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SolveByTheVariableOverTheRootOfAnEvenQuartic​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

A rational function beside a half-odd power of an even quartic, R(x) P^(m/2) with
P = a + b x^2 + c x^4, as a rational function of t = x/sqrt(P) where it is
one: Rubi's (a - c x^4)/((a d + a e x^2 + c d x^4) sqrt(a + b x^2 + c x^4)) is
int dt/(d + (a e - b d) t^2), and sqrt(a + b x^2 + c x^4)/(a d - c d x^4) is
int dt/(d ((1 - b t^2)^2 - 4 a c t^4)).

Remarks

dt/dx = (a - c x^4)/P^(3/2), so the integrand is H dt for
H = R P^((m + 3)/2)/(a - c x^4), a rational function of x. With y = x^2,
t^2 = y/(a + b y + c y^2) is 1/(c w + b) for w = y + a/(c y); so
H is a rational function of t where it is even, H = h(x^2), and h is a
rational function of w -- the same at y and at a/(c y), the two roots of
Y^2 - w Y + a/c. That is read by writing the numerator and the denominator of
h modulo that quadratic, each as u_1 y + u_0 with the u_i polynomials
in w: the quotient is free of y exactly where n_1 d_0 = n_0 d_1, decided as a value,
and is then n_1/d_1. An elliptic integral does not pass: 1/sqrt(P) has
h(a/(c y)) = -h(y).
 Each step is an identity for the principal root wherever P is not zero, of either sign:
P^(m/2) P^(3/2) = P^((m + 3)/2) for an odd m, t^2 = x^2/P, and the derivative
of P^(-1/2) on an interval where P is negative. These were declined, as no rule
writes the variable over the root; with numbers for coefficients too,
x^2/((1 + x^4) sqrt(x^4 - 1)) is -int t^2/(1 + 4 t^4) dt. Asked late and at
the question asked, so that every rule reading the integrand as written goes first, and
the answer is differentiated back at sampled points before it is returned.
https://github.com/asc-community/AngouriMath/issues/233

























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