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How can I make Mathematica understand these two integrals?

$$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{i t}(u) \frac{du}{u} = K_{i t}(x) K_{i t}(y) $$

There is some discussion about the first form in this question: (4728) incorrect intergrate results. I've run across the second form in the literature under the name "MacDonald identity."

Stating explicitly that all variables are real and positive does not seem to help Integrate evaluate these forms. Is there something else I can do to help Mathematica concerning these integrals? I have some experience working with integrals related to modified Bessel functions, but I'm at a loss here. Any tips for success with these forms or other things to try are appreciated.

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  • $\begingroup$ you could up vote this question (:-)) mathematica.stackexchange.com/questions/6169/… $\endgroup$
    – chris
    Commented Nov 12, 2014 at 8:45
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    $\begingroup$ "Integrate`InverseIntegrate[Exp[-x Cosh[t]], {t, 0, Infinity}, Assumptions -> Re[x] > 0]" $\endgroup$
    – Dr. belisarius
    Commented Nov 12, 2014 at 13:12
  • $\begingroup$ @belisarius Yes I saw this 'experimental' command in the other question.I actually assumed that experimental meant it wasn't in the retail version. It does not find the answer to the second integral I posted. $\endgroup$
    – Kevin Driscoll
    Commented Nov 12, 2014 at 19:58

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