764 lines
38 KiB
Python
764 lines
38 KiB
Python
"""Most of these tests come from the examples in Bronstein's book."""
|
|
from sympy.core.function import (Function, Lambda, diff, expand_log)
|
|
from sympy.core.numbers import (I, Rational, pi)
|
|
from sympy.core.relational import Ne
|
|
from sympy.core.singleton import S
|
|
from sympy.core.symbol import (Symbol, symbols)
|
|
from sympy.functions.elementary.exponential import (exp, log)
|
|
from sympy.functions.elementary.miscellaneous import sqrt
|
|
from sympy.functions.elementary.piecewise import Piecewise
|
|
from sympy.functions.elementary.trigonometric import (atan, sin, tan)
|
|
from sympy.polys.polytools import (Poly, cancel, factor)
|
|
from sympy.integrals.risch import (gcdex_diophantine, frac_in, as_poly_1t,
|
|
derivation, splitfactor, splitfactor_sqf, canonical_representation,
|
|
hermite_reduce, polynomial_reduce, residue_reduce, residue_reduce_to_basic,
|
|
integrate_primitive, integrate_hyperexponential_polynomial,
|
|
integrate_hyperexponential, integrate_hypertangent_polynomial,
|
|
integrate_nonlinear_no_specials, integer_powers, DifferentialExtension,
|
|
risch_integrate, DecrementLevel, NonElementaryIntegral, recognize_log_derivative,
|
|
recognize_derivative, laurent_series)
|
|
from sympy.testing.pytest import raises
|
|
|
|
from sympy.abc import x, t, nu, z, a, y
|
|
t0, t1, t2 = symbols('t:3')
|
|
i = Symbol('i')
|
|
|
|
def test_gcdex_diophantine():
|
|
assert gcdex_diophantine(Poly(x**4 - 2*x**3 - 6*x**2 + 12*x + 15),
|
|
Poly(x**3 + x**2 - 4*x - 4), Poly(x**2 - 1)) == \
|
|
(Poly((-x**2 + 4*x - 3)/5), Poly((x**3 - 7*x**2 + 16*x - 10)/5))
|
|
assert gcdex_diophantine(Poly(x**3 + 6*x + 7), Poly(x**2 + 3*x + 2), Poly(x + 1)) == \
|
|
(Poly(1/13, x, domain='QQ'), Poly(-1/13*x + 3/13, x, domain='QQ'))
|
|
|
|
|
|
def test_frac_in():
|
|
assert frac_in(Poly((x + 1)/x*t, t), x) == \
|
|
(Poly(t*x + t, x), Poly(x, x))
|
|
assert frac_in((x + 1)/x*t, x) == \
|
|
(Poly(t*x + t, x), Poly(x, x))
|
|
assert frac_in((Poly((x + 1)/x*t, t), Poly(t + 1, t)), x) == \
|
|
(Poly(t*x + t, x), Poly((1 + t)*x, x))
|
|
raises(ValueError, lambda: frac_in((x + 1)/log(x)*t, x))
|
|
assert frac_in(Poly((2 + 2*x + x*(1 + x))/(1 + x)**2, t), x, cancel=True) == \
|
|
(Poly(x + 2, x), Poly(x + 1, x))
|
|
|
|
|
|
def test_as_poly_1t():
|
|
assert as_poly_1t(2/t + t, t, z) in [
|
|
Poly(t + 2*z, t, z), Poly(t + 2*z, z, t)]
|
|
assert as_poly_1t(2/t + 3/t**2, t, z) in [
|
|
Poly(2*z + 3*z**2, t, z), Poly(2*z + 3*z**2, z, t)]
|
|
assert as_poly_1t(2/((exp(2) + 1)*t), t, z) in [
|
|
Poly(2/(exp(2) + 1)*z, t, z), Poly(2/(exp(2) + 1)*z, z, t)]
|
|
assert as_poly_1t(2/((exp(2) + 1)*t) + t, t, z) in [
|
|
Poly(t + 2/(exp(2) + 1)*z, t, z), Poly(t + 2/(exp(2) + 1)*z, z, t)]
|
|
assert as_poly_1t(S.Zero, t, z) == Poly(0, t, z)
|
|
|
|
|
|
def test_derivation():
|
|
p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
|
|
(2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
|
|
assert derivation(p, DE) == Poly(-20*x**4*t**6 + (2*x**3 + 16*x**4)*t**5 +
|
|
(21*x**2 + 12*x**3)*t**4 + (x*Rational(7, 2) - 25*x**2 - 12*x**3)*t**3 +
|
|
(-5 - x*Rational(15, 2) + 7*x**2)*t**2 - (3 - 8*x - 10*x**2 - 4*x**3)/(2*x)*t +
|
|
(1 - 4*x**2)/(2*x), t)
|
|
assert derivation(Poly(1, t), DE) == Poly(0, t)
|
|
assert derivation(Poly(t, t), DE) == DE.d
|
|
assert derivation(Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t), DE) == \
|
|
Poly(-2*t**3 - 4/x*t**2 - (5 - 2*x)/(2*x**2)*t - (1 - 2*x)/(2*x**3), t, domain='ZZ(x)')
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(t, t)]})
|
|
assert derivation(Poly(x*t*t1, t), DE) == Poly(t*t1 + x*t*t1 + t, t)
|
|
assert derivation(Poly(x*t*t1, t), DE, coefficientD=True) == \
|
|
Poly((1 + t1)*t, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
|
|
assert derivation(Poly(x, x), DE) == Poly(1, x)
|
|
# Test basic option
|
|
assert derivation((x + 1)/(x - 1), DE, basic=True) == -2/(1 - 2*x + x**2)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
|
|
assert derivation((t + 1)/(t - 1), DE, basic=True) == -2*t/(1 - 2*t + t**2)
|
|
assert derivation(t + 1, DE, basic=True) == t
|
|
|
|
|
|
def test_splitfactor():
|
|
