Transcendental Function Integration

SymbolicIntegration.jl implements the Risch algorithm for integrating elementary transcendental functions.

Supported Functions

Exponential Functions

using SymbolicIntegration, Symbolics
@variables x

integrate(exp(x), x)        # exp(x)
integrate(exp(2*x), x)      # (1//2)*exp(2*x)
integrate(x*exp(x), x)      # -exp(x) + x*exp(x)

Logarithmic Functions

integrate(log(x), x)        # -x + x*log(x)
integrate(1/(x*log(x)), x)  # log(log(x))
integrate(log(x)^2, x)      # x*log(x)^2 - 2*x*log(x) + 2*x

Trigonometric Functions

Basic trigonometric functions are transformed to exponential form:

integrate(sin(x), x)   # Transformed via half-angle formulas
integrate(cos(x), x)   # Transformed via half-angle formulas  
integrate(tan(x), x)   # Uses differential field extension

Hyperbolic Functions

Hyperbolic functions are transformed to exponential form:

integrate(sinh(x), x)  # Equivalent to (exp(x) - exp(-x))/2
integrate(cosh(x), x)  # Equivalent to (exp(x) + exp(-x))/2
integrate(tanh(x), x)  # Transformed to exponential form

Algorithm: The Risch Method

The Risch algorithm builds a tower of differential fields to handle transcendental extensions systematically.

Differential Field Tower

For an integrand like exp(x^2) * log(x), the algorithm constructs:

1. Base field: ℚ(x) with derivation d/dx
2. First extension: ℚ(x, log(x)) with D(log(x)) = 1/x
3. Second extension: ℚ(x, log(x), exp(x^2)) with D(exp(x^2)) = 2*x*exp(x^2)

Integration Steps

1. Field Tower Construction: Build the appropriate differential field tower
2. Canonical Form: Transform the integrand to canonical form in the tower
3. Residue Computation: Apply the Risch algorithm recursively
4. Result Assembly: Convert back to symbolic form

Implementation Details

Function Transformations

The algorithm transforms complex functions to simpler forms:

• Trigonometric functions → Half-angle formulas with tan(x/2)
• Hyperbolic functions → Exponential expressions
• Inverse functions → Differential field extensions

Example: sin(x) Integration

# sin(x) is transformed to:
# 2*tan(x/2) / (1 + tan(x/2)^2)
# Then integrated using the Risch algorithm

Advanced Usage

Direct Algorithm Access

You can access the lower-level algorithms directly:

# Use the Risch algorithm directly
using SymbolicIntegration
# ... (advanced example would go here)

Custom Derivations

# Create custom differential field extensions
# ... (advanced example would go here)  

Limitations

• No algebraic functions (√x, x^(1/3), etc.)
• Some complex trigonometric cases may not be handled
• Non-elementary integrals return unevaluated forms