A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

Question

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)?

Answer 1

Hint: Use the formula $(79)$ from this MathWorld page: $$J_0(z)=\frac1\pi\int_0^\pi e^{i\,z\cos\theta}\,d\theta$$ and then change the order of integration.

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Result: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]2\right)}xdx=\arcsin\frac{\sqrt{2+\sqrt[3]4+\sqrt[3]{16}}-\sqrt{2+\sqrt[3]4-\sqrt[3]{16}}}2$$