This is an integral from Anders Losberg in 1960 related to concrete pavement theory:
$$\int_0^\infty\frac{J_0(\alpha r)J_1(\alpha a)}{\alpha(1+\alpha^3\ell_e^3)}d\alpha$$
I'm wondering how to solve it so that I can more quickly numerically evaluate a larger expression in which it occurs several times. (For what it's worth, $a$ is a positive real constant, $r$ varies over a set of about eight fixed positive reals, and $\ell_e$ will vary over positive reals up to (usually) 100 or so.)
I'm not a professional mathematician, and it's been years since I cracked any integral by hand, but I am a professional programmer, so I tried throwing Mathematica (10.0.2) at it. Unfortunately, the evaluation did not complete in a timely manner.
So some of my more ancient neurons got excited about trying to use integration by parts. I literally dusted off my old calc textbook and refreshed myself. I then read the chapter about integration by parts.
I wrote the following Mathematica code to automate "trying out" some potential $u$-$dv$ breakdowns:
$Assumptions = r > 0 && a > 0 && l > 0;
integrand = BesselJ[0, aa r] BesselJ[1, aa a]/(aa (1 + aa^3 l^3));
terms = List @@ integrand;
parts = Times @@@ Subsets[terms, {1, Length@terms - 1}];
pairs = Thread@{parts, Reverse@parts};
ibpPairs = {Integrate[#2, {aa, 0, Infinity}], D[#1, aa]} & @@@ pairs;
ibpResults = Integrate[#, {aa, 0, Infinity}] & /@ Times @@@ ibpPairs;
TableForm[
ArrayFlatten@{{pairs, ibpPairs, List /@ ibpResults}},
TableHeadings -> {None, {"u", "dv", "v", "du", "int(v*du)"}}
]
Unfortunately, none of the $v$-$du$ integrations there at the end evaluated in a timely manner either.
I tried it all again, this time using the indefinite integral. And this time, all integrations evaluated in a timely manner... but remained in integral form. So Mathematica may be flummoxed.
(Note that I'm going to try using Mathematica against the definite integral again, but with a more generous allotment of time for it to complete. In the meantime, I thought it might help to broach the integral here. Of course, I'll post an answer on Mathematica's behalf if it manages to spit out something useful.)
Is there a solution to this integral? If so, how is it obtained and what is it?
Note that I'm not asking whether Mathematica can somehow solve the integral. I'm looking for any solution.
In case I've fallen into posing an XY problem question, I'll also note that I'm glad to hear comments and redirection about alternative ways to speed up numerical evaluation of this integral, e.g. by using series expansions somehow (just a guess that I'm not sure how to begin applying).
Update 2015-03-26:
Not directly related, but I did forward a suggestion to the symbolic math group at Wolfram to try to add this integral to the collection of those that Mathematica can handle. Maybe it can't be done, and/or maybe the customer significance factor just isn't there for them to spend time on it, but if they choose to investigate maybe something interesting will result.
Despite my enchantment with the not-quite-exact periodicity of the integrand's oscillations (at least for some $r$, $a$, and $\ell_e$), I doubt I'll develop the math chops to seriously attempt a solution for some time to come... But who knows, it might happen.