For each $x>0$ define, $$F(x)=\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt$$
Is it the case that $F$ is positive for all $x>0$? This appears to be the case, when I do approximations. This kernel is not decreasing, since $\tanh$ is zero at zero, and so a proof appears to be very complicated. Does anyone know of a reference or a proof? Or a counter-example?