p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
|
|
(2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t, field=True)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
|
|
assert splitfactor(p, DE) == (Poly(4*x**4*t**3 + (-8*x**3 - 4*x**4)*t**2 +
|
|
(4*x**2 + 8*x**3)*t - 4*x**2, t, domain='ZZ(x)'),
|
|
Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t, domain='ZZ(x)'))
|
|
assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
|
|
r = Poly(-4*x**4*z**2 + 4*x**6*z**2 - z*x**3 - 4*x**5*z**3 + 4*x**3*z**3 + x**4 + z*x**5 - x**6, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
|
|
assert splitfactor(r, DE, coefficientD=True) == \
|
|
(Poly(x*z - x**2 - z*x**3 + x**4, t), Poly(-x**2 + 4*x**2*z**2, t))
|
|
assert splitfactor_sqf(r, DE, coefficientD=True) == \
|
|
(((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
|
|
assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
|
|
assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1),), ())
|
|
|
|
|
|
def test_canonical_representation():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
|
|
assert canonical_representation(Poly(x - t, t), Poly(t**2, t), DE) == \
|
|
(Poly(0, t, domain='ZZ[x]'), (Poly(0, t, domain='QQ[x]'),
|
|
Poly(1, t, domain='ZZ')), (Poly(-t + x, t, domain='QQ[x]'),
|
|
Poly(t**2, t)))
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
|
|
assert canonical_representation(Poly(t**5 + t**3 + x**2*t + 1, t),
|
|
Poly((t**2 + 1)**3, t), DE) == \
|
|
(Poly(0, t, domain='ZZ[x]'), (Poly(t**5 + t**3 + x**2*t + 1, t, domain='QQ[x]'),
|
|
Poly(t**6 + 3*t**4 + 3*t**2 + 1, t, domain='QQ')),
|
|
(Poly(0, t, domain='QQ[x]'), Poly(1, t, domain='QQ')))
|
|
|
|
|
|
def test_hermite_reduce():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
|
|
|
|
assert hermite_reduce(Poly(x - t, t), Poly(t**2, t), DE) == \
|
|
((Poly(-x, t, domain='QQ[x]'), Poly(t, t, domain='QQ[x]')),
|
|
(Poly(0, t, domain='QQ[x]'), Poly(1, t, domain='QQ[x]')),
|
|
(Poly(-x, t, domain='QQ[x]'), Poly(1, t, domain='QQ[x]')))
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
|
|
|
|
assert hermite_reduce(
|
|
Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 - nu**2)*t - x**5/4, t),
|
|
Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t), DE) == \
|
|
((Poly(-x**2 - 4, t, domain='ZZ(x,nu)'), Poly(4*t**2 + 2*x**2 + 4, t, domain='ZZ(x,nu)')),
|
|
(Poly((-2*nu**2 - x**4)*t - (2*x**3 + 2*x), t, domain='ZZ(x,nu)'),
|
|
Poly(2*x**2*t**2 + x**4 + 2*x**2, t, domain='ZZ(x,nu)')),
|
|
(Poly(x*t + 1, t, domain='ZZ(x,nu)'), Poly(x, t, domain='ZZ(x,nu)')))
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
|
|
|
|
a = Poly((-2 + 3*x)*t**3 + (-1 + x)*t**2 + (-4*x + 2*x**2)*t + x**2, t)
|
|
d = Poly(x*t**6 - 4*x**2*t**5 + 6*x**3*t**4 - 4*x**4*t**3 + x**5*t**2, t)
|
|
|
|
assert hermite_reduce(a, d, DE) == \
|
|
((Poly(3*t**2 + t + 3*x, t, domain='ZZ(x)'),
|
|
Poly(3*t**4 - 9*x*t**3 + 9*x**2*t**2 - 3*x**3*t, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')))
|
|
|
|
assert hermite_reduce(
|
|
Poly(-t**2 + 2*t + 2, t, domain='ZZ(x)'),
|
|
Poly(-x*t**2 + 2*x*t - x, t, domain='ZZ(x)'), DE) == \
|
|
((Poly(3, t, domain='ZZ(x)'), Poly(t - 1, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
|
|
(Poly(1, t, domain='ZZ(x)'), Poly(x, t, domain='ZZ(x)')))
|
|
|
|
assert hermite_reduce(
|
|
Poly(-x**2*t**6 + (-1 - 2*x**3 + x**4)*t**3 + (-3 - 3*x**4)*t**2 -
|
|
2*x*t - x - 3*x**2, t, domain='ZZ(x)'),
|
|
Poly(x**4*t**6 - 2*x**2*t**3 + 1, t, domain='ZZ(x)'), DE) == \
|
|
((Poly(x**3*t + x**4 + 1, t, domain='ZZ(x)'), Poly(x**3*t**3 - x, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
|
|
(Poly(-1, t, domain='ZZ(x)'), Poly(x**2, t, domain='ZZ(x)')))
|
|
|
|
assert hermite_reduce(
|
|
Poly((-2 + 3*x)*t**3 + (-1 + x)*t**2 + (-4*x + 2*x**2)*t + x**2, t),
|
|
Poly(x*t**6 - 4*x**2*t**5 + 6*x**3*t**4 - 4*x**4*t**3 + x**5*t**2, t), DE) == \
|
|
((Poly(3*t**2 + t + 3*x, t, domain='ZZ(x)'),
|
|
Poly(3*t**4 - 9*x*t**3 + 9*x**2*t**2 - 3*x**3*t, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
|
|
(Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')))
|
|
|
|
|
|
def test_polynomial_reduce():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
|
