def test_collect_5():
"""Collect with respect to a tuple."""
assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [
z*(1 + a + x**2*y**4) + x**2*y**4,
z*(1 + a) + x**2*y**4*(1 + z)]
assert collect((1 + (x + y) + (x + y)**2).expand(),
[x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2
def test_collect_5():
"""Collect with respect to a tuple"""
assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [
z*(1 + a + x**2*y**4) + x**2*y**4,
z*(1 + a) + x**2*y**4*(1 + z) ]
assert collect((1 + (x + y) + (x + y)**2).expand(),
[x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2
def test_collect_4():
"""Collect with respect to a power"""
assert collect(a * x**c + b * x**c, x**c) == x**c * (a + b)
# issue sympy/sympy#6096: 2 stays with c (unless c is integer or x is positive0
assert collect(a * x**(2 * c) + b * x**(2 * c),
x**c) == x**(2 * c) * (a + b)
def test_collect_func():
f = ((x + a + 1)**3).expand()
assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \
x*(3*a**2 + 6*a + 3) + 1
assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \
(a + 1)**3
assert collect(f, x, evaluate=False) == {
1: a**3 + 3 * a**2 + 3 * a + 1,
x: 3 * a**2 + 6 * a + 3,
x**2: 3 * a + 3,
x**3: 1
}
def test_collect_D_0():
D = Derivative
f = Function('f')
fxx = D(f(x), x, x)
# collect does not distinguish nested derivatives, so it returns
# -- (a + b)*D(D(f, x), x)
assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx
def test_collect_func_xfail():
# XXX: this test will pass when automatic constant distribution is removed (issue sympy/sympy#4596)
assert collect(f, x, factor, evaluate=False) == {
1: (a + 1)**3,
x: 3 * (a + 1)**2,
x**2: 3 * (a + 1),
x**3: 1
}
def test_collect_D_0():
D = Derivative
f = Function('f')
fxx = D(f(x), x, x)
# collect does not distinguish nested derivatives, so it returns
# -- (a + b)*D(D(f, x), x)
assert collect(a * fxx + b * fxx, fxx) == (a + b) * fxx
def test_collect_func():
f = ((x + a + 1)**3).expand()
assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \
x*(3*a**2 + 6*a + 3) + 1
assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \
(a + 1)**3
assert collect(f, x, evaluate=False) == {
1: a**3 + 3*a**2 + 3*a + 1,
x: 3*a**2 + 6*a + 3, x**2: 3*a + 3,
x**3: 1
}
assert collect(f, x, factor,
evaluate=False) == {1: (a + 1)**3, x: 3*(a + 1)**2,
x**2: Mul(3, a + 1, evaluate=False),
x**3: 1}
def test_collect_Wild():
"""Collect with respect to functions with Wild argument"""
f = Function('f')
w1 = Wild('.1')
w2 = Wild('.2')
assert collect(f(x) + a*f(x), f(w1)) == (1 + a)*f(x)
assert collect(f(x, y) + a*f(x, y), f(w1)) == f(x, y) + a*f(x, y)
assert collect(f(x, y) + a*f(x, y), f(w1, w2)) == (1 + a)*f(x, y)
assert collect(f(x, y) + a*f(x, y), f(w1, w1)) == f(x, y) + a*f(x, y)
assert collect(f(x, x) + a*f(x, x), f(w1, w1)) == (1 + a)*f(x, x)
assert collect(a*(x + 1)**y + (x + 1)**y, w1**y) == (1 + a)*(x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
a*(x + 1)**y + (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
(1 + a)*(x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**w2) == (1 + a)*(x + 1)**y
def test_collect_Wild():
"""Collect with respect to functions with Wild argument"""
f = Function('f')
w1 = Wild('.1')
w2 = Wild('.2')
assert collect(f(x) + a * f(x), f(w1)) == (1 + a) * f(x)
assert collect(f(x, y) + a * f(x, y), f(w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w2)) == (1 + a) * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, x) + a * f(x, x), f(w1, w1)) == (1 + a) * f(x, x)
assert collect(a * (x + 1)**y + (x + 1)**y, w1**y) == (1 + a) * (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
a*(x + 1)**y + (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
