/**
 * Risch Layer 3 — Hermite reduction + Rothstein–Trager / residue formula
 * for rational functions whose denominator has a degree-≥3 irreducible
 * factor or a repeated positive-discriminant quadratic.
 *
 * After Ostrogradsky–Hermite peels the rational part (any repeated factor),
 * the remaining square-free integrand is integrated by Rothstein–Trager
 * when every residue is rational, otherwise by the residue formula
 * (companion-matrix roots of the denominator, conjugate pairs folded into
 * real `log` + `atan2`). Correctness is verified by differentiation; the
 * string form is not contractual.
 */
import { type IntPoly } from '../typed/factorization/integer-poly.js';
/** Bareiss fraction-free determinant over ℤ. */
export declare function detZ(matrix: bigint[][]): bigint;
/** Sylvester resultant of two integer polynomials. */
export declare function resultantZ(a: IntPoly, b: IntPoly): bigint;
/**
 * Rothstein–Trager resultant `R(t) = Res_x(A − t B', B)`, recovered by
 * interpolating integer samples `R(k)` at `k = 0..N`.
 */
export declare function rothsteinResultant(A: IntPoly, B: IntPoly): IntPoly;
/**
 * Ostrogradsky–Hermite: `R/Q = (P/Q1)' + S/Q2` with `Q1 = gcd(Q,Q')`,
 * `Q2 = Q/Q1` square-free. Returns the rational part `P/Q1` as a string
 * (or `'0'`) and the remaining square-free numerator `S`.
 */
export declare function hermiteReduce(R: IntPoly, Q: IntPoly, v: string): {
    rational: string;
    squareFreeNumer: IntPoly;
    squareFreeDenom: IntPoly;
};
/**
 * Residue-formula antiderivative of a square-free proper `A/B`:
 * `Σ (A(r)/B'(r)) log(x − r)`, with conjugate pairs folded into a real
 * `log` + `atan2` pair so the result is evaluable over ℝ.
 */
export declare function residueIntegral(A: IntPoly, B: IntPoly, v: string): string;
/**
 * Exact Rothstein–Trager when every residue is rational; otherwise the
 * residue formula. `A/B` must be proper and `B` square-free.
 */
export declare function rothsteinTrager(A: IntPoly, B: IntPoly, v: string): string;
/**
 * Full Layer-3 pipeline for a proper remainder `R/Q` (already split off the
 * polynomial part). Hermite-reduces repeated factors, then integrates the
 * square-free remainder.
 */
export declare function integrateLayer3(R: IntPoly, Q: IntPoly, v: string): string;
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