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SolveARationalFunctionOfASineOverPowersOfLinearsInIt(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)

Method (no overloads)

Summary

A rational function of a sine whose denominator in it is written as powers of
linears -- of the sine itself, of 1 ± sin(x) from an even power of the
cosine, and of a + b sin(x) with symbols in it: Rubi's
cot(x)^6/(a + b sin(x))^3 is (1 - s^2)^3/(s^6 (a + b s)^3) in
s = sin(x). Split over those factors by the residues at their roots, every
piece is a constant over a power of one linear in the function, which the rules for
csc(x)^n, 1/(1 ± sin(x))^n and 1/(a + b sin(x))^n answer.

Remarks

Under the half-angle tangent, which took these before, every linear but the function
itself is a quadratic in t -- a + b s is (a t^2 + 2 b t + a)/(1 + t^2) --
and the cube of a symbolic one is split by Newton's iteration modulo its powers, with
the symbols in every round. Rubi's cot(x)^6/(a + b sin(x))^3,
tan(x)^4/(a + b sin(x))^3 and 1/((a + b sin(x))^3 (c + d sin(x))^3) ran
past thirty seconds there. In s each block is a Taylor series at a root.
Not of a cosine: a + b cos(x) is ((a + b) + (a - b) t^2)/(1 + t^2), even in
t, and the half-angle answers tan(x)^6/(a + b cos(x))^3 in two seconds and
ten thousand characters, where the pieces here, each a piecewise of its own, came to
three hundred thousand.
Only where the half-angle is that slow: a linear with a symbolic root to the third power
or more, and the half-angle's denominator past the eleventh degree, which is 2 d - k for a
denominator of degree d in the function with s^k among its factors.
cot(x)^4/(a + b sin(x))^3, of the tenth, and cos(x)^6 csc(x)^5/(a + b sin(x))^3,
of the eleventh, are answered there within a second, and are left to it; past it,
csc(x)^2 sec(x)^2/(a + b sin(x))^3 took seven seconds there and a quarter of one here.
The decomposition is checked at sampled points by the split, and the sum of the
integrated pieces by its derivative before it is returned. A piece is over one linear,
which this declines, so nothing it asks comes back here.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

Table of contents

AngouriMath

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

SolveARationalFunctionOfASineOverPowersOfLinearsInIt(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)