(1 + a)*(x + 1)**y
assert collect(a * (x + 1)**y + (x + 1)**y, w1**w2) == (1 + a) * (x + 1)**y
def test_collect_1():
"""Collect with respect to a Symbol"""
assert collect( x + y*x, x ) == x * (1 + y)
assert collect( x + x**2, x ) == x + x**2
assert collect( x**2 + y*x**2, x ) == (x**2)*(1 + y)
assert collect( x**2 + y*x, x ) == x*y + x**2
assert collect( 2*x**2 + y*x**2 + 3*x*y, [x] ) == x**2*(2 + y) + 3*x*y
assert collect( 2*x**2 + y*x**2 + 3*x*y, [y] ) == 2*x**2 + y*(x**2 + 3*x)
assert collect( ((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \
x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \
x**3*(4*(1 + y)).expand() + x**4
# symbols can be given as any iterable
expr = x + y
assert collect(expr, expr.free_symbols) == expr
def test_collect_1():
# Collect with respect to a Symbol.
assert collect(x + y*x, x) == x * (1 + y)
assert collect(x + x**2, x) == x + x**2
assert collect(x**2 + y*x**2, x) == (x**2)*(1 + y)
assert collect(x**2 + y*x, x) == x*y + x**2
assert collect(2*x**2 + y*x**2 + 3*x*y, [x]) == x**2*(2 + y) + 3*x*y
assert collect(2*x**2 + y*x**2 + 3*x*y, [y]) == 2*x**2 + y*(x**2 + 3*x)
assert collect(((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \
x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \
x**3*(4*(1 + y)).expand() + x**4
# symbols can be given as any iterable
expr = x + y
assert collect(expr, expr.free_symbols) == expr
def test_gcd_terms():
f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + \
(2*x + 2)*(x + 6)/(5*x**2 + 5)
assert _gcd_terms(f) == (Rational(6, 5) * ((1 + x) / (1 + x**2)), 5 + x, 1)
assert _gcd_terms(Add.make_args(f)) == \
(Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
newf = Rational(6, 5) * ((1 + x) * (5 + x) / (1 + x**2))
assert gcd_terms(f) == newf
args = Add.make_args(f)
# non-Basic sequences of terms treated as terms of Add
assert gcd_terms(list(args)) == newf
assert gcd_terms(tuple(args)) == newf
assert gcd_terms(set(args)) == newf
# but a Basic sequence is treated as a container
assert gcd_terms(Tuple(*args)) != newf
assert gcd_terms(Basic(Tuple(1, 3*y + 3*x*y), Tuple(1, 3))) == \
Basic((1, 3*y*(x + 1)), (1, 3))
# but we shouldn't change keys of a dictionary or some may be lost
assert gcd_terms(Dict((x*(1 + y), 2), (x + x*y, y + x*y))) == \
Dict({x*(y + 1): 2, x + x*y: y*(1 + x)})
assert gcd_terms((2 * x + 2)**3 +
(2 * x + 2)**2) == 4 * (x + 1)**2 * (2 * x + 3)
assert gcd_terms(0) == 0
assert gcd_terms(1) == 1
assert gcd_terms(x) == x
assert gcd_terms(2 + 2 * x) == Mul(2, 1 + x, evaluate=False)
arg = x * (2 * x + 4 * y)
garg = 2 * x * (x + 2 * y)
assert gcd_terms(arg) == garg
assert gcd_terms(sin(arg)) == sin(garg)
# issue sympy/sympy#6139-like
alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
a = alpha**2 - alpha * x**2 + alpha + x**3 - x * (alpha + 1)
rep = {
alpha: (1 + sqrt(5)) / 2 + alpha1 * x + alpha2 * x**2 + alpha3 * x**3
}
s = (a / (x - alpha)).subs(rep).series(x, 0, 1)
assert simplify(collect(s, x)) == -sqrt(5) / 2 - Rational(3, 2) + O(x)
# issue sympy/sympy#5917
assert _gcd_terms([Integer(0), Integer(0)]) == (0, 0, 1)
assert _gcd_terms([2 * x + 4]) == (2, x + 2, 1)
eq = x / (x + 1 / x)
assert gcd_terms(eq, fraction=False) == eq
def test_collect_3():
"""Collect with respect to a product"""
f = Function('f')
assert collect(-x/8 + x*y, -x) == x*(y - Rational(1, 8))
assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2)
assert collect( x*y + a*x*y, x*y) == x*y*(1 + a)
assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
x**2*log(x)**2*(a + b)
# with respect to a product of three symbols
assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
def test_collect_3():
