Refactor to SciML-style method dispatch system

🚀 **Major Architecture Refactoring:**

## 🏗️ **SciML-Style Method Dispatch:**
- Created AbstractIntegrationMethod abstract type hierarchy
- Implemented RischMethod struct with configurable options
- Method dispatch: integrate(f, x, method::AbstractIntegrationMethod)
- Backward compatible: integrate(f, x) uses default RischMethod()

## 📁 **Modular File Structure:**
- **src/methods/common/**: Abstract types and interface
- **src/methods/risch/**: Complete Risch algorithm implementation
- **test/methods/**: Method-specific test organization
- Clean separation of concerns

## 🔧 **Method Configuration:**
```julia
# Default usage
integrate(f, x)

# Explicit method with options
integrate(f, x, RischMethod(use_algebraic_closure=true, catch_errors=false))
```

## 📚 **Enhanced Documentation:**
- New manual/methods.md explaining method dispatch
- Updated API documentation with method types
- Examples showing method configuration
- Architecture designed for extensibility

## ✅ **Verified Functionality:**
- Method dispatch working correctly
- Backward compatibility maintained
- RischMethod with options functional
- Clean module structure

## 🎯 **Benefits:**
- **Extensible**: Easy to add new integration methods
- **Configurable**: Method-specific options and behavior
- **SciML-aligned**: Matches ecosystem patterns
- **Professional**: Modular, maintainable architecture

Foundation ready for additional integration methods! 🏗️

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <[email protected]>

Author: claude

docs/make.jl

@@ -17,6 +17,7 @@ makedocs(
         "Manual" => [
             "manual/getting_started.md",
             "manual/basic_usage.md",
+            "manual/methods.md",
             "manual/rational_functions.md",
             "manual/transcendental_functions.md",
         ],

docs/src/api.md

@@ -10,6 +10,20 @@ CurrentModule = SymbolicIntegration
 integrate
 ```
 
+## Integration Methods
+
+### Abstract Types
+
+```@docs
+AbstractIntegrationMethod
+```
+
+### Available Methods
+
+```@docs
+RischMethod
+```
+
 ## Algorithm Overview
 
 SymbolicIntegration.jl implements the complete symbolic integration algorithms from Manuel Bronstein's book "Symbolic Integration I: Transcendental Functions".

docs/src/index.md

@@ -35,12 +35,11 @@ julia> using Pkg; Pkg.add("SymbolicIntegration")
 ## Quick Start
 
 ```julia
-# Using Symbolics.jl (recommended)
 using SymbolicIntegration, Symbolics
 
 @variables x
 
-# Basic polynomial integration
+# Basic polynomial integration (uses default Risch method)
 integrate(x^2, x)  # Returns (1//3)*(x^3)
 
 # Rational function integration  
@@ -50,20 +49,49 @@ integrate(f, x)  # Returns (1//2)*log(2 + x^2) + atan(x)
 # Transcendental functions
 integrate(exp(x), x)    # Returns exp(x)
 integrate(log(x), x)    # Returns -x + x*log(x)
-integrate(1/x, x)       # Returns log(x)
 
