, 1, 17, 1, 6, 5, 2, 2, 59, 1, 8, 3, 42, 15, 5, 1, 1, 2, 3, 1, 2, 12, 2, 3, 1, 2, 8, 1, 4, 2, 2, 3, 1, 2, 5, 1, 1, 18, 1, 1, 4, 2, 2, 1, 1, 2, 2, 2, 1, 20, 5, 10]$$

(i.e. $0 + 1/(1 + 1/(3+1/(4+1/\ldots)))$), which shows no sign of terminating; there's no reason to think it is rational, nor is it unusually well approximated by rationals.