Module rational 

Rational function integration via partial fraction decomposition

Implements integration of rational functions P(x)/Q(x) using:

1. Polynomial long division if deg(P) >= deg(Q)
2. Factor denominator into linear and irreducible quadratic factors
3. Decompose into partial fractions using Heaviside’s method
4. Integrate each term using standard formulas

Mathematical Background

For P(x)/Q(x) where deg(P) < deg(Q), factor Q(x) as:

Q(x) = (x-r₁)^n₁ · ... · (x-rₖ)^nₖ · (x²+p₁x+q₁)^m₁ · ...

Then the partial fraction decomposition is:

P(x)/Q(x) = Σᵢ Σⱼ₌₁ⁿⁱ Aᵢⱼ/(x-rᵢ)ʲ + Σᵢ Σⱼ₌₁ᵐⁱ (Bᵢⱼx+Cᵢⱼ)/(x²+pᵢx+qᵢ)ʲ

Integration Formulas

Linear terms:

• ∫A/(x-r) dx = A·ln|x-r| + C
• ∫A/(x-r)ⁿ dx = -A/((n-1)(x-r)^(n-1)) + C for n > 1

Quadratic terms (irreducible x²+px+q with p²-4q < 0):

• Complete the square: x²+px+q = (x+p/2)² + a² where a² = q - p²/4
• ∫(Bx+C)/(x²+px+q) dx = (B/2)·ln|x²+px+q| + ((C-Bp/2)/a)·arctan((x+p/2)/a) + C

Implementation Status

Fully Implemented:

• Simple linear factors (x-r) via cover-up method
• Repeated linear factors (x-r)^n via Heaviside’s method with derivatives
• Simple irreducible quadratics (x²+px+q) with proper coefficient extraction
• Repeated irreducible quadratics (x²+px+q)² via Ostrogradsky’s reduction formula

Not Yet Implemented:

• Repeated irreducible quadratics (x²+px+q)^m with m > 2 (Can be generalized using recursive Ostrogradsky reduction)
• Automatic polynomial factorization (assumes factored form)

References

This implementation follows the approaches in:

• Heaviside’s cover-up method and derivative technique for repeated poles
• Ostrogradsky’s reduction formula for repeated quadratics
• Stewart, Calculus (8th ed), Chapter 7
• Bronstein, “Symbolic Integration I”

Modules

helpers

Helper utilities for rational function integration

linear

Integration of linear factors in partial fraction decomposition

quadratic

Integration of irreducible quadratic factors in partial fraction decomposition

Structs

LinearTerm

PartialFractionDecomposition

QuadraticTerm

Functions

extract_numerator_denominator

Extract numerator and denominator from rational expression

integrate_rational

Integrate rational function P(x)/Q(x) via partial fractions

is_rational_function

Check if expression is a rational function P(x)/Q(x)