|
assert polynomial_reduce(Poly(1 + x*t + t**2, t), DE) == \
|
|
(Poly(t, t), Poly(x*t, t))
|
|
assert polynomial_reduce(Poly(0, t), DE) == \
|
|
(Poly(0, t), Poly(0, t))
|
|
|
|
|
|
def test_laurent_series():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
|
|
a = Poly(36, t)
|
|
d = Poly((t - 2)*(t**2 - 1)**2, t)
|
|
F = Poly(t**2 - 1, t)
|
|
n = 2
|
|
assert laurent_series(a, d, F, n, DE) == \
|
|
(Poly(-3*t**3 + 3*t**2 - 6*t - 8, t), Poly(t**5 + t**4 - 2*t**3 - 2*t**2 + t + 1, t),
|
|
[Poly(-3*t**3 - 6*t**2, t, domain='QQ'), Poly(2*t**6 + 6*t**5 - 8*t**3, t, domain='QQ')])
|
|
|
|
|
|
def test_recognize_derivative():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, t)]})
|
|
a = Poly(36, t)
|
|
d = Poly((t - 2)*(t**2 - 1)**2, t)
|
|
assert recognize_derivative(a, d, DE) == False
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
|
|
a = Poly(2, t)
|
|
d = Poly(t**2 - 1, t)
|
|
assert recognize_derivative(a, d, DE) == False
|
|
assert recognize_derivative(Poly(x*t, t), Poly(1, t), DE) == True
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
|
|
assert recognize_derivative(Poly(t, t), Poly(1, t), DE) == True
|
|
|
|
|
|
def test_recognize_log_derivative():
|
|
|
|
a = Poly(2*x**2 + 4*x*t - 2*t - x**2*t, t)
|
|
d = Poly((2*x + t)*(t + x**2), t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
|
|
assert recognize_log_derivative(a, d, DE, z) == True
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
|
|
assert recognize_log_derivative(Poly(t + 1, t), Poly(t + x, t), DE) == True
|
|
assert recognize_log_derivative(Poly(2, t), Poly(t**2 - 1, t), DE) == True
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
|
|
assert recognize_log_derivative(Poly(1, x), Poly(x**2 - 2, x), DE) == False
|
|
assert recognize_log_derivative(Poly(1, x), Poly(x**2 + x, x), DE) == True
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
|
|
assert recognize_log_derivative(Poly(1, t), Poly(t**2 - 2, t), DE) == False
|
|
assert recognize_log_derivative(Poly(1, t), Poly(t**2 + t, t), DE) == False
|
|
|
|
|
|
def test_residue_reduce():
|
|
a = Poly(2*t**2 - t - x**2, t)
|
|
d = Poly(t**3 - x**2*t, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)], 'Tfuncs': [log]})
|
|
assert residue_reduce(a, d, DE, z, invert=False) == \
|
|
([(Poly(z**2 - Rational(1, 4), z, domain='ZZ(x)'),
|
|
Poly((1 + 3*x*z - 6*z**2 - 2*x**2 + 4*x**2*z**2)*t - x*z + x**2 +
|
|
2*x**2*z**2 - 2*z*x**3, t, domain='ZZ(z, x)'))], False)
|
|
assert residue_reduce(a, d, DE, z, invert=True) == \
|
|
([(Poly(z**2 - Rational(1, 4), z, domain='ZZ(x)'), Poly(t + 2*x*z, t))], False)
|
|
assert residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t,), DE, z, invert=False) == \
|
|
([(Poly(z**2 - 1, z, domain='QQ'), Poly(-2*z*t/x - 2/x, t, domain='ZZ(z,x)'))], True)
|
|
ans = residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t), DE, z, invert=True)
|
|
assert ans == ([(Poly(z**2 - 1, z, domain='QQ'), Poly(t + z, t))], True)
|
|
assert residue_reduce_to_basic(ans[0], DE, z) == -log(-1 + log(x)) + log(1 + log(x))
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
|
|
# TODO: Skip or make faster
|
|
assert residue_reduce(Poly((-2*nu**2 - x**4)/(2*x**2)*t - (1 + x**2)/x, t),
|
|
Poly(t**2 + 1 + x**2/2, t), DE, z) == \
|
|
([(Poly(z + S.Half, z, domain='QQ'), Poly(t**2 + 1 + x**2/2, t,
|
|
domain='ZZ(x,nu)'))], True)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
|
|
assert residue_reduce(Poly(-2*x*t + 1 - x**2, t),
|
|
Poly(t**2 + 2*x*t + 1 + x**2, t), DE, z) == \
|
|
([(Poly(z**2 + Rational(1, 4), z), Poly(t + x + 2*z, t))], True)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
|
|
assert residue_reduce(Poly(t, t), Poly(t + sqrt(2), t), DE, z) == \
|
|
([(Poly(z - 1, z, domain='QQ'), Poly(t + sqrt(2), t))], True)
|
|
|
|
|
|
def test_integrate_hyperexponential():
|
|
# TODO: Add tests for integrate_hyperexponential() from the book
|
|
a = Poly((1 + 2*t1 + t1**2 + 2*t1**3)*t**2 + (1 + t1**2)*t + 1 + t1**2, t)
|
|
d = Poly(1, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t1**2, t1),
|
|
Poly(t*(1 + t1**2), t)], 'Tfuncs': [tan, Lambda(i, exp(tan(i)))]})
|
|
assert integrate_hyperexponential(a, d, DE) == \
|
