Module rational 

Rational function integration using Hermite reduction algorithm

Implements the Hermite reduction method for integrating rational functions. This is a core component of the Risch algorithm for symbolic integration.

Mathematical Background

For a rational function R(x) = P(x)/Q(x), the integral decomposes as:

∫ R(x) dx = polynomial_part + ∑ cᵢ ln|qᵢ(x)| + ∫ remaining_rational

where:

• polynomial_part comes from polynomial long division if deg(P) ≥ deg(Q)
• logarithmic terms arise from square-free factorization of denominator
• remaining_rational is a proper rational function with square-free denominator

Algorithm Steps

1. Polynomial Division: If deg(P) ≥ deg(Q), divide to get quotient + remainder
2. Square-Free Factorization: Factor Q = q₁·q₂²·q₃³·… into square-free parts
3. Hermite Reduction: Extract logarithmic terms using GCD operations
4. Partial Fractions: Decompose remaining rational part

References

• Bronstein, M. (2005). Symbolic Integration I: Transcendental Functions
• Geddes, K. et al. (1992). Algorithms for Computer Algebra

Structs

RationalIntegral

Result of rational function integration

Functions

assemble_integral

Assemble final integral expression from RationalIntegral

integrate_rational

Integrate a rational function P(x)/Q(x)