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SolveAnEllipticLookingQuotientOfBinomials(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A root of a quadratic binomial beside another, 1/((A + B x^2)^(1/3) (C + D x^2)) with B C + 3 A D = 0 or B C - 9 A D = 0, and
1/((A + B x^2)^(1/4) (C + D x^2)) with B C - 2 A D = 0: the ratios at which
the integral, elliptic otherwise, is elementary, in closed form.

Remarks

With y the root, a = A^(1/3), and q = sqrt(B/A) or sqrt(-B/A) by its sign:

B C + 3 A D = 0, B/A > 0:  -2^(1/3) q/(4 a D) (atan(a q x/(a + 2^(1/3) y)) - atan(q x)/3
                               - (artanh(sqrt(3) (a - 2^(1/3) y)/(a q x)) + artanh(sqrt(3)/(q x)))/sqrt(3))
                 B/A < 0:   2^(1/3) q/(4 a D) (artanh(a q x/(a + 2^(1/3) y)) - artanh(q x)/3
                               + (atan(sqrt(3) (a - 2^(1/3) y)/(a q x)) + atan(sqrt(3)/(q x)))/sqrt(3))
B C - 9 A D = 0, B/A > 0:   q/(12 a D) (atan((a - y)^2/(3 a^2 q x)) + atan(q x/3)
                               - sqrt(3) artanh(sqrt(3) (a - y)/(a q x)))
                 B/A < 0:   q/(12 a D) (artanh((a - y)^2/(3 a^2 q x)) - artanh(q x/3)
                               - sqrt(3) atan(sqrt(3) (a - y)/(a q x)))

and for the fourth root, with s = sqrt(A + B x^2) and r = sqrt(|B|), times
B/D, which takes C + D x^2 to 2 A + B x^2:

A > 0, B > 0:  -(atan(A^(3/4) (1 + s/sqrt(A))/(x y r)) + artanh(A^(3/4) (1 - s/sqrt(A))/(x y r)))/(2 A^(3/4) r)
A > 0, B < 0:   (atan(A^(3/4) (1 - s/sqrt(A))/(x y r)) + artanh(A^(3/4) (1 + s/sqrt(A))/(x y r)))/(2 A^(3/4) r)
A < 0:          -(atan(u) + artanh(u))/(2 (-A)^(3/4) sqrt(2) r),   u = x r/((-A)^(1/4) y sqrt(2))

Rubi's, from its 1.1.2.3, written for any constants with the ratio; each was checked by
differentiating it at points in every sign case. The derivative of each uses only that
the roots are roots, so for a sign that is a symbol's both forms are given, each where
it holds.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

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

SolveAnEllipticLookingQuotientOfBinomials(AngouriMath.Entity,AngouriMath.Entity.Variable)