I require the following integral involving the modified Bessel functions of the first and second kinds of order one

$$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \quad\text{where} \quad c \ge b$$

For the case $b=c$, Mathematica gives the result

$$I(a, b) = \frac{1}{m} E(m) + \left( 1-\frac{1}{m} \right) K(m), \quad\text{where} \quad m = - \frac{4 b^2}{a^2}$$

where $K(m)$ and $E(m)$ are the complete elliptic integral of the first and second kinds respectively (as defined by Mathematica).

How to solve the general case? Any help is much appreciated.

Please note this is a cross-post from Math.SE where it received no answers after 3 months.