def test_prde_cancel_liouvillian():
### 1. case == 'primitive'
# used when integrating f = log(x) - log(x - 1)
# Not taken from 'the' book
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 / x, t)]})
p0 = Poly(0, t, field=True)
h, A = prde_cancel_liouvillian(Poly(
-1 / (x - 1), t), [Poly(-x + 1, t), Poly(1, t)], 1, DE)
V = A.nullspace()
h == [
p0, p0,
Poly((x - 1) * t, t), p0, p0, p0, p0, p0, p0, p0,
Poly(x - 1, t),
Poly(-x**2 + x, t), p0, p0, p0, p0
]
assert A.rank() == 16
assert (Matrix([h]) * V[0][:16, :]) == Matrix(
[[Poly(0, t, domain='QQ(x)')]])
### 2. case == 'exp'
# used when integrating log(x/exp(x) + 1)
# Not taken from book
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t, t)]})
assert prde_cancel_liouvillian(Poly(0, t, domain='QQ[x]'), [Poly(1, t, domain='QQ(x)')], 0, DE) == \
([Poly(1, t, domain='QQ'), Poly(x, t)], Matrix([[-1, 0, 1]]))
def test_prde_cancel_liouvillian():
### 1. case == 'primitive'
# used when integrating f = log(x) - log(x - 1)
# Not taken from 'the' book
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
p0 = Poly(0, t, field=True)
h, A = prde_cancel_liouvillian(Poly(-1/(x - 1), t), [Poly(-x + 1, t), Poly(1, t)], 1, DE)
V = A.nullspace()
h == [p0, p0, Poly((x - 1)*t, t), p0, p0, p0, p0, p0, p0, p0, Poly(x - 1, t), Poly(-x**2 + x, t), p0, p0, p0, p0]
assert A.rank() == 16
assert (Matrix([h])*V[0][:16, :]) == Matrix([[Poly(0, t, domain='QQ(x)')]])
### 2. case == 'exp'
# used when integrating log(x/exp(x) + 1)
# Not taken from book
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t, t)]})
assert prde_cancel_liouvillian(Poly(0, t, domain='QQ[x]'), [Poly(1, t, domain='QQ(x)')], 0, DE) == \
([Poly(1, t, domain='QQ'), Poly(x, t)], Matrix([[-1, 0, 1]]))