# Collect with respect to a product.
f = Function('f')
assert collect(-x/8 + x*y, -x) == x*(y - Rational(1, 8))
assert collect(1 + x*(y**2), x*y) == 1 + x*(y**2)
assert collect(x*y + a*x*y, x*y) == x*y*(1 + a)
assert collect(1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
x**2*log(x)**2*(a + b)
# with respect to a product of three symbols
assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
def test_gcd_terms():
f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + \
(2*x + 2)*(x + 6)/(5*x**2 + 5)
assert _gcd_terms(f) == (Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
assert _gcd_terms(Add.make_args(f)) == \
(Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
newf = Rational(6, 5)*((1 + x)*(5 + x)/(1 + x**2))
assert gcd_terms(f) == newf
args = Add.make_args(f)
# non-Basic sequences of terms treated as terms of Add
assert gcd_terms(list(args)) == newf
assert gcd_terms(tuple(args)) == newf
assert gcd_terms(set(args)) == newf
# but a Basic sequence is treated as a container
assert gcd_terms(Tuple(*args)) != newf
assert gcd_terms(Basic(Tuple(1, 3*y + 3*x*y), Tuple(1, 3))) == \
Basic((1, 3*y*(x + 1)), (1, 3))
# but we shouldn't change keys of a dictionary or some may be lost
assert gcd_terms(Dict((x*(1 + y), 2), (x + x*y, y + x*y))) == \
Dict({x*(y + 1): 2, x + x*y: y*(1 + x)})
assert gcd_terms((2*x + 2)**3 + (2*x + 2)**2) == 4*(x + 1)**2*(2*x + 3)
assert gcd_terms(0) == 0
assert gcd_terms(1) == 1
assert gcd_terms(x) == x
assert gcd_terms(2 + 2*x) == Mul(2, 1 + x, evaluate=False)
arg = x*(2*x + 4*y)
garg = 2*x*(x + 2*y)
assert gcd_terms(arg) == garg
assert gcd_terms(sin(arg)) == sin(garg)
# issue sympy/sympy#6139-like
alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
a = alpha**2 - alpha*x**2 + alpha + x**3 - x*(alpha + 1)
rep = {alpha: (1 + sqrt(5))/2 + alpha1*x + alpha2*x**2 + alpha3*x**3}
s = (a/(x - alpha)).subs(rep).series(x, 0, 1)
assert simplify(collect(s, x)) == -sqrt(5)/2 - Rational(3, 2) + O(x)
# issue sympy/sympy#5917
assert _gcd_terms([Integer(0), Integer(0)]) == (0, 0, 1)
assert _gcd_terms([2*x + 4]) == (2, x + 2, 1)
eq = x/(x + 1/x)
assert gcd_terms(eq, fraction=False) == eq
def test_collect_D():
D = Derivative
f = Function('f')
fx = D(f(x), x)
fxx = D(f(x), x, x)
assert collect(a * fx + b * fx, fx) == (a + b) * fx
assert collect(a * D(fx, x) + b * D(fx, x), fx) == (a + b) * D(fx, x)
assert collect(a * fxx + b * fxx, fx) == (a + b) * D(fx, x)
# issue sympy/sympy#4784
assert collect(5 * f(x) + 3 * fx, fx) == 5 * f(x) + 3 * fx
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
def test_collect_D():
D = Derivative
f = Function('f')
fx = D(f(x), x)
fxx = D(f(x), x, x)
assert collect(a*fx + b*fx, fx) == (a + b)*fx
assert collect(a*D(fx, x) + b*D(fx, x), fx) == (a + b)*D(fx, x)
assert collect(a*fxx + b*fxx, fx) == (a + b)*D(fx, x)
# issue sympy/sympy#4784
assert collect(5*f(x) + 3*fx, fx) == 5*f(x) + 3*fx
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
def test_collect_order():
assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3))
assert collect(t + t*x + x**2 + O(x**3), t) == \
t*(1 + x + O(x**3)) + x**2 + O(x**3)
f = a*x + b*x + c*x**2 + d*x**2 + O(x**3)
g = x*(a + b) + x**2*(c + d) + O(x**3)
assert collect(f, x) == g
assert collect(f, x, distribute_order_term=False) == g
f = sin(a + b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)]) == \
sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \
sin(a)*cos(b).series(b, 0, 10).removeO() + \
cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10)
def test_collect_order():
assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3))
assert collect(t + t*x + x**2 + O(x**3), t) == \
t*(1 + x + O(x**3)) + x**2 + O(x**3)
f = a*x + b*x + c*x**2 + d*x**2 + O(x**3)
g = x*(a + b) + x**2*(c + d) + O(x**3)
assert collect(f, x) == g
assert collect(f, x, distribute_order_term=False) == g
f = sin(a + b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)]) == \
sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \
sin(a)*cos(b).series(b, 0, 10).removeO() + \
cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10)
def log_to_real(h, q, x, t):
"""
Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
Examples
========
>>> from diofant.integrals.rationaltools import log_to_real
>>> from diofant.abc import x, y
>>> from diofant import Poly, sqrt, Rational
>>> log_to_real(Poly(x + 3*y/2 + Rational(1, 2), x, domain='QQ[y]'),
... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
>>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
... Poly(-2*y + 1, y, domain='ZZ'), x, y)
log(x**2 - 1)/2
See Also
========
log_to_atan
"""
from diofant import collect
u, v = symbols('u,v', cls=Dummy)
H = h.as_expr().subs({t: u + I * v}).expand()
Q = q.as_expr().subs({t: u + I * v}).expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(Integer(1), Integer(0)), H_map.get(I, Integer(0))
c, d = Q_map.get(Integer(1), Integer(0)), Q_map.get(I, Integer(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, filter='R')
if len(R_u) != R.count_roots():
return
result = Integer(0)
for r_u in R_u.keys():
C = Poly(c.subs({u: r_u}), v)
R_v = roots(C, filter='R')
if len(R_v) != C.count_roots():
return
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u: r_u, v: r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u: r_u, v: r_v}), x)
B = Poly(b.subs({u: r_u, v: r_v}), x)
AB = (A**2 + B**2).as_expr()
result += r_u * log(AB) + r_v * log_to_atan(A, B)
R_q = roots(q, filter='R')
if len(R_q) != q.count_roots():
return
for r in R_q.keys():
result += r * log(h.as_expr().subs(t, r))
return result
def residue(expr, x, x0):
"""
Finds the residue of ``expr`` at the point ``x=x0``.
The residue is defined [1]_ as the coefficient of `1/(x - x_0)`
in the power series expansion around `x=x_0`.