|
(exp(2*tan(x))*tan(x) + exp(tan(x)), 1 + t1**2, True)
|
|
a = Poly((t1**3 + (x + 1)*t1**2 + t1 + x + 2)*t, t)
|
|
assert integrate_hyperexponential(a, d, DE) == \
|
|
((x + tan(x))*exp(tan(x)), 0, True)
|
|
|
|
a = Poly(t, t)
|
|
d = Poly(1, t)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*x*t, t)],
|
|
'Tfuncs': [Lambda(i, exp(x**2))]})
|
|
|
|
assert integrate_hyperexponential(a, d, DE) == \
|
|
(0, NonElementaryIntegral(exp(x**2), x), False)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
|
|
assert integrate_hyperexponential(a, d, DE) == (exp(x), 0, True)
|
|
|
|
a = Poly(25*t**6 - 10*t**5 + 7*t**4 - 8*t**3 + 13*t**2 + 2*t - 1, t)
|
|
d = Poly(25*t**6 + 35*t**4 + 11*t**2 + 1, t)
|
|
assert integrate_hyperexponential(a, d, DE) == \
|
|
(-(11 - 10*exp(x))/(5 + 25*exp(2*x)) + log(1 + exp(2*x)), -1, True)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(t0*t, t)],
|
|
'Tfuncs': [exp, Lambda(i, exp(exp(i)))]})
|
|
assert integrate_hyperexponential(Poly(2*t0*t**2, t), Poly(1, t), DE) == (exp(2*exp(x)), 0, True)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(-t0*t, t)],
|
|
'Tfuncs': [exp, Lambda(i, exp(-exp(i)))]})
|
|
assert integrate_hyperexponential(Poly(-27*exp(9) - 162*t0*exp(9) +
|
|
27*x*t0*exp(9), t), Poly((36*exp(18) + x**2*exp(18) - 12*x*exp(18))*t, t), DE) == \
|
|
(27*exp(exp(x))/(-6*exp(9) + x*exp(9)), 0, True)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
|
|
assert integrate_hyperexponential(Poly(x**2/2*t, t), Poly(1, t), DE) == \
|
|
((2 - 2*x + x**2)*exp(x)/2, 0, True)
|
|
assert integrate_hyperexponential(Poly(1 + t, t), Poly(t, t), DE) == \
|
|
(-exp(-x), 1, True) # x - exp(-x)
|
|
assert integrate_hyperexponential(Poly(x, t), Poly(t + 1, t), DE) == \
|
|
(0, NonElementaryIntegral(x/(1 + exp(x)), x), False)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)],
|
|
'Tfuncs': [log, Lambda(i, exp(i**2))]})
|
|
|
|
elem, nonelem, b = integrate_hyperexponential(Poly((8*x**7 - 12*x**5 + 6*x**3 - x)*t1**4 +
|
|
(8*t0*x**7 - 8*t0*x**6 - 4*t0*x**5 + 2*t0*x**3 + 2*t0*x**2 - t0*x +
|
|
24*x**8 - 36*x**6 - 4*x**5 + 22*x**4 + 4*x**3 - 7*x**2 - x + 1)*t1**3
|
|
+ (8*t0*x**8 - 4*t0*x**6 - 16*t0*x**5 - 2*t0*x**4 + 12*t0*x**3 +
|
|
t0*x**2 - 2*t0*x + 24*x**9 - 36*x**7 - 8*x**6 + 22*x**5 + 12*x**4 -
|
|
7*x**3 - 6*x**2 + x + 1)*t1**2 + (8*t0*x**8 - 8*t0*x**6 - 16*t0*x**5 +
|
|
6*t0*x**4 + 10*t0*x**3 - 2*t0*x**2 - t0*x + 8*x**10 - 12*x**8 - 4*x**7
|
|
+ 2*x**6 + 12*x**5 + 3*x**4 - 9*x**3 - x**2 + 2*x)*t1 + 8*t0*x**7 -
|
|
12*t0*x**6 - 4*t0*x**5 + 8*t0*x**4 - t0*x**2 - 4*x**7 + 4*x**6 +
|
|
4*x**5 - 4*x**4 - x**3 + x**2, t1), Poly((8*x**7 - 12*x**5 + 6*x**3 -
|
|
x)*t1**4 + (24*x**8 + 8*x**7 - 36*x**6 - 12*x**5 + 18*x**4 + 6*x**3 -
|
|
3*x**2 - x)*t1**3 + (24*x**9 + 24*x**8 - 36*x**7 - 36*x**6 + 18*x**5 +
|
|
18*x**4 - 3*x**3 - 3*x**2)*t1**2 + (8*x**10 + 24*x**9 - 12*x**8 -
|
|
36*x**7 + 6*x**6 + 18*x**5 - x**4 - 3*x**3)*t1 + 8*x**10 - 12*x**8 +
|
|
6*x**6 - x**4, t1), DE)
|
|
|
|
assert factor(elem) == -((x - 1)*log(x)/((x + exp(x**2))*(2*x**2 - 1)))
|
|
assert (nonelem, b) == (NonElementaryIntegral(exp(x**2)/(exp(x**2) + 1), x), False)
|
|
|
|
def test_integrate_hyperexponential_polynomial():
|
|
# Without proper cancellation within integrate_hyperexponential_polynomial(),
|
|
# this will take a long time to complete, and will return a complicated
|
|
# expression
|
|
p = Poly((-28*x**11*t0 - 6*x**8*t0 + 6*x**9*t0 - 15*x**8*t0**2 +
|
|
15*x**7*t0**2 + 84*x**10*t0**2 - 140*x**9*t0**3 - 20*x**6*t0**3 +
|
|
20*x**7*t0**3 - 15*x**6*t0**4 + 15*x**5*t0**4 + 140*x**8*t0**4 -
|
|
84*x**7*t0**5 - 6*x**4*t0**5 + 6*x**5*t0**5 + x**3*t0**6 - x**4*t0**6 +
|
|
28*x**6*t0**6 - 4*x**5*t0**7 + x**9 - x**10 + 4*x**12)/(-8*x**11*t0 +
|
|
28*x**10*t0**2 - 56*x**9*t0**3 + 70*x**8*t0**4 - 56*x**7*t0**5 +
|
|
28*x**6*t0**6 - 8*x**5*t0**7 + x**4*t0**8 + x**12)*t1**2 +
|
|
(-28*x**11*t0 - 12*x**8*t0 + 12*x**9*t0 - 30*x**8*t0**2 +
|
|
30*x**7*t0**2 + 84*x**10*t0**2 - 140*x**9*t0**3 - 40*x**6*t0**3 +
|
|
40*x**7*t0**3 - 30*x**6*t0**4 + 30*x**5*t0**4 + 140*x**8*t0**4 -
|
|
84*x**7*t0**5 - 12*x**4*t0**5 + 12*x**5*t0**5 - 2*x**4*t0**6 +
|
|
2*x**3*t0**6 + 28*x**6*t0**6 - 4*x**5*t0**7 + 2*x**9 - 2*x**10 +
