Within this AoPS thread it is asked to evaluate the following integral
$$\mathfrak I~=~\int_0^\infty \frac{x\sin x}{\cos x+\cosh^2 x}\mathrm dx\tag1$$
In order to be precise there is also a possible closed-form conjectured which is given by
$$\mathfrak I~=~G-\frac12\tag2$$
But as it is pointed out within the linked thread this seems to be only a reasonable approximation off after the $5$th decimal digit.
I have to admit that it is highly improbable that there exists a nice looking closed-form for $(1)$ since the integrand involves polynomials, trigonometric aswell as hyperbolic functions. I am not even sure how to get started, i.e. which substitution to choose or which technique at all to start with.
A related, but perhaps more handable integral, would be the following
$$\mathfrak J~=~\int_0^\infty \frac{\sin x}{\cos x+\cosh^2 x}\mathrm dx\tag{1$'$}$$
Out of experience I could imagine that $(1')$ may has a closed-form in terms of known constants $($or series$)$ since it only contains the two closely connected trigonometric and hyperbolic functions.
Is it in fact possible to deduce a closed-form for $(1)$ and $(1')$? For myself I cannot offer an approach since everything I tried was not helpful at all hence I was not even able to perform one or two steps in order to simplify the given integrals. I would be glad to see a full solution or even attempts in evaluating $(1)$ and $(1')$ since I have no idea how to deal with such integrands.
Thanks in advance!
EDIT
Out of pure chance I just stumbled upon a related MSE question dealing with the integral
$$\int_0^\infty\frac{x\sin^2x}{\cosh x+\cos x}\mathrm dx=1$$
Which on the other hand motivates me to believe that there may be a closed-form for $(1)$.