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

 Method (no overloads)

Summary

Bioche's first two rules for the hyperbolic functions: a rational function of
sinh(y) and cosh(y) that is odd in the hyperbolic sine is a rational
function of u = cosh(y) times sinh(y) dy = du, with sinh^2 as
u^2 - 1; one odd in the hyperbolic cosine is one of u = sinh(y) times
cosh(y) dy = du, with cosh^2 as u^2 + 1. Radicals of polynomials
in the two even in the function are admitted as coefficients, as in
SolveByBiochesOddSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean).

Remarks

The hyperbolic functions are not nodes: the library writes sinh(y) as
(e^y - e^-y)/2 and tanh(y) as (e^(2y) - 1)/(e^(2y) + 1), so the
six are read back by their spellings, for the argument each exponential names, and
nothing else in x may remain. Rubi's sinh(x)^3/(a + b cosh(x)^2) is
(u^2 - 1)/(a + b u^2) here, where under u = e^x it is a rational
function of a symbolic palindromic quartic and under u = tanh(x/2) of one of
degree eight, and both were searches past the budget; cosh(x)/(a + b tanh(x)^2),
sech(x)/(a + b sinh(x)^2)^(3/2) and csch(x)^5/(a + b cosh(x)^2) likewise.
In front of the substitution search, which with a symbolic coefficient spends the
budget on u = e^x. Exact: cosh is positive, so u = sinh(y) is a
bijection of the line, and u = cosh(y) one on each side of zero.
https://github.com/asc-community/AngouriMath/issues/718

























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