# TranscendentalHermiteIntegration(F, UP)ΒΆ

- F: Field
- UP: UnivariatePolynomialCategory F

Hermite integration, transcendental case.

- HermiteIntegrate: (Fraction UP, UP -> UP) -> Record(answer: Fraction UP, logpart: Fraction UP, specpart: Fraction UP, polypart: UP)
`HermiteIntegrate(f, D)`

returns`[g, h, s, p]`

such that`f = Dg + h + s + p`

,`h`

has a squarefree denominator normal`w`

.`r`

.`t`

.`D`

, and all the squarefree factors of the denominator of`s`

are special`w`

.`r`

.`t`

.`D`

. Furthermore,`h`

and`s`

have no polynomial parts.`D`

is the derivation to use on UP.

- HermiteIntegrate: (Fraction UP, UP -> UP, UP) -> Record(answer: Fraction UP, logpart: Fraction UP, specpart: Fraction UP, polypart: UP)
`HermiteIntegrate(f, D, d0)`

returns`[g, h, s, p]`

such that`f = Dg + g*d0 + h + s + p`

,`h`

has a squarefree denominator normal`w`

.`r`

.`t`

.`D`

, and all the squarefree factors of the denominator of`s`

are special`w`

.`r`

.`t`

.`D`

. Furthermore,`h`

and`s`

have no polynomial parts.`D`

is the derivation to use on UP.