Today I came across the integral


$\int\_a^\infty e^{-bx} I\_n(x) dx$


where $I\_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, but I am afraid no closed-form solution exists otherwise. Correct me if I am wrong!



A Bessel integral
I suspect your fears are justified. You are basically trying to integrate the integrand from $0$ to $a$ (since, as you have noted, and Mathematica confirms, the integral from $0$ has a closed form) You are thus trying to evaluate the indefinite integral of a Bessel function times an exponential, which does not exist in closed form (unless you call it a special function, and thus, presto, closed form, as @Yemon notes.)


