# ParametricTranscendentalIntegration(F, UP)¶

intpar.spad line 466 [edit on github]

F: Field

This package implements parametric integration in transcendental case.

- diffextint: (List UP -> List Record(ratpart: F, coeffs: Vector F), Matrix F -> List Vector F, List Fraction UP) -> List Record(ratpart: F, coeffs: Vector F)
`diffextint(ext, csolve, [g1, ..., gn])`

is like primextint and expextint but for differentialy transcendental extensions.

- expextint: (UP -> UP, (Integer, List F) -> List Record(ratpart: F, coeffs: Vector F), Matrix F -> List Vector F, List Fraction UP) -> List Record(ratpart: Fraction UP, coeffs: Vector F)
`expextint(', rde, csolve, [g1, ..., gn])`

returns a basis of solution of the homogeneous system`h' + c1*g1 + ... + cn*gn = 0`

Argument foo is an parametric`rde`

solver on`F`

.`csolve`

is solver over constants.

- logextint: (UP -> UP, UP -> Factored UP, Matrix F -> List Vector Fraction Integer, List UP -> Record(logands: List Fraction UP, basis: List Vector Fraction Integer), List Fraction UP) -> Record(logands: List Fraction UP, basis: List Vector Fraction Integer)
`logextint(der, ufactor, csolve, rec, [g1, ..., gn])`

returns [[`u1`

, …, um], bas] giving basis of solution of the homogeneous systym`c1*g1 + ... + cn*gn + c_{n+1}u1'/u1 + ... c_{n+m}um'/um = 0`

- monologextint: (List UP, Matrix F -> List Vector Fraction Integer, List F -> Record(logands: List F, basis: List Vector Fraction Integer)) -> Record(logands: List Fraction UP, basis: List Vector Fraction Integer)
`monologextint(lup, csolve, rec)`

is a helper for logextint

- primextint: (UP -> UP, List F -> List Record(ratpart: F, coeffs: Vector F), Matrix F -> List Vector F, List Fraction UP) -> List Record(ratpart: Fraction UP, coeffs: Vector F)
`primextint(', ext, csolve, [g1, ..., gn])`

returns a basis of solutions of the homogeneous system`h' + c1*g1 + ... + cn*gn = 0`

. Argument`ext`

is an extended integration function on`F`

.`csolve`

is solver over constants.