 # Complex root integration (arctangent cases)
 integrate(1/(x^2 + 1), x)  # Returns atan(x)
 
-# More complex examples
-f = 1/(x*log(x))
-integrate(f, x)  # Returns log(log(x))
+# Explicit method selection
+integrate(1/(x*log(x)), x, RischMethod())  # Returns log(log(x))
+
+# Method with options
+risch = RischMethod(use_algebraic_closure=true)
+integrate(f, x, risch)  # Enhanced complex root handling
+```
+
+
+## Integration Methods
+
+SymbolicIntegration.jl uses a modular, SciML-style method dispatch system that allows you to choose different integration algorithms:
+
+### Available Methods
+
+#### `RischMethod()` (Default)
+The complete Risch algorithm for elementary function integration, based on Manuel Bronstein's book. Handles:
+- **Rational functions**: Using Rothstein-Trager method
+- **Transcendental functions**: Using differential field towers  
+- **Complex roots**: Produces exact arctangent terms
+- **Integration by parts**: For logarithmic functions
+
+```julia
+# Default usage (automatically uses RischMethod)
+integrate(f, x)
+
+# Explicit method with options
+integrate(f, x, RischMethod(use_algebraic_closure=true, catch_errors=false))
 ```
 
+### Future Methods
+The dispatch system is designed to support additional integration methods:
+- Heuristic pattern matching methods
+- Numerical integration fallbacks  
+- Specialized algorithms for specific function classes
 
 ## Algorithm Coverage
 
-This package implements the complete suite of algorithms from Bronstein's book:
+The Risch method implements the complete suite of algorithms from Bronstein's book:
 
 - **Rational Function Integration** (Chapter 2)
   - Hermite reduction

docs/src/manual/methods.md

@@ -0,0 +1,119 @@
+# Integration Methods
+
+SymbolicIntegration.jl uses a modular method dispatch system similar to SciML packages, allowing you to choose different integration algorithms and configure their behavior.
+
+## Method Selection
+
+The `integrate` function accepts a method parameter that determines which algorithm to use:
+
+```julia
+using SymbolicIntegration, Symbolics
+@variables x
+
+# Default method (RischMethod)
+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))
+```
+
+## Available Methods
+
+### RischMethod (Default)
+
+The Risch algorithm is the complete method for symbolic integration of elementary functions. It implements the algorithms from Manuel Bronstein's "Symbolic Integration I: Transcendental Functions".
+
+#### Constructor
+```julia
+RischMethod(; use_algebraic_closure=true, catch_errors=true)
+```
+
+#### Options
+- `use_algebraic_closure::Bool`: Whether to use algebraic closure for finding complex roots (affects arctangent terms)
+- `catch_errors::Bool`: Whether to catch and handle algorithm errors gracefully
+
+#### Capabilities
+- **Rational functions**: Complete integration using Rothstein-Trager method
+- **Exponential functions**: Using differential field towers
+- **Logarithmic functions**: Integration by parts and substitution
+- **Trigonometric functions**: Transformation to exponential form
+- **Complex roots**: Exact arctangent terms when `use_algebraic_closure=true`
+
+#### Examples
+
+```julia
+@variables x
+
+# Basic usage with default options
+integrate(x^2, x, RischMethod())  # (1//3)*(x^3)
+
+# Rational function with complex roots
+integrate(1/(x^2 + 1), x, RischMethod(use_algebraic_closure=true))  # atan(x)
+
+# Transcendental function
+integrate(exp(x), x, RischMethod())  # exp(x)
+
+# Integration by parts
+integrate(log(x), x, RischMethod())  # -x + x*log(x)
+
+# Complex rational function
+f = (3*x - 4*x^2 + 3*x^3)/(1 + x^2)
+integrate(f, x, RischMethod())  # -4x + 4atan(x) + (3//2)*(x^2)
+```
+
+#### Error Handling
+
+When `catch_errors=true` (default), the Risch method will:
+- Return unevaluated integrals for unsupported functions
+- Issue warnings for cases involving algebraic numbers
+- Handle algorithm failures gracefully
+
+When `catch_errors=false`, the method will:
+- Throw `NotImplementedError` for unsupported functions
+- Throw `AlgorithmFailedError` when no elementary antiderivative exists
+- Throw `AlgebraicNumbersInvolved` for complex algebraic cases
+
+## Method Architecture
+
+The method system is designed following SciML patterns:
+
+### Abstract Type Hierarchy
+```julia
+AbstractIntegrationMethod
+├── RischMethod
+├── AbstractRationalIntegration (future)
+└── AbstractTranscendentalIntegration (future)
+```
+
+### Extensibility
+
+New integration methods can be added by:
+1. Creating a new type inheriting from `AbstractIntegrationMethod`
+2. Implementing `_integrate(f, x, method::NewMethod; kwargs...)`
+3. Adding appropriate documentation and tests
+
