def test_sympyissue_3449():
# test if powers are simplified correctly
# see also issue sympy/sympy#3995
assert cbrt(x)**2 == x**Rational(2, 3)
assert (x**3)**Rational(2, 5) == Pow(x**3, Rational(2, 5), evaluate=False)
a = Symbol('a', extended_real=True)
b = Symbol('b', extended_real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert cbrt(a**3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == cbrt(Number(25))
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == (a**(2*x/3)) != (a**x)**Rational(2, 3)
def test_add2():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m, n, p, q = tensor_indices('m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
A = tensorhead('A', [Lorentz] * 3, [[3]])
t1 = 2 * R(m, n, p, q) - R(m, q, n, p) + R(m, p, n, q)
t2 = t1 * A(-n, -p, -q)
assert t2 == 0
t1 = Rational(2, 3) * R(m, n, p, q) - Rational(1, 3) * R(
m, q, n, p) + Rational(1, 3) * R(m, p, n, q)
t2 = t1 * A(-n, -p, -q)
assert t2 == 0
t = A(m, -m, n) + A(n, p, -p)
assert t == 0
def _print_Mul(self, expr):
prec = precedence(expr)
c, e = expr.as_coeff_Mul()
if c < 0:
expr = _keep_coeff(-c, e)
sign = "-"
else:
sign = ""
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order != 'none':
args = expr.as_ordered_factors()
else:
# use make_args in case expr was something like -x -> x
args = Mul.make_args(expr)
multiple_ones = len([x for x in args if x == S.One]) > 1
# Gather args for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1 or multiple_ones:
a.append(Rational(item.p))
if item.q != 1:
b.append(Rational(item.q))
else:
a.append(item)
a = a or [S.One]
a_str = [self.parenthesize(x, prec) for x in a]
b_str = [self.parenthesize(x, prec) for x in b]
if len(b) == 0:
return sign + '*'.join(a_str)
elif len(b) == 1:
return sign + '*'.join(a_str) + "/" + b_str[0]
else:
return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str)
def test_Pow():
assert mcode(x**3) == "x^3"
assert mcode(x**(y**3)) == "x^(y^3)"
assert mcode(1/(f(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"(3.5*f[x])^(-x + y^x)/(x^2 + y)"
assert mcode(x**-1.0) == 'x^(-1.0)'
assert mcode(x**Rational(2, 3)) == 'x^(2/3)'
def test_pow_as_base_exp():
assert (oo**(2 - x)).as_base_exp() == (oo, 2 - x)
assert (oo**(x - 2)).as_base_exp() == (oo, x - 2)
p = Rational(1, 2)**x
assert p.base, p.exp == p.as_base_exp() == (Integer(2), -x)
# issue sympy/sympy#8344:
assert Pow(1, 2, evaluate=False).as_base_exp() == (Integer(1), Integer(2))
def test_sympyissue_6990():
x = Symbol('x')
a = Symbol('a')
b = Symbol('b')
assert (sqrt(a + b*x + x**2)).series(x, 0, 3).removeO() == \
b*x/(2*sqrt(a)) + x**2*(1/(2*sqrt(a)) -
b**2/(8*a**Rational(3, 2))) + sqrt(a)
def test_simplex():
pytest.raises(ValueError, lambda: simplex([1], [[1, 2], [2, 3]], [4, 7]))
pytest.raises(ValueError, lambda: simplex([1, 3], [[1, 2], [2, 3]], [4]))
pytest.raises(InfeasibleProblem,
lambda: simplex([1, 3], [[1, 2], [2, 3]], [-3, 1]))
assert simplex([2, 3, 2], [[2, 1, 1], [1, 2, 1], [0, 0, 1]],
[4, 7, 5]) == (11, (0, 3, 1))
assert simplex([2, 3, 4], [[3, 2, 1], [2, 5, 3]],
[10, 15]) == (20, (0, 0, 5))
assert simplex([Rational(1, 2), 3, 1, 1],
[[1, 1, 1, 1], [-2, -1, 1, 1], [0, 1, 0, -1]],
[40, 10, 10]) == (90, (0, 25, 0, 15))
assert simplex([2, 1], [[-1, 1], [1, -2]], [1, 2]) == (oo, (oo, oo))
assert simplex([1, 2, 3, -2],
[[3, -2, 1, 1], [-2, 1, 10, -1], [2, 0, 0, 1]],
[1, -2, 3]) == (-4, (0, 1, 0, 3))
pytest.raises(
InfeasibleProblem, lambda: simplex(
[1, 2, 3, -2], [[3, -2, 1, 1], [-2, 1, 10, -1], [2, 0, 0, 1],
