Refactor to SciML-style method dispatch system

[ChrisRackauckas-Claude]

🚀 Major Architecture Refactoring: SciML-Style Method Dispatch

This PR transforms SymbolicIntegration.jl into a modern, extensible package using SciML-style method dispatch patterns, making it ready for multiple integration algorithms.

🏗️ SciML-Style Method Dispatch System

Clean Method Interface

# Default Risch method
integrate(f, x)

# Explicit method selection  
integrate(f, x, RischMethod())

# Method with custom options
integrate(f, x, RischMethod(use_algebraic_closure=true, catch_errors=false))

Abstract Type Hierarchy

AbstractIntegrationMethod
├── RischMethod (implemented)
├── AbstractRationalIntegration (future)
└── AbstractTranscendentalIntegration (future)

📁 Professional File Organization

Modular Source Structure

src/
├── SymbolicIntegration.jl       # Main module with all imports at top
├── methods/
│   ├── common/
│   │   ├── abstract_types.jl    # Type hierarchy & method traits
│   │   └── interface.jl         # Main dispatch interface
│   └── risch/
│       ├── risch_method.jl      # Method implementation
│       ├── rational_functions.jl
│       ├── transcendental_functions.jl
│       ├── differential_fields.jl
│       └── ... (all Risch components)
└── frontend.jl                  # Internal expression analysis

Organized Test Structure

test/
├── runtests.jl                  # Simple include dispatcher
├── test_core.jl                 # Core functionality & method dispatch
└── methods/
    ├── common/
    │   └── test_stewart_examples.jl
    └── risch/
        ├── test_rational_integration.jl
        ├── test_complex_fields.jl
        └── ... (method-specific tests)

🔧 RischMethod Configuration

Constructor & Options

RischMethod(; 
    use_algebraic_closure=true,    # Enable complex root finding
    catch_errors=true              # Graceful error handling
)

Method Traits (Instance-Based)

method_supports_rational(method::RischMethod)      # true
method_supports_transcendental(method::RischMethod) # true

📚 Enhanced Documentation

New Architecture Guide

• manual/methods.md: Complete method dispatch documentation
• Abstract type hierarchy: How the system works
• Extension guide: Adding new integration methods
• Configuration examples: Method options and use cases

Updated API Reference

• Method types: AbstractIntegrationMethod, RischMethod
• Dispatch patterns: Different usage examples
• Professional documentation: Following SciML standards

🧹 Architecture Cleanup

Dependency Management

• All imports at top level: Single source in main module
• No redundant imports: Clean file organization
• Proper export structure: Only public API exported

Code Quality

• Removed @pure annotations: Using instance methods instead
• Simplified test structure: runtests.jl just includes others
• No backward compatibility complexity: Clean, focused design
• Professional standards: Following Julia/SciML best practices

✅ Functionality Verified

Method dispatch working:

• ✅ Default integration: integrate(x^2, x) → (1//3)*(x^3)
• ✅ Explicit method: integrate(x^2, x, RischMethod()) → (1//3)*(x^3)
• ✅ Method options: RischMethod(use_algebraic_closure=false) working
• ✅ Core algorithms: Polynomial, exponential, logarithmic integration

🎯 Future Extensibility

This architecture enables:

• Additional methods: Heuristic, numerical, specialized algorithms
• Method composition: Combining different approaches
• Community plugins: Third-party method implementations
• Performance tuning: Method-specific optimizations

🔗 SciML Ecosystem Alignment

Follows established SciML patterns:

• Abstract type hierarchies for algorithm organization
• Method dispatch for algorithm selection
• Configurable structs for method options
• Instance-based traits for method capabilities
• Modular architecture for extensibility

Foundation ready for a comprehensive symbolic integration ecosystem! 🏗️

🤖 Generated with Claude Code