I have a difficult integral again, which I would really prefer to be able to express analytically:
$$Y_{nlL}=\int_0^1 \sin \pi n r J_0( \gamma_l r )J_0(\gamma_L r) dr$$
Here $n=1,2,3,4,\dots$ and $\gamma_l$ is the $l$th zero of $J_0(x)$.
The integrand looks nice, but it's highly oscillatory for large values of the parameters, and since there are $3$ of them, precomputing it numerically for all the cases up to some $N_m,L_m$ takes very long.
I have found a lot of examples with a single Bessel function and the limits up to $\infty$ (see this question for example), but I don't see how they help in this case.
A series expansion of one of the functions is not the best option either, since the expression for the terms would remain complicated, if they even exist.
If the community thinks I should, I will provide some work of my own, but I would really prefer a hint in the correct direction without the need to do a lot of useless manipulations.
In the end, it would be great to get an expression in terms of a finite number of any special functions Mathematica can handle. Fast converging single series is an option too of course, but I don't see how it could work with an oscillatory integrand.
The motivation for this: my attempts to approximate a much more difficult double integral have led to this. Using numerical pre-computation of $Y_{nlL}$, I have achieved a good accuracy for the original integral, but what's more important: a very fast evaluation.
One way I could see is to use one of the integral representations for the Bessel function:
$$J_0(a)=\frac{2}{\pi} \int_0^1 \frac{\cos a u}{\sqrt{1-u^2}}du$$
This could allow me to easily take the integral over $r$, but I would be left with a double integral instead, with the integrand still oscillatory.
I suppose I could explicitly try Levin's quadrature for highly oscillatory integrals, but it needs integrand in the form:
$$f(x) e^{i g(x)}$$
So if I had to transform both the Bessel functions into this form, I would again be left with a double integral for each step of the quadrature.