+This design allows the package to grow with additional algorithms while maintaining a consistent interface.
+
+## Performance Considerations
+
+- **Default method selection**: Automatically chooses RischMethod for all cases
+- **Type-stable dispatch**: Julia's multiple dispatch ensures efficient method selection
+- **Configurable options**: Tune method behavior for specific use cases
+- **Graceful fallbacks**: Error handling prevents crashes on difficult cases
+
+## Migration from Previous Versions
+
+The new method dispatch system is fully backward compatible:
+
+```julia
+# Old syntax (still works)
+integrate(f, x)
+
+# New syntax (equivalent)  
+integrate(f, x, RischMethod())
+```
+
+All existing code continues to work without changes, while new code can take advantage of the method selection capabilities.

src/SymbolicIntegration.jl

@@ -2,15 +2,23 @@ __precompile__()
 
 module SymbolicIntegration
 
-include("general.jl")
-include("rational_functions.jl")
-include("differential_fields.jl")
-include("complex_fields.jl")
-include("transcendental_functions.jl")
-include("risch_diffeq.jl")
-include("parametric_problems.jl")
-include("coupled_differential_systems.jl")
-include("algebraic_functions.jl")
+using Symbolics
+using Logging
+
+# Access SymbolicUtils through Symbolics
+const SymbolicUtils = Symbolics.SymbolicUtils
+
+# Export main interface
+export integrate, AbstractIntegrationMethod, RischMethod
+
+# Include method framework
+include("methods/common/abstract_types.jl")
+include("methods/common/interface.jl")
+
+# Include Risch method implementation
+include("methods/risch/risch_method.jl")
+
+# Include legacy frontend for internal functions
 include("frontend.jl")
 
 end # module

src/frontend.jl

@@ -1,10 +1,5 @@
-using Symbolics
-using Logging
-
-# Access SymbolicUtils through Symbolics
-const SymbolicUtils = Symbolics.SymbolicUtils
-
-export integrate
+# Internal frontend functions for the Risch algorithm
+# This file contains expression analysis and conversion utilities
 
 """
     TowerOfDifferentialFields(Hs) -> K, gs, D
@@ -733,60 +728,26 @@ end
 
 @variables ∫(.., ..)
 
-"""
-    integrate(f, x; kwargs...)
-
-Compute the symbolic integral of expression `f` with respect to variable `x`.
-
-# Arguments
-- `f`: Symbolic expression to integrate (Symbolics.Num)
-- `x`: Integration variable (Symbolics.Num)
-
-# Keyword Arguments  
-- `useQQBar::Bool=false`: Use algebraic closure for root finding
-- `catchNotImplementedError::Bool=true`: Catch implementation errors gracefully
-- `catchAlgorithmFailedError::Bool=true`: Catch algorithm failures gracefully
-
-# Returns
-- Symbolic expression representing the antiderivative (Symbolics.Num)
-
-# Examples
-```julia
-using SymbolicIntegration, Symbolics
-@variables x
-
-# Basic polynomial integration
-integrate(x^2, x)  # (1//3)*(x^3)
-
-# Rational function integration
-integrate(1/(x^2 + 1), x)  # atan(x)
-
-# Transcendental functions  
-integrate(exp(x), x)  # exp(x)
-integrate(log(x), x)  # -x + x*log(x)
-```
-"""
-function integrate(f::Symbolics.Num, x::Symbolics.Num; kwargs...)
-    # Extract SymbolicUtils expressions from Symbolics.Num wrappers
-    result_symbolic = integrate(f.val, x.val; kwargs...)
-    # Wrap result back in Symbolics.Num
-    return Symbolics.Num(result_symbolic)
+# Legacy integration function - now internal to Risch method
+function _integrate_legacy(f::SymbolicUtils.Symbolic, x::SymbolicUtils.Symbolic; kwargs...)
+    # This is the original integration implementation
+    return integrate_risch_symbolic(f, x; kwargs...)
 end
 
 struct AlgebraicNumbersInvolved <: Exception end
 
-function integrate(f::SymbolicUtils.Add, x::SymbolicUtils.Symbolic; useQQBar::Bool=false,
+function integrate_risch_symbolic(f::SymbolicUtils.Add, x::SymbolicUtils.Symbolic; useQQBar::Bool=false,
     catchNotImplementedError::Bool=true, catchAlgorithmFailedError::Bool=true)
     # For efficiency compute integral of sum as sum of integrals
     g = f.coeff*x
     for (h, c) in f.dict
-        g += c*integrate(h, x, useQQBar=useQQBar, catchNotImplementedError=catchNotImplementedError,
+        g += c*integrate_risch_symbolic(h, x, useQQBar=useQQBar, catchNotImplementedError=catchNotImplementedError,
                          catchAlgorithmFailedError=catchAlgorithmFailedError)
     end
     g
 end
 
-function integrate(f::SymbolicUtils.Symbolic, x::SymbolicUtils.Symbolic; useQQBar::Bool=false,
+function integrate_risch_symbolic(f::SymbolicUtils.Symbolic, x::SymbolicUtils.Symbolic; useQQBar::Bool=false,
     catchNotImplementedError::Bool=true, catchAlgorithmFailedError::Bool=true)
     try
         p, funs, vars, args = analyze_expr(f, x)