I've found the following identity.
$$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$
I could verify it by using CAS, and calculate the integrals in term of exponential and trigonometric integrals, then using identities between them. However, I think there is a more elegent way to prove it.
How could we prove this identity?
Also would be nice to see some references.