Somewhat Irreproducible Integrate Results

Question

Backslide introduced in v10 and persisting through v10.3.1.

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In the course of considering question 102922, I encountered erratic results from a particular integration. It is illustrated as follows. With a new session of Mathematica

$Version
(* 10.3.0 for Microsoft Windows (64-bit) (October 9, 2015) *)

run

Do[ans = Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, 
        Assumptions -> {l > 0, R > 0, n ∈ Integers}]; Print[ans], {i, 6}]

Integate returns unevaluated the first time called. Thereafter, it returns

(* ConditionalExpression[(2^(-1 + n)*Sqrt[Pi]*R^(-1 - n)*Gamma[(1 + l + n)/2])/
    Gamma[(2 + l - n)/2], n < 1 && l + n > -1] *)

as expected. A variant on this strange behavior, again in a new session of Mathematica, is

Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, Assumptions -> {l > 0, R > 0}]
(* returns expected result *)
Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, Assumptions -> {l > 0, R > 0, 
    n ∈ Integers}]
(* returns unevaluated *)
Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, Assumptions -> {l > 0, R > 0, 
    n ∈ Integers}]
(* returns unevaluated *)
Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, Assumptions -> {l > 0, R > 0, 
    n ∈ Integers}]
(* returns expected result, as at the beginning of the question*)

There are many possible combinations of Assumptions for this calculation, and most behave similarly, at first returning unevaluated and later returning evaluated. Moreover, with Assumptions cases a, b, and c, Integrate run with the sets in that order typically returns unevaluated in the first two cases, and evaluated in the third, and shuffling the three cases likewise typically also returns the first two cases (for example, b and c) unevaluated and the third (in the same example, a) evaluated.

I know of two work-arounds. One, due to Bob Hanlon, is to use GenerateConditions -> False which gives the evaluated result but at the cost of no conditions. The second is illustrated at the beginning of this question. It, too, seems unsatisfactory. So, can this behavior be explained, and what is a better work-around? Thanks.

Answer 1

The reference provided above by Daniel Lichtblau, and the comments therein by Szabolcs and him, suggested that Integrate may be returning unevaluated simply because it timed out, which appears to occur after about 60 seconds. The obvious solution, then, is to give it more time.

TimeConstrained[
    Integrate[k^n SphericalBesselJ[l, R*k], {k, 0, ∞}, 
        Assumptions -> {l > 0, R > 0, n ∈ Integers}], 120]

which returns with the desired answer after nearly 100 seconds on my PC.

(As explained by Daniel Lichtblau and Szabolcs in the reference, a second call to Integrate after it times out may provide an answer even though the first call did not, because Integrate stores intermediate results and, therefore, may proceed more rapidly when called the second time. Consequently, the comparison made in the comment above by xzczd between v9 and v10 may reflect that v10 is slower than v9 or has a smaller default time limit in this instance.)