Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$\int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k,$$ where $$J_0(kr)$$ is the Bessel function of the first kind and where the interest is focused on where $$k$$ is small.

By expanding the hyperbolic function about $$k=0$$, $$\frac{\cosh(ky)}{\cosh(k)} = 1 + \textit{O}(k^2),$$ and the preceding integral reduces to $$\int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} k\mathrm{d} k = K_0(\xi r),$$ where $$K_0(\xi r)$$ is the modified Bessel function of the second kind.

The problem is the approximation for the hyperbolic function is valid only about $$k=0$$, while the integral is over the $$k\in(0,\infty)$$. How can I justify this controversy?

Or, perhaps it is acceptable to evaluate $$\int_{0}^{1} \frac{J_0(kr)}{k^2+\xi^2} k\mathrm{d} k,$$ instead, but how to do this?

Thanking you and, please, could you help me...

Wang Zhe