I am wondering how to compute the following integral: $$\int_{x_0}^{\infty} K_{i\alpha}(x) dx$$ where $K_{i\alpha}(x)$ is the modified Bessel function of the second kind, also known as the Bessel function of imaginary argument. Has anyone a reference that performs this integral in this way? I looked for it in NIST Digital Library of Mathematical Functions and other sources, with no success.
In fact, Ref. 1 says that I should find some further relations in Ref. [A. Apelblat (1983) Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.], but I have no access to this reference.
Assuming $t>0$ , Mathematica returns
$$\int_{t}^{\infty} K_{i\alpha}(x)\,dx=2^{-1+i \alpha } \,\Gamma (i \alpha -1)\, t^{1-i \alpha } \, _1F_2\left(\frac{1-i \alpha }{2};\frac{3-i \alpha }{2},1-i \alpha ;\frac{t^2}{4}\right)+$$ $$2^{-1-i \alpha }\, \Gamma (-i \alpha -1)\, t^{1+i \alpha } \, _1F_2\left(\frac{1+i \alpha }{2};\frac{3+i \alpha }{2},1+i \alpha ;\frac{t^2}{4}\right)+$$ $$\frac{\pi}{2} \, \text{sech}\left(\frac{\pi \alpha }{2}\right)$$