|
|
4*x**12)/(-8*x**11*t0 + 28*x**10*t0**2 - 56*x**9*t0**3 +
|
|
70*x**8*t0**4 - 56*x**7*t0**5 + 28*x**6*t0**6 - 8*x**5*t0**7 +
|
|
x**4*t0**8 + x**12)*t1 + (-2*x**2*t0 + 2*x**3*t0 + x*t0**2 -
|
|
x**2*t0**2 + x**3 - x**4)/(-4*x**5*t0 + 6*x**4*t0**2 - 4*x**3*t0**3 +
|
|
x**2*t0**4 + x**6), t1, z, expand=False)
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)]})
|
|
assert integrate_hyperexponential_polynomial(p, DE, z) == (
|
|
Poly((x - t0)*t1**2 + (-2*t0 + 2*x)*t1, t1), Poly(-2*x*t0 + x**2 +
|
|
t0**2, t1), True)
|
|
|
|
DE = DifferentialExtension(extension={'D':[Poly(1, x), Poly(t0, t0)]})
|
|
assert integrate_hyperexponential_polynomial(Poly(0, t0), DE, z) == (
|
|
Poly(0, t0), Poly(1, t0), True)
|
|
|
|
|
|
def test_integrate_hyperexponential_returns_piecewise():
|
|
a, b = symbols('a b')
|
|
DE = DifferentialExtension(a**x, x)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
(exp(x*log(a))/log(a), Ne(log(a), 0)), (x, True)), 0, True)
|
|
DE = DifferentialExtension(a**(b*x), x)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
(exp(b*x*log(a))/(b*log(a)), Ne(b*log(a), 0)), (x, True)), 0, True)
|
|
DE = DifferentialExtension(exp(a*x), x)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
(exp(a*x)/a, Ne(a, 0)), (x, True)), 0, True)
|
|
DE = DifferentialExtension(x*exp(a*x), x)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
((a*x - 1)*exp(a*x)/a**2, Ne(a**2, 0)), (x**2/2, True)), 0, True)
|
|
DE = DifferentialExtension(x**2*exp(a*x), x)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
((x**2*a**2 - 2*a*x + 2)*exp(a*x)/a**3, Ne(a**3, 0)),
|
|
(x**3/3, True)), 0, True)
|
|
DE = DifferentialExtension(x**y + z, y)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
(exp(log(x)*y)/log(x), Ne(log(x), 0)), (y, True)), z, True)
|
|
DE = DifferentialExtension(x**y + z + x**(2*y), y)
|
|
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
|
|
((exp(2*log(x)*y)*log(x) +
|
|
2*exp(log(x)*y)*log(x))/(2*log(x)**2), Ne(2*log(x)**2, 0)),
|
|
(2*y, True),
|
|
), z, True)
|
|
# TODO: Add a test where two different parts of the extension use a
|
|
# Piecewise, like y**x + z**x.
|
|
|
|
|
|
def test_issue_13947():
|
|
a, t, s = symbols('a t s')
|
|
assert risch_integrate(2**(-pi)/(2**t + 1), t) == \
|
|
2**(-pi)*t - 2**(-pi)*log(2**t + 1)/log(2)
|
|
assert risch_integrate(a**(t - s)/(a**t + 1), t) == \
|
|
exp(-s*log(a))*log(a**t + 1)/log(a)
|
|
|
|
|
|
def test_integrate_primitive():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)],
|
|
'Tfuncs': [log]})
|
|
assert integrate_primitive(Poly(t, t), Poly(1, t), DE) == (x*log(x), -1, True)
|
|
assert integrate_primitive(Poly(x, t), Poly(t, t), DE) == (0, NonElementaryIntegral(x/log(x), x), False)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x + 1), t2)],
|
|
'Tfuncs': [log, Lambda(i, log(i + 1))]})
|
|
assert integrate_primitive(Poly(t1, t2), Poly(t2, t2), DE) == \
|
|
(0, NonElementaryIntegral(log(x)/log(1 + x), x), False)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x*t1), t2)],
|
|
'Tfuncs': [log, Lambda(i, log(log(i)))]})
|
|
assert integrate_primitive(Poly(t2, t2), Poly(t1, t2), DE) == \
|
|
(0, NonElementaryIntegral(log(log(x))/log(x), x), False)
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0)],
|
|
'Tfuncs': [log]})
|
|
assert integrate_primitive(Poly(x**2*t0**3 + (3*x**2 + x)*t0**2 + (3*x**2
|
|
+ 2*x)*t0 + x**2 + x, t0), Poly(x**2*t0**4 + 4*x**2*t0**3 + 6*x**2*t0**2 +
|
|
4*x**2*t0 + x**2, t0), DE) == \
|
|
(-1/(log(x) + 1), NonElementaryIntegral(1/(log(x) + 1), x), False)
|
|
|
|
def test_integrate_hypertangent_polynomial():
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
|
|
assert integrate_hypertangent_polynomial(Poly(t**2 + x*t + 1, t), DE) == \
|
|
(Poly(t, t), Poly(x/2, t))
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(a*(t**2 + 1), t)]})
|
|
assert integrate_hypertangent_polynomial(Poly(t**5, t), DE) == \
|
|
(Poly(1/(4*a)*t**4 - 1/(2*a)*t**2, t), Poly(1/(2*a), t))
|
|
|
|
|
|
def test_integrate_nonlinear_no_specials():
|
|
a, d, = Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 -
|
|
nu**2)*t - x**5/4, t), Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t)
