Is it possible to evaluate the integral
$$\int\frac{\operatorname{Ai}'(x+a)}{\operatorname{Ai}(x+a)}xdx$$
where $\operatorname{Ai}(x)$ is an Airy function, and $a$ is a constant, in terms of special functions?
Edit: since it was pointed out that since $\operatorname{Ai}(x)$ will have zeros that make this poorly defined, can we instead find a closed form for an integral of the form
$$\int_c^x\frac{\operatorname{Ai}'(t-a)}{\operatorname{Ai}(t-a)}tdt,$$ where $c>a$, so that we avoid these singularities?