Consider the following integral
$$I\left(a, b\right)=\int\limits _0^{+\infty}\cos\left(x\right)I_1\left(ax\right)K_1\left(bx\right)dx,$$
where $a$ and $b$ are parameters such that $b\geq a \geq 0$, and $I_\nu\left(x\right)$ and $K_\nu\left(x\right)$ are the modified Bessel functions of the first and second kind, respectively.
Is it possible to express this integral in terms of elementary and basic special functions?
Similar issue has already been addressed; the result involves complete elliptic integrals.
Best Answer
$$\int_0^{\infty } \cos (x) I_1(a x) K_1(b x) \, dx=\frac{Q_{\frac{1}{2}}\left(\frac{a^2+b^2+1}{2 a b}\right)}{2 \sqrt{a b}}$$
where:$Q_{\frac{1}{2}}\left(\frac{a^2+b^2+1}{2 a b}\right)$ is Legendre Q Function.
from Book: Table of Integrals, Series, and Products 8th Edition on page: 726 in:
6.672.4