Consider integral involving the modified Bessel functions of the first and second kinds (of order one) and sine function
$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x$
For the case $b=c$, Mathematica gives the result
$I(a, b, b) = \frac{1}{m} E(m) + \left( 1-\frac{1}{m} \right) K(m), \qquad m = - \frac{4 b^2}{a^2}$
where $K(m)$ and $E(m)$ are the complete elliptic integral of the first and second kinds respectively (in Mathematica notation).
How can one derive the general result for $b \ne c$?