Add more special-function rules

[AshtonSBradley]

Summary

• Adds rule-based integration support for more registered special functions.
• Extends the existing Bessel special-function rules with Hankel H1/H2 analogues.
• Adds focused regression coverage for Hankel, Airy, error functions, exponential integral, and sine/cosine integral primitives.

Explanation

This PR adds low-risk special-function rules for cases that are already represented in the rule-based integration machinery.

The Hankel rules mirror the existing Bessel J/Y monomial recurrence structure, covering direct order-raising primitives and the same even-step recurrence form used by the current Bessel rules.

It also adds direct primitives for Airy derivative identities and the Airy differential equation cases:

• ∫ airyaiprime(a*x) dx
• ∫ airybiprime(a*x) dx
• ∫ x*airyai(x) dx
• ∫ x*airybi(x) dx

Finally, it adds integration-by-parts primitives for linear arguments of erf, erfi, expinti, sinint, and cosint. These use standard derivative identities and do not add new dependencies or public API.

Tests

• julia --project=. test/methods/rule_based/test_special_functions.jl
  • Passed: 22/22
• julia --project=. test/runtests.jl
  • Passed with exit code 0: 195 passed, 1 broken
  • The runner also reported one existing RuleBased baseline improvement to tighten separately.

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 54.51%. Comparing base (0f6cd70) to head (3625e0e).
⚠️ Report is 6 commits behind head on main.
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Additional details and impacted files

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  Coverage   54.51%   54.51%           
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  Files          22       23    +1     
  Lines        4309     4309           
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  Hits         2349     2349           
  Misses       1960     1960           

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[Bumblebee00]

Thank for adding these rules I think they are really useful. Just one correction, I think is better to put in the file src/methods/rule_based/rules2/8 Special functions/8.1 Bessel functions.jl only the rules for the bessel functions, and to put the airy functions in another file (maybe 8.1 Airy functions.jl), then erf, erfi, expinti, in another and also sinint, and cosint in another, to keep it ordered. Could you do this modification? then is ready to merge

Then 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?

[AshtonSBradley]

Thank for adding these rules I think they are really useful. Just one correction, I think is better to put in the file src/methods/rule_based/rules2/8 Special functions/8.1 Bessel functions.jl only the rules for the bessel functions, and to put the airy functions in another file (maybe 8.1 Airy functions.jl), then erf, erfi, expinti, in another and also sinint, and cosint in another, to keep it ordered. Could you do this modification? then is ready to merge

Done

Then I am actually curious on why you added them, you use them in your research/work?

Research - I would like to see first class open source symbolic integration available to the math/physics community.

did you create these rules manually yourself or you translated them from the Mathematica ones?

I added these manually. They’re small standard identities chosen to match the existing rule style here, not translated from the Mathematica rules.