|
|
# f(x) == phi_nu(x), the logarithmic derivative of J_v, the Bessel function,
|
|
# which has no specials (see Chapter 5, note 4 of Bronstein's book).
|
|
f = Function('phi_nu')
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x),
|
|
Poly(-t**2 - t/x - (1 - nu**2/x**2), t)], 'Tfuncs': [f]})
|
|
assert integrate_nonlinear_no_specials(a, d, DE) == \
|
|
(-log(1 + f(x)**2 + x**2/2)/2 + (- 4 - x**2)/(4 + 2*x**2 + 4*f(x)**2), True)
|
|
assert integrate_nonlinear_no_specials(Poly(t, t), Poly(1, t), DE) == \
|
|
(0, False)
|
|
|
|
|
|
def test_integer_powers():
|
|
assert integer_powers([x, x/2, x**2 + 1, x*Rational(2, 3)]) == [
|
|
(x/6, [(x, 6), (x/2, 3), (x*Rational(2, 3), 4)]),
|
|
(1 + x**2, [(1 + x**2, 1)])]
|
|
|
|
|
|
def test_DifferentialExtension_exp():
|
|
assert DifferentialExtension(exp(x) + exp(x**2), x)._important_attrs == \
|
|
(Poly(t1 + t0, t1), Poly(1, t1), [Poly(1, x,), Poly(t0, t0),
|
|
Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
|
|
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
|
|
assert DifferentialExtension(exp(x) + exp(2*x), x)._important_attrs == \
|
|
(Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0, t0)], [x, t0],
|
|
[Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
|
|
assert DifferentialExtension(exp(x) + exp(x/2), x)._important_attrs == \
|
|
(Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)],
|
|
[x, t0], [Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
|
|
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2), x)._important_attrs == \
|
|
(Poly((1 + t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
|
|
Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
|
|
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
|
|
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2 + 1), x)._important_attrs == \
|
|
(Poly((1 + S.Exp1*t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x),
|
|
Poly(t0, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
|
|
Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
|
|
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2), x)._important_attrs == \
|
|
(Poly((t0 + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
|
|
Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1],
|
|
[Lambda(i, exp(i/2)), Lambda(i, exp(i**2))],
|
|
[(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'], [None, x/2, x**2])
|
|
assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2 + 3), x)._important_attrs == \
|
|
(Poly((t0*exp(3) + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
|
|
Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i/2)),
|
|
Lambda(i, exp(i**2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'],
|
|
[None, x/2, x**2])
|
|
assert DifferentialExtension(sqrt(exp(x)), x)._important_attrs == \
|
|
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
|
|
[Lambda(i, exp(i/2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp'], [None, x/2])
|
|
|
|
assert DifferentialExtension(exp(x/2), x)._important_attrs == \
|
|
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
|
|
[Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
|
|
|
|
|
|
def test_DifferentialExtension_log():
|
|
assert DifferentialExtension(log(x)*log(x + 1)*log(2*x**2 + 2*x), x)._important_attrs == \
|
|
(Poly(t0*t1**2 + (t0*log(2) + t0**2)*t1, t1), Poly(1, t1),
|
|
[Poly(1, x), Poly(1/x, t0),
|
|
Poly(1/(x + 1), t1, expand=False)], [x, t0, t1],
|
|
[Lambda(i, log(i)), Lambda(i, log(i + 1))], [], [None, 'log', 'log'],
|
|
[None, x, x + 1])
|
|
assert DifferentialExtension(x**x*log(x), x)._important_attrs == \
|
|
(Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
|
|
Poly((1 + t0)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)),
|
|
Lambda(i, exp(t0*i))], [(exp(x*log(x)), x**x)], [None, 'log', 'exp'],
|
|
[None, x, t0*x])
|
|
|
|
|
|
def test_DifferentialExtension_symlog():
|
|
# See comment on test_risch_integrate below
|
|
assert DifferentialExtension(log(x**x), x)._important_attrs == \
|
|
(Poly(t0*x, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0), Poly((t0 +
|
|
1)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i*t0))],
|
|
[(exp(x*log(x)), x**x)], [None, 'log', 'exp'], [None, x, t0*x])
|
|
assert DifferentialExtension(log(x**y), x)._important_attrs == \
|
|
(Poly(y*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
|
|
[Lambda(i, log(i))], [(y*log(x), log(x**y))], [None, 'log'],
|
|
[None, x])
|
|
assert DifferentialExtension(log(sqrt(x)), x)._important_attrs == \
|
|
(Poly(t0, t0), Poly(2, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
|
|
[Lambda(i, log(i))], [(log(x)/2, log(sqrt(x)))], [None, 'log'],
|
|
[None, x])
|
|
|
|
|
|
def test_DifferentialExtension_handle_first():
|
|
assert DifferentialExtension(exp(x)*log(x), x, handle_first='log')._important_attrs == \
|
|
(Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
|
|
Poly(t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i))],
|
|
[], [None, 'log', 'exp'], [None, x, x])
|
|
assert DifferentialExtension(exp(x)*log(x), x, handle_first='exp')._important_attrs == \
|
|
(Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
|
|
Poly(1/x, t1)], [x, t0, t1], [Lambda(i, exp(i)), Lambda(i, log(i))],
|
|
[], [None, 'exp', 'log'], [None, x, x])
|
|
|
|
# This one must have the log first, regardless of what we set it to
|
|
# (because the log is inside of the exponential: x**x == exp(x*log(x)))
|
|
assert DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
|
|
handle_first='exp')._important_attrs == \
|
|
DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
|
|
handle_first='log')._important_attrs == \
|
|
(Poly((-1 + x - x*t0**2)*t1, t1), Poly(x, t1),
|
|
[Poly(1, x), Poly(1/x, t0), Poly((1 + t0)*t1, t1)], [x, t0, t1],
|
|
[Lambda(i, log(i)), Lambda(i, exp(t0*i))], [(exp(x*log(x)), x**x)],
|
|
[None, 'log', 'exp'], [None, x, t0*x])
|
|
|
|
|
|
def test_DifferentialExtension_all_attrs():
|
|
# Test 'unimportant' attributes
|
|
DE = DifferentialExtension(exp(x)*log(x), x, handle_first='exp')
|
|
assert DE.f == exp(x)*log(x)
|
|
assert DE.newf == t0*t1
|
|
assert DE.x == x
|
|
assert DE.cases == ['base', 'exp', 'primitive']
|
|
assert DE.case == 'primitive'
|
|
|
|
assert DE.level == -1
|
|
assert DE.t == t1 == DE.T[DE.level]
|
|
assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
|
|
raises(ValueError, lambda: DE.increment_level())
|
|
DE.decrement_level()
|
|
assert DE.level == -2
|
|
assert DE.t == t0 == DE.T[DE.level]
|
|
assert DE.d == Poly(t0, t0) == DE.D[DE.level]
|
|
assert DE.case == 'exp'
|
|
DE.decrement_level()
|
|
assert DE.level == -3
|
|
assert DE.t == x == DE.T[DE.level] == DE.x
|
|
assert DE.d == Poly(1, x) == DE.D[DE.level]
|
|
assert DE.case == 'base'
|
|
raises(ValueError, lambda: DE.decrement_level())
|
|
DE.increment_level()
|
|
DE.increment_level()
|
|
assert DE.level == -1
|
|
assert DE.t == t1 == DE.T[DE.level]
|
|
assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
|
|
assert DE.case == 'primitive'
|
|
|
|
# Test methods
|
|
assert DE.indices('log') == [2]
|
|
assert DE.indices('exp') == [1]
|
|
|
|
|
|
def test_DifferentialExtension_extension_flag():
|
|
raises(ValueError, lambda: DifferentialExtension(extension={'T': [x, t]}))
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
|
|
assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
|
|
None, None, None, None)
|
|
assert DE.d == Poly(t, t)
|
|
assert DE.t == t
|
|
assert DE.level == -1
|
|
assert DE.cases == ['base', 'exp']
|
|
assert DE.x == x
|
|
assert DE.case == 'exp'
|
|
|
|
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)],
|
|
'exts': [None, 'exp'], 'extargs': [None, x]})
|
|
assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
|
|
None, None, [None, 'exp'], [None, x])
|
|
raises(ValueError, lambda: DifferentialExtension())
|
|
|
|
|
|
def test_DifferentialExtension_misc():
|
|
# Odd ends
|
|
assert DifferentialExtension(sin(y)*exp(x), x)._important_attrs == \
|
|
(Poly(sin(y)*t0, t0, domain='ZZ[sin(y)]'), Poly(1, t0, domain='ZZ'),
|
|
[Poly(1, x, domain='ZZ'), Poly(t0, t0, domain='ZZ')], [x, t0],
|
|
[Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
|
|
raises(NotImplementedError, lambda: DifferentialExtension(sin(x), x))
|
|
assert DifferentialExtension(10**x, x)._important_attrs == \
|
|
(Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(log(10)*t0, t0)], [x, t0],
|
|
[Lambda(i, exp(i*log(10)))], [(exp(x*log(10)), 10**x)], [None, 'exp'],
|
|
[None, x*log(10)])
|
|
assert DifferentialExtension(log(x) + log(x**2), x)._important_attrs in [
|
|
(Poly(3*t0, t0), Poly(2, t0), [Poly(1, x), Poly(2/x, t0)], [x, t0],
|
|
[Lambda(i, log(i**2))], [], [None, ], [], [1], [x**2]),
|
|
(Poly(3*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
|
|
[Lambda(i, log(i))], [], [None, 'log'], [None, x])]
|
|
assert DifferentialExtension(S.Zero, x)._important_attrs == \
|
|
(Poly(0, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
|
|
assert DifferentialExtension(tan(atan(x).rewrite(log)), x)._important_attrs == \
|
|
(Poly(x, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
|
|
|
|
|
|
def test_DifferentialExtension_Rothstein():
|
|
# Rothstein's integral
|
|
f = (2581284541*exp(x) + 1757211400)/(39916800*exp(3*x) +
|
|
119750400*exp(x)**2 + 119750400*exp(x) + 39916800)*exp(1/(exp(x) + 1) - 10*x)
|
|
assert DifferentialExtension(f, x)._important_attrs == \
|
|
(Poly((1757211400 + 2581284541*t0)*t1, t1), Poly(39916800 +
|
|
119750400*t0 + 119750400*t0**2 + 39916800*t0**3, t1),
|
|
[Poly(1, x), Poly(t0, t0), Poly(-(10 + 21*t0 + 10*t0**2)/(1 + 2*t0 +
|
|
t0**2)*t1, t1, domain='ZZ(t0)')], [x, t0, t1],
|
|
[Lambda(i, exp(i)), Lambda(i, exp(1/(t0 + 1) - 10*i))], [],
|
|
[None, 'exp', 'exp'], [None, x, 1/(t0 + 1) - 10*x])
|
|
|
|
|
|
class _TestingException(Exception):
|
|
"""Dummy Exception class for testing."""
|
|
pass
|
|
|
|
|
|
def test_DecrementLevel():
|
|
DE = DifferentialExtension(x*log(exp(x) + 1), x)
|
|
assert DE.level == -1
|
|
assert DE.t == t1
|
|
assert DE.d == Poly(t0/(t0 + 1), t1)
|
|
assert DE.case == 'primitive'
|
|
|
|
with DecrementLevel(DE):
|
|
assert DE.level == -2
|
|
assert DE.t == t0
|
|
assert DE.d == Poly(t0, t0)
|
|
assert DE.case == 'exp'
|
|
|
|
with DecrementLevel(DE):
|
|
assert DE.level == -3
|
|
assert DE.t == x
|
|
assert DE.d == Poly(1, x)
|
|
assert DE.case == 'base'
|
|
|
|
assert DE.level == -2
|
|
assert DE.t == t0
|
|
assert DE.d == Poly(t0, t0)
|
|
assert DE.case == 'exp'
|
|
|
|
assert DE.level == -1
|
|
assert DE.t == t1
|
|
assert DE.d == Poly(t0/(t0 + 1), t1)
|
|
assert DE.case == 'primitive'
|
|
|
|
# Test that __exit__ is called after an exception correctly
|
|
try:
|
|
with DecrementLevel(DE):
|
|
raise _TestingException
|
|
except _TestingException:
|
|
pass
|
|
else:
|
|
raise AssertionError("Did not raise.")
|
|
|
|
assert DE.level == -1
|
|
assert DE.t == t1
|
|
assert DE.d == Poly(t0/(t0 + 1), t1)
|
|
assert DE.case == 'primitive'
|
|
|
|
|
|
def test_risch_integrate():
|
|
assert risch_integrate(t0*exp(x), x) == t0*exp(x)
|
|
assert risch_integrate(sin(x), x, rewrite_complex=True) == -exp(I*x)/2 - exp(-I*x)/2
|
|
|
|
# From my GSoC writeup
|
|
assert risch_integrate((1 + 2*x**2 + x**4 + 2*x**3*exp(2*x**2))/
|
|
(x**4*exp(x**2) + 2*x**2*exp(x**2) + exp(x**2)), x) == \
|
|
NonElementaryIntegral(exp(-x**2), x) + exp(x**2)/(1 + x**2)
|
|
|
|
|
|
assert risch_integrate(0, x) == 0
|
|
|
|
# also tests prde_cancel()
|
|
e1 = log(x/exp(x) + 1)
|
|
ans1 = risch_integrate(e1, x)
|
|
assert ans1 == (x*log(x*exp(-x) + 1) + NonElementaryIntegral((x**2 - x)/(x + exp(x)), x))
|
|
assert cancel(diff(ans1, x) - e1) == 0
|
|
|
|
# also tests issue #10798
|
|
e2 = (log(-1/y)/2 - log(1/y)/2)/y - (log(1 - 1/y)/2 - log(1 + 1/y)/2)/y
|
|
ans2 = risch_integrate(e2, y)
|
|
assert ans2 == log(1/y)*log(1 - 1/y)/2 - log(1/y)*log(1 + 1/y)/2 + \
|
|
NonElementaryIntegral((I*pi*y**2 - 2*y*log(1/y) - I*pi)/(2*y**3 - 2*y), y)
|
|
assert expand_log(cancel(diff(ans2, y) - e2), force=True) == 0
|
|
|
|
# These are tested here in addition to in test_DifferentialExtension above
|
|
# (symlogs) to test that backsubs works correctly. The integrals should be
|
|
# written in terms of the original logarithms in the integrands.
|
|
|
|
# XXX: Unfortunately, making backsubs work on this one is a little
|
|
# trickier, because x**x is converted to exp(x*log(x)), and so log(x**x)
|
|
# is converted to x*log(x). (x**2*log(x)).subs(x*log(x), log(x**x)) is
|
|
# smart enough, the issue is that these splits happen at different places
|
|
# in the algorithm. Maybe a heuristic is in order
|
|
assert risch_integrate(log(x**x), x) == x**2*log(x)/2 - x**2/4
|
|
|
|
assert risch_integrate(log(x**y), x) == x*log(x**y) - x*y
|
|
assert risch_integrate(log(sqrt(x)), x) == x*log(sqrt(x)) - x/2
|
|
|
|
|
|
def test_risch_integrate_float():
|
|
assert risch_integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) == -2.4*exp(8*x) - 12.0*exp(5*x)
|
|
|
|
|
|
def test_NonElementaryIntegral():
|
|
assert isinstance(risch_integrate(exp(x**2), x), NonElementaryIntegral)
|
|
assert isinstance(risch_integrate(x**x*log(x), x), NonElementaryIntegral)
|
|
# Make sure methods of Integral still give back a NonElementaryIntegral
|
|
assert isinstance(NonElementaryIntegral(x**x*t0, x).subs(t0, log(x)), NonElementaryIntegral)
|
|
|
|
|
|
def test_xtothex():
|
|
a = risch_integrate(x**x, x)
|
|
assert a == NonElementaryIntegral(x**x, x)
|
|
assert isinstance(a, NonElementaryIntegral)
|
|
|
|
|
|
def test_DifferentialExtension_equality():
|
|
DE1 = DE2 = DifferentialExtension(log(x), x)
|
|
assert DE1 == DE2
|
|
|
|
|
|
def test_DifferentialExtension_printing():
|
|
DE = DifferentialExtension(exp(2*x**2) + log(exp(x**2) + 1), x)
|
|
assert repr(DE) == ("DifferentialExtension(dict([('f', exp(2*x**2) + log(exp(x**2) + 1)), "
|
|
"('x', x), ('T', [x, t0, t1]), ('D', [Poly(1, x, domain='ZZ'), Poly(2*x*t0, t0, domain='ZZ[x]'), "
|
|
"Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')]), ('fa', Poly(t1 + t0**2, t1, domain='ZZ[t0]')), "
|
|
"('fd', Poly(1, t1, domain='ZZ')), ('Tfuncs', [Lambda(i, exp(i**2)), Lambda(i, log(t0 + 1))]), "
|
|
"('backsubs', []), ('exts', [None, 'exp', 'log']), ('extargs', [None, x**2, t0 + 1]), "
|
|
"('cases', ['base', 'exp', 'primitive']), ('case', 'primitive'), ('t', t1), "
|
|
"('d', Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')), ('newf', t0**2 + t1), ('level', -1), "
|
|
"('dummy', False)]))")
|
|
|
|
assert str(DE) == ("DifferentialExtension({fa=Poly(t1 + t0**2, t1, domain='ZZ[t0]'), "
|
|
"fd=Poly(1, t1, domain='ZZ'), D=[Poly(1, x, domain='ZZ'), Poly(2*x*t0, t0, domain='ZZ[x]'), "
|
|
"Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')]})")
|
|
|
|
|
|
def test_issue_23948():
|
|
f = (
|
|
( (-2*x**5 + 28*x**4 - 144*x**3 + 324*x**2 - 270*x)*log(x)**2
|
|
+(-4*x**6 + 56*x**5 - 288*x**4 + 648*x**3 - 540*x**2)*log(x)
|
|
+(2*x**5 - 28*x**4 + 144*x**3 - 324*x**2 + 270*x)*exp(x)
|
|
+(2*x**5 - 28*x**4 + 144*x**3 - 324*x**2 + 270*x)*log(5)
|
|
-2*x**7 + 26*x**6 - 116*x**5 + 180*x**4 + 54*x**3 - 270*x**2
|
|
)*log(-log(x)**2 - 2*x*log(x) + exp(x) + log(5) - x**2 - x)**2
|
|
+( (4*x**5 - 44*x**4 + 168*x**3 - 216*x**2 - 108*x + 324)*log(x)
|
|
+(-2*x**5 + 24*x**4 - 108*x**3 + 216*x**2 - 162*x)*exp(x)
|
|
+4*x**6 - 42*x**5 + 144*x**4 - 108*x**3 - 324*x**2 + 486*x
|
|
)*log(-log(x)**2 - 2*x*log(x) + exp(x) + log(5) - x**2 - x)
|
|
)/(x*exp(x)**2*log(x)**2 + 2*x**2*exp(x)**2*log(x) - x*exp(x)**3
|
|
+(-x*log(5) + x**3 + x**2)*exp(x)**2)
|
|
|
|
F = ((x**4 - 12*x**3 + 54*x**2 - 108*x + 81)*exp(-2*x)
|
|
*log(-x**2 - 2*x*log(x) - x + exp(x) - log(x)**2 + log(5))**2)
|
|
|
|
assert risch_integrate(f, x) == F
|