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CubeRootLogarithmAnsatz

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Description

Summary

A rational function of x and one cube root y = P(x)^(1/3) of a polynomial,
integrated by an ansatz over logarithms of L - y for linear L, their
conjugates, and a rational part in y and y^2.

Remarks

The curve y^3 = P(x) has no rational parametrisation for a cubic P, so no
substitution makes these rational; the ones that are elementary at all -- Welz's
x/((1 + x)(1 - x^3)^(1/3)), 1/(x^3 - 3x^2 + 7x - 5)^(1/3),
(x - 1)/((x + 1)(2 + x^3)^(1/3)) -- are integrated by logarithms of
L(x) - y, with L linear, and of their conjugates L - w y for the
cube roots of unity w, which pair into ln(L^2 + L y + y^2) and
arctan((2L + y)/(sqrt(3) y)). The norm (L - y)(L - wy)(L - w^2 y) = L^3 - P is where the logarithm has its poles, so the L that can occur are the ones
whose cube agrees with P at the places the integrand has poles: the linear
function through two points of the curve, or tangent to it at one. Each of those is
a candidate here -- the tangent at a rational pole, at infinity, and at a root of
P; the line through a pole and a root, or two poles -- and for an irreducible
quadratic factor Q of the denominator with P constant modulo Q,
the cube roots of that constant times a cube root of unity of Q[x]/(Q), which
is where ln(2^(1/3) - y), ln(2^(1/3) x + y) and
ln(2^(1/3)(x - 1) - y) in Welz's answers over 1 - x + x^2 come from.
 The constants those bring in are cube roots of rationals, 2^(1/3) most often,
so the coefficients live in Q(c) with c^3 rational; the linear system is
solved in that field exactly, as polynomials in c reduced modulo c^3 - m,
rather than numerically -- a symbol for c would not be told from zero at
c^3 - 2, and a floating c would give floating coefficients. One such
constant per integrand; two independent ones would need their compositum and are
declined. With symbols in the coefficients and every constant rational, the general
symbolic solver is used instead.
 The identity sum c_k g_k' = f is an identity in the function field
Q(x)[y]/(y^3 - P): everything is brought over one denominator in x, the
numerators reduced by y^3 = P to A + B y + C y^2, and the three
coefficient polynomials in x matched term by term. The answer is checked against
the integrand at sampled points before it is returned.
https://github.com/asc-community/AngouriMath/issues/718

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