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Hermite reduction for a function in a monomial extension.

Specifically, for derivation \(D\) and function \(f\), returns \(g\), \(h\), and \(r\) such that

\[ f = Dg + h + r \]

where \(h\) is simple and \(r\) is reduced.

For example, with derivation \(D\) and monomial \(t\) such that \(Dt = t^2 + 1\), it is the case that \(\frac{t^4 + t + 1}{t^2} = Dg + h + r\), where \(g = -\frac{1}{t}\), \(h = \frac{1}{t}\), and \(r = t^2 - 1\).

>>> let deriv = extend (const 0) (power 2 + 1 :: IndexedPolynomial)
>>> hermiteReduce deriv $ fromPolynomials (power 4 + power 1 + 1) (power 2)
([Function ((-1)) (x)],Function (1) (x),Function (x^2 + (-1)) (1))

Since it is the case that \(\frac{dt}{dx} = t^2 + 1\) for \(t = \tan x\), this implies that

\[ \int \frac{\tan^4 x + \tan x + 1}{\tan^2 x} \, dx = -\frac{1}{\tan x} + \int \left( \frac{1}{\tan x} + \tan^2 x - 1\right) \, dx \]

\(g\) is returned as a list of functions which sum to \(g\) instead of a single function, because the former could sometimes be simpler to read.

Polynomial reduction in a monomial extension with the given derivation and polynomial. Specifically, for derivation \(D\) and polynomial \(p\), returns polynomials \(g\) and \(h\) such that

\[ p = Dg + h \]

where the degree of \(h\) is less than the degree of \(Dt\), if the latter is larger than 0.

For example, with derivation \(D\) such that \(Dt = t^2 + 1\), it is the case that \(3t^4 + 3t^3 + 4 = D\left( t^3 + \frac{3}{2}t^2 - 3t \right) - 3t + 7\).

>>> let derivation = extend (const 0) (power 2 + 1 :: IndexedPolynomial)
>>> polynomialReduce derivation $ 3 * power 4 + 3 * power 3 + 4
(x^3 + (3 % 2)x^2 + (-3)x,(-3)x + 7)

Resultant reduction of a simple function in a monomial extension.

Specifically, for a derivation \(D\) on \(k(t)\) and a simple function \(f \in k(t)\), returns logarithmic terms \((s_i, S_i)\) such that

\[ g = \sum_i \sum_{\alpha \mid s_i(\alpha) = 0} \alpha \log S_i(\alpha, t) \]

and also whether \(f\) has an elemantary integral. If \(f\) has an elementary integral, then there is a polynomial \(h\) of the monomial \(t\) such that \(f - Dg = h\).

For example, for derivation \(D=\frac{d}{dx}\) and simple function \(f = \frac{t^2+2t+1}{t^2-t+1}\),

>>> residueReduce differentiate $ fromPolynomials (power 2 + 2 * power 1 + 1) (2 * power 2 - 2 * power 1 - 1 :: IndexedPolynomial)
Just ([(x^2 + ((-3) % 2)x + ((-3) % 16),[(0,(4 % 9)x + (1 % 3)),(1,((-8) % 9)x + (2 % 3))])],True)

so that

\[ g = \sum_{\alpha \mid \alpha^2-\frac{3}{2}\alpha-\frac{3}{16} = 0} \log \left( (\frac{2}{3}-\frac{8}{9}\alpha)t + \frac{4}{9}\alpha+\frac{1}{3} \right) \]

and \(\int f \, dx\) has an elementary integral, which happens to be

\[ \int f \, dx = g + \int h \, dx \]

where \(h = f-Dg\) is a polynomial of the monomial \(t\).