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Got a tough one and I'm just not seeing the path here. I need to find the close form expression of:

The integral from zero to infinity:

∫x^{λ}* cos(2ax) * [K_{v}(x)]^{2}dx

where K_{v}(x) is the modified Bessel function of the second kind of order v and argument x. If it helps, the value of v=1/3 and the value of λ=2/3

The result will have the form of a hypergeometric function_{2}F_{1}

I've just been racking my brain for too long with this one. If anyone has some experience with Bessel functions, any help would be appreciated.

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# Evaluate Integral of Bessel K Function

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