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SolveByRischNormanAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A rational function of x and of exponentials and logarithms built over it,
integrated by the Risch-Norman ansatz: F = P/Q + sum c_j ln(q_j) with P a
polynomial of unknown coefficients in x and the transcendental monomials, Q tried from the integrand's denominator, and the q_j its factors.

Remarks

Hearn's e^(1 - x e^(x^2) + 2x^2)(x + 2x^3)/(1 - x e^(x^2))^2 is
(e^(1 - x e^(x^2))/(1 - x e^(x^2)))', and his
e^(x^2)/x + 2x e^(x^2) ln(x) + (ln(x) - 2)/(x + ln(x)^2)^2 + (1 + 1/x + 2 ln(x)/x)/(x + ln(x)^2) is (e^(x^2) ln(x) - ln(x)/(x + ln(x)^2) + ln(x + ln(x)^2))'. Neither had an
antiderivative: the first is an exponential of something with an exponential in it,
and the second a sum whose terms are not elementary apart -- e^(x^2)/x is
not -- so that every split loses it. With t_1 = e^(x^2), t_2 = ln(x),
t_3 = e^(1 - x t_1 + 2x^2) for indeterminates, each with the derivative it has
-- 2x t_1, 1/x, (4x - t_1 - 2x^2 t_1) t_3 -- the integrand is a
rational function of x and the t, and Liouville's theorem says its
elementary antiderivative, where there is one, is a rational function of the same
plus logarithms with constant coefficients. The rational part's denominator divides
the integrand's with each factor's power lowered by one, the exponential monomials
excepted, which may stand to any power; the logarithms' arguments are the
denominator's factors. So F' = f, over one denominator, is a polynomial
identity in x and the t, linear in the unknown coefficients of P and the c_j: one equation per monomial, solved by the elimination the other
ansätze use. The identity is exact, so a solution is an answer and its absence a
decline; the derivative of what comes out is checked against the integrand at
sampled points all the same. This is the parallel Risch algorithm of Norman and
Moore as a heuristic, with the degrees bounded as the tower ansätze bound theirs.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

SolveByRischNormanAnsatz​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveByRischNormanAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable)