def test_apart_extension():
f = 2 / (x**2 + 1)
g = I / (x + I) - I / (x - I)
assert apart(f, extension=I) == g
assert apart(f, gaussian=True) == g
f = x / ((x - 2) * (x + I))
assert factor(together(apart(f))) == f
def test_apart_extension():
f = 2/(x**2 + 1)
g = I/(x + I) - I/(x - I)
assert apart(f, extension=I) == g
assert apart(f, gaussian=True) == g
f = x/((x - 2)*(x + I))
assert factor(together(apart(f))) == f
def test_Limits_simple_3a():
# issue sympy/sympy#3513
assert together(limit((x**2 - (a + 1)*x + a)/(x**3 - a**3), x, a)) == \
(a - 1)/(3*a**2) # 196
def test_together():
assert together(0) == 0
assert together(1) == 1
assert together(x * y * z) == x * y * z
assert together(x + y) == x + y
assert together(1 / x) == 1 / x
assert together(1 / x + 1) == (x + 1) / x
assert together(1 / x + 3) == (3 * x + 1) / x
assert together(1 / x + x) == (x**2 + 1) / x
assert together((1 / x + 1, 1 / x + 3)) == ((x + 1) / x, (3 * x + 1) / x)
assert together(1 / x + Rational(1, 2)) == (x + 2) / (2 * x)
assert together(Rational(1, 2) + x / 2) == Mul(Rational(1, 2),
x + 1,
evaluate=False)
assert together(1 / x + 2 / y) == (2 * x + y) / (y * x)
assert together(1 / (1 + 1 / x)) == x / (1 + x)
assert together(x / (1 + 1 / x)) == x**2 / (1 + x)
assert together(1 / x + 1 / y +
1 / z) == (x * y + x * z + y * z) / (x * y * z)
assert together(
1 / (1 + x + 1 / y + 1 / z)) == y * z / (y + z + y * z + x * y * z)
assert together(1 / (x * y) + 1 /
(x * y)**2) == y**(-2) * x**(-2) * (1 + x * y)
assert together(1 / (x * y) + 1 /
(x * y)**4) == y**(-4) * x**(-4) * (1 + x**3 * y**3)
assert together(1 / (x**7 * y) + 1 /
(x * y)**4) == y**(-4) * x**(-7) * (x**3 + y**3)
assert together(5/(2 + 6/(3 + 7/(4 + 8/(5 + 9/x))))) == \
Rational(5, 2)*((171 + 119*x)/(279 + 203*x))
assert together(1 + 1 / (x + 1)**2) == (1 + (x + 1)**2) / (x + 1)**2
assert together(1 + 1 / (x * (1 + x))) == (1 + x * (1 + x)) / (x * (1 + x))
assert together(1 / (x * (x + 1)) + 1 /
(x * (x + 2))) == (3 + 2 * x) / (x * (1 + x) * (2 + x))
assert together(1 + 1 /
(2 * x + 2)**2) == (4 * (x + 1)**2 + 1) / (4 * (x + 1)**2)
assert together(sin(1 / x + 1 / y)) == sin(1 / x + 1 / y)
assert together(sin(1 / x + 1 / y), deep=True) == sin((x + y) / (x * y))
assert together(1 / exp(x) + 1 / (x * exp(x))) == (1 + x) / (x * exp(x))
assert together(1 / exp(2 * x) + 1 /
(x * exp(3 * x))) == (1 + exp(x) * x) / (x * exp(3 * x))
assert together(Integral(1 / x + 1 / y, x)) == Integral((x + y) / (x * y),
x)
assert together(Eq(1 / x + 1 / y, 1 + 1 / z)) == Eq((x + y) / (x * y),
(z + 1) / z)
assert together((A * B)**-1 + (B * A)**-1) == (A * B)**-1 + (B * A)**-1