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(
(x, Eq(log(a), 0)), (exp(x*log(a))/log(a), True)), 0, True)
DE = DifferentialExtension(a**(b*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x, Eq(b*log(a), 0)), (exp(b*x*log(a))/(b*log(a)), True)), 0, True)
DE = DifferentialExtension(exp(a*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x, Eq(a, 0)), (exp(a*x)/a, True)), 0, 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 x: exp(tan(x))]})
assert integrate_hyperexponential(a, d, DE) == \
(exp(2*tan(x))*tan(x) + exp(tan(x)), 1 + t1**2, True)
# exp(2*tan(x))*tan(x) + tan(x) + exp(tan(x))
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 x: 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) == \
(-(55 - 50*exp(x))/(25 + 125*exp(2*x)) + log(1 + exp(2*x)), -1, True)
# -(55 - 50*exp(x))/(25 + 125*exp(2*x)) - x + log(1 + exp(2*x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(t0*t, t)],
'Tfuncs': [exp, lambda x: exp(exp(x))]})
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 x: exp(-exp(x))]})
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)
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(
(x, Eq(log(a), 0)), (exp(x*log(a))/log(a), True)), 0, True)
DE = DifferentialExtension(a**(b*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x, Eq(b*log(a), 0)), (exp(b*x*log(a))/(b*log(a)), True)), 0, True)
DE = DifferentialExtension(exp(a*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x, Eq(a, 0)), (exp(a*x)/a, True)), 0, True)
DE = DifferentialExtension(x*exp(a*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x**2/2, Eq(a**3, 0)), ((x*a**2 - a)*exp(a*x)/a**3, True)), 0, True)
DE = DifferentialExtension(x**2*exp(a*x), x)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
(x**3/3, Eq(a**6, 0)),
((x**2*a**5 - 2*x*a**4 + 2*a**3)*exp(a*x)/a**6, True)), 0, True)
DE = DifferentialExtension(x**y + z, y)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise((y,
Eq(log(x), 0)), (exp(log(x)*y)/log(x), True)), z, True)
DE = DifferentialExtension(x**y + z + x**(2*y), y)
assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise((2*y,
Eq(2*log(x)**2, 0)), ((exp(2*log(x)*y)*log(x) +
2*exp(log(x)*y)*log(x))/(2*log(x)**2), True)), z, 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)
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)
# exp(2*tan(x))*tan(x) + tan(x) + exp(tan(x))
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)
# -(55 - 50*exp(x))/(25 + 125*exp(2*x)) - x + log(1 + exp(2*x))
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():
# 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)