I am stuck with this integral: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx,$$ where $J_0$ is the Bessel function of the first kind.
Is it possible to express this integral in a closed form (preferably, using elementary functions, Bessel functions, integers and basic constants)?