This notion is essential for the Residue Theorem [2]_
Examples
========
>>> from diofant import residue, sin
>>> from diofant.abc import x
>>> residue(1/x, x, 0)
1
>>> residue(1/x**2, x, 0)
0
>>> residue(2/sin(x), x, 0)
2
References
==========
.. [1] http://en.wikipedia.org/wiki/Residue_%28complex_analysis%29
.. [2] http://en.wikipedia.org/wiki/Residue_theorem
"""
# The current implementation uses series expansion to
# calculate it. A more general implementation is explained in
# the section 5.6 of the Bronstein's book {M. Bronstein:
# Symbolic Integration I, Springer Verlag (2005)}. For purely
# rational functions, the algorithm is much easier. See
# sections 2.4, 2.5, and 2.7 (this section actually gives an
# algorithm for computing any Laurent series coefficient for
# a rational function). The theory in section 2.4 will help to
# understand why the resultant works in the general algorithm.
# For the definition of a resultant, see section 1.4 (and any
# previous sections for more review).
from diofant import collect, Mul, Order
expr = sympify(expr)
if x0 != 0:
expr = expr.subs(x, x + x0)
for n in [1, 2, 4, 8, 16, 32]:
s = expr.nseries(x, n=n)
if not s.has(Order) or s.getn() >= 0:
break
s = collect(s.removeO(), x)
if s.is_Add:
args = s.args
else:
args = [s]
res = Integer(0)
for arg in args:
c, m = arg.as_coeff_mul(x)
m = Mul(*m)
if not (m == 1 or m == x or (m.is_Pow and m.exp.is_Integer)):
raise NotImplementedError('term of unexpected form: %s' % m)
if m == 1 / x:
res += c
return res
def test_collect():
assert collect(A*B - B*A, A) == A*B - B*A
assert collect(A*B - B*A, B) == A*B - B*A
assert collect(A*B - B*A, x) == A*B - B*A
def test_collect():
assert collect(A * B - B * A, A) == A * B - B * A
assert collect(A * B - B * A, B) == A * B - B * A
assert collect(A * B - B * A, x) == A * B - B * A
def test_sympyissue_5615():
aA, Re, D = symbols('aA Re D')
e = ((D**3 * a + b * aA**3) / Re).expand()
assert collect(e, [aA**3 / Re, a]) == e
def test_collect_4():
"""Collect with respect to a power"""
assert collect(a*x**c + b*x**c, x**c) == x**c*(a + b)
# issue sympy/sympy#6096: 2 stays with c (unless c is integer or x is positive0
assert collect(a*x**(2*c) + b*x**(2*c), x**c) == x**(2*c)*(a + b)
def test_collect_issues():
D = Derivative
f = Function('f')
e = (1 + x*D(f(x), x) + D(f(x), x))/f(x)
assert collect(e.expand(), f(x).diff(x)) != e
def test_sympyissue_6097():
assert collect(a * y**(2.0 * x) + b * y**(2.0 * x),
y**x) == y**(2.0 * x) * (a + b)
assert collect(a * 2**(2.0 * x) + b * 2**(2.0 * x),
2**x) == 2**(2.0 * x) * (a + b)
def test_sympyissue_6097():
assert collect(a*y**(2.0*x) + b*y**(2.0*x), y**x) == y**(2.0*x)*(a + b)
assert collect(a*2**(2.0*x) + b*2**(2.0*x), 2**x) == 2**(2.0*x)*(a + b)
def test_collect_issues():
D = Derivative
f = Function('f')
e = (1 + x * D(f(x), x) + D(f(x), x)) / f(x)
assert collect(e.expand(), f(x).diff(x)) != e
def test_sympyissue_5615():
aA, Re, D = symbols('aA Re D')
e = ((D**3*a + b*aA**3)/Re).expand()
assert collect(e, [aA**3/Re, a]) == e
def test_collect_2():
"""Collect with respect to a sum"""
assert collect(a * (cos(x) + sin(x)) + b * (cos(x) + sin(x)),
sin(x) + cos(x)) == (a + b) * (cos(x) + sin(x))
def test_collect_2():
"""Collect with respect to a sum"""
assert collect(a*(cos(x) + sin(x)) + b*(cos(x) + sin(x)),
sin(x) + cos(x)) == (a + b)*(cos(x) + sin(x))