AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug ~~Donate~~ Contacts

SolveALinearOverTheQuadraticFactorOfADifferenceOfCubesBesideACubeRoot(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A linear over the quadratic factor of a difference of cubes, beside the cube root of a
binomial in x^3, (x + s)/((x^2 + s x + s^2) y) with y = (a + b x^3)^(1/3):
for c a cube root of a + b s^3, its integral is
ln((c - y)/(c x/s - y))/(2 c) + (atan((1 + 2 c x/(s y))/sqrt(3)) + atan((1 + 2 y/c)/sqrt(3)))/(sqrt(3) c).
Welz's (1 + x)/((1 + x + x^2) (a + b x^3)^(1/3)).

Remarks

Multiplied above and below by s - x it is (s^2 - x^2)/((s^3 - x^3) y): the
first term is dt/(s^3 - (a + b s^3) t^3) under t = x/y, the second a function
of x^3 beside its derivative, and the sum of their integrals is Rubi's, at
s = 1, and the form above at any s under x = s u. Both logarithms
of Rubi's vanish at x = s, where the integrand has no pole, and written apart they
jump there by an imaginary constant; their quotient tends to -b s^3/a and does
not. The identity uses no more about c than c^3 = a + b s^3, and holds for
every cube root; the principal one is y at x = s, so that the two
logarithms vanish together there whatever the sign of a + b s^3. With the real
root at a + b s^3 < 0, 1 + 2 y/c is sqrt(3) i at x = s, an
arctangent's singularity, and the answer jumped there; with the principal one it was
continuous on either side of the binomial's real zero, at thirty digits, at every sign
of a, b and a + b s^3 tried. The linear has to be x + s: any other leaves
x/((s^3 - x^3) y), which is not elementary -- Rubi's answer to Welz's
(2 + x)/((1 + x + x^2) (2 + x^3)^(1/3)) has an Appell function in it.
With numbers in the binomial the ansatz over logarithms of L - y answers most of
these, and is asked first; with symbols in it, as Welz's, it and every rule declined
them. The answer is differentiated back at sampled points.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

Table of contents

AngouriMath

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

SolveALinearOverTheQuadraticFactorOfADifferenceOfCubesBesideACubeRoot(AngouriMath.Entity,AngouriMath.Entity.Variable)