How big is the integral $\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx$ I can't seem to get Maple to approximate the integral 
$$\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx.$$
Could somebody tell me why?
This integral "should be" well-defined. (My reasons are not mathematical. The  book I'm reading suggest that this integral makes sense.) Do note that the denominator of the integrand explodes at $x=0$, but this should not be a problem... 
Can we give an upper bound for this integral?

Using the following inequality:
$$
  \frac{x}{\sqrt{\cosh{x}-1}} = \sqrt{2} \frac{x/2}{\sinh(x/2)} \leqslant \sqrt{2}
$$
It is easy to work out the upper bound:
$$
 \int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} \mathrm{d}x <
   \sqrt{2} \int_0^\infty \exp(-x^2/4)\cosh(x) \mathrm{d}x = \sqrt{2 \pi} \mathrm{e} \approx 6.8 
$$
