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Why can't Mathematica integrate

r = Integrate[Exp[-x Cosh[t]], {t, 0, Infinity}];
r = Assuming[Element[x, Reals], Simplify[r]];
Together[r]

From Wikipedia, it should be $K_0(x)$, the modified Bessel function of the second kind for $\alpha =0$, since$$K_\alpha(x) = \int_0^\infty \exp(-x\cosh t) \cosh(\alpha t) \,dt.$$ And there is definitely a BesselK function in Mathematica. It seems like a similar problem to here.

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    $\begingroup$ It has been like that for a while; I'm not sure why. In any case, try making the substitution $u=\cosh t$ before using Integrate[]. $\endgroup$ – J. M. will be back soon Feb 27 '16 at 4:23
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    $\begingroup$ To elaborate on the comment by @J.M., Integrate[Exp[-x u]/Sqrt[u^2 - 1], {u, 1, Infinity}, Assumptions -> x > 0] gives the desired result. $\endgroup$ – bbgodfrey Feb 27 '16 at 5:00
  • $\begingroup$ Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. $\endgroup$ – bbgodfrey Feb 27 '16 at 5:01

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