Add initial Bessel rule support

[AshtonSBradley]

Summary

• Adds initial rule-based support for Bessel integrals through SpecialFunctions.jl.
• Teaches the Mathematica rule/test translators about Bessel, Hankel, and Airy function names.
• Adds generic product distribution for algebraic sums multiplied by supported special functions.

Explanation

This adds a new Bessel rule file covering direct power-weighted Bessel integrals for besselj, bessely, besseli, and besselk, plus recurrence reductions for even positive order gaps such as x^3*besselj(0, a*x).

The implementation also makes special-function handling less Bessel-specific: products containing an x-dependent sum and a supported special-function factor are distributed before integration. That lets algebraic combinations like (x + x^3)*besselj(0, a*x) reuse the simpler Bessel rules, and it also applies to other known special functions such as Airy functions.

SpecialFunctions is added as a direct dependency because rule results now explicitly emit SpecialFunctions.besselj, bessely, besseli, besselk, Hankel, and Airy calls.

Tests

• julia --project=@runic -m Runic --inplace src/methods/rule_based/rules2/8\ Special\ functions/8.1\ Bessel\ functions.jl test/methods/rule_based/test_special_functions.jl completed successfully.
• julia --project=. test/methods/rule_based/test_special_functions.jl → 11 passed.
• env TEST_GROUP=easy julia --project=. test/runtests.jl → 182 passed, 1 broken, 183 total.

Notes

• This does not attempt the odd-gap Bessel cases that require broader special-function results such as Struve or hypergeometric forms.
• No public API is changed beyond adding the direct SpecialFunctions dependency.

[codecov-commenter]

⚠️ Please install the to ensure uploads and comments are reliably processed by Codecov.

Codecov Report

❌ Patch coverage is 93.33333% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 54.51%. Comparing base (85a221a) to head (0f6cd70).
⚠️ Report is 37 commits behind head on main.

Files with missing lines	Patch %	Lines
...iscellaneous/0.1 Integrand simplification rules.jl	0.00%	1 Missing ⚠️
❗ Your organization needs to install the Codecov GitHub app to enable full functionality.

Additional details and impacted files

@@             Coverage Diff             @@
##             main     #109       +/-   ##
===========================================
+ Coverage   13.91%   54.51%   +40.60%     
===========================================
  Files          22       22               
  Lines        4291     4309       +18     
===========================================
+ Hits          597     2349     +1752     
+ Misses       3694     1960     -1734     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[Bumblebee00]

This looks great, thank you for contributing and taking the time to add new rules @AshtonSBradley ! I am actually curious on why you added them, you use them in your research/work? did you create these rules manually yourself or you translated them from the Mathematica ones?

Some small notes:

• in test/methods/rule_based/test_rule2.jl there is a testset called "special functions in rules" but is not meant to test the special functions form SpecialFunctions.jl, is meant to test only exp and sqrt. I called it special because they are shorthand notation for powers, and this complicates a bit the mathcing logic. So the test you added are not needded, i removed them and changed the name to a clearer one (here)
• I dont really like the addition of rule 0_1_1 in the src/methods/rule_based/rules2/9 Miscellaneous/0.1 Integrand simplification rules.jl file, because those are rules executed for every expression and for most expressions this rule is useless. I moved it to src/methods/rule_based/rules2/8 Special functions/8.1 Bessel functions.jl with the index 8_1_0 so still before the other special functions rules (the tests still pass)
• would be cool to also add rules for functions like besselj0 becasue this is whats happening now:

julia> integrate(x * SpecialFunctions.besselj(0, 2x))
(1//2)*x*besselj(1, 2x)

julia> integrate(x * SpecialFunctions.besselj0(2x))
∫(x*besselj0(2x), x)

but i actually dont know it that's really useful

• test/methods/rule_based/test_special_functions.jl is ok for now because testing is a bit messy but the way to add tests is to insert them in a file in test/test_files add that filename to the list in test/rundifficulttests.jl

[Bumblebee00]

Also @ChrisRackauckas the next times please wait for also my review before merging into main 😃

[ChrisRackauckas]

Okay yup, always happy to have other folks maintain. I didn't know if you were still around. But I also thought this was posted like weeks ago, and now I see it was just yesterday 😅 time is nebulous.