I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds

$$I = \int_0^{\infty} \frac{\cos(ax)}{x} I_0(bx) K_1(cx) \mathrm{d}x$$

where $a, b, c$ are real and positive numbers.

It does not appear in Gradshteyn and Ryzhik, and it is not known by Mathematica.

Can any mathematicians suggest a method or trick to evaluate this integral exactly or approximately?