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According to http://people.math.sfu.ca/~cbm/aands/page_376.htm, I know that

$\int_0^\infty e^{-z\cosh t}\cosh vt~dt=K_v(z)$ and $\int_0^\infty e^{-z\cosh t}\sinh^vt~dt=\dfrac{2^\frac{v}{2}~\Gamma\left(\dfrac{v+1}{2}\right)K_\frac{v}{2}(z)}{\sqrt\pi z^\frac{v}{2}}$ .

How about $\int_0^\infty e^{-z\cosh t}\sinh^at\cosh bt~dt$ , especially when $a$ and $b$ are not integers?

Does some formulae in http://dlmf.nist.gov/5.12 helpful?

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