[0, -1, 2, 0]], [1, -2, 3, -3]))
def test_Pow():
assert mcode(x**3) == "x.^3"
assert mcode(x**(y**3)) == "x.^(y.^3)"
assert mcode(x**Rational(2, 3)) == 'x.^(2/3)'
g = implemented_function('g', Lambda(x, 2 * x))
assert mcode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"(3.5*2*x).^(-x + y.^x)./(x.^2 + y)"
def test_diofant_parser():
x = Symbol('x')
inputs = {
'2*x':
2 * x,
'3.00':
Float(3),
'22/7':
Rational(22, 7),
'2+3j':
2 + 3 * I,
'exp(x)':
exp(x),
'-(2)':
-Integer(2),
'[-1, -2, 3]': [Integer(-1), Integer(-2),
Integer(3)],
'Symbol("x").free_symbols':
x.free_symbols,
'Float(Integer(3).evalf(3))':
3.00,
'factorint(12, visual=True)':
Mul(Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir="-")':
Limit(sin(x), x, 0, dir='-'),
}
for text, result in inputs.items():
assert parse_expr(text) == result
def test_rationalize():
inputs = {
'0.123': Rational(123, 1000)
}
transformations = standard_transformations + (rationalize,)
for text, result in inputs.items():
assert parse_expr(text, transformations=transformations) == result
def test_doit_with_MatrixBase():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).doit() == X
assert MatPow(X, 2).doit() == X**2
assert MatPow(X, -1).doit() == X.inv()
assert MatPow(X, -2).doit() == (X.inv())**2
# less expensive than testing on a 2x2
assert MatPow(ImmutableMatrix([4]), Rational(1, 2)).doit() == ImmutableMatrix([2])
def test_minimize_linear():
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z <= 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) == (-11, {
x: 0,
y: 3,
z: 1
})
assert minimize([
-2 * x - 3 * y - 2 * z, 2 * x + y + z <= 4, x + 2 * y + z < 7, z <= 5,
x >= 0, y >= 0, z >= 0
], x, y, z) is None
assert minimize([
-2 * x - 3 * y - 4 * z, 3 * x + 2 * y + z <= 10,
2 * x + 5 * y + 3 * z <= 15, x >= 0, y >= 0, z >= 0
], x, y, z) == (-20, {
x: 0,
y: 0,
z: 5
})
assert maximize([
12 * x + 40 * y, x + y <= 15, x + 3 * y <= 36, x <= 10, x >= 0, y >= 0
], x, y) == (480, {
x: 0,
y: 12
})
assert minimize([
-2 * x - 3 * y - 4 * z,
Eq(3 * x + 2 * y + z, 10),
Eq(2 * x + 5 * y + 3 * z, 15), x >= 0, y >= 0, z >= 0
], x, y, z) == (Rational(-130, 7), {
x: Rational(15, 7),
y: 0,
z: Rational(25, 7)
})
def _eval_Eq(self, other):
# RootOf represents a Root, so if other is that root, it should set
# the expression to zero *and* it should be in the interval of the
# RootOf instance. It must also be a number that agrees with the
# is_real value of the RootOf instance.
if type(self) == type(other):
return sympify(self.__eq__(other))
if not (other.is_number and not other.has(AppliedUndef)):
return S.false
if not other.is_finite:
return S.false
z = self.expr.subs(self.expr.free_symbols.pop(), other).is_zero
if z is False: # all roots will make z True but we don't know
# whether this is the right root if z is True
return S.false
o = other.is_extended_real, other.is_imaginary
s = self.is_extended_real, self.is_imaginary
if o != s and None not in o and None not in s:
return S.false
i = self._get_interval()
was = i.a, i.b
need = [True] * 2
# make sure it would be distinct from others
while any(need):
i = i.refine()
a, b = i.a, i.b
if need[0] and a != was[0]:
need[0] = False
if need[1] and b != was[1]:
need[1] = False
re, im = other.as_real_imag()
if not im:
if self.is_extended_real:
a, b = [Rational(str(i)) for i in (a, b)]
return sympify(a < other and other < b)
return S.false
if self.is_extended_real:
return S.false
z = r1, r2, i1, i2 = [
Rational(str(j)) for j in (i.ax, i.bx, i.ay, i.by)
]
return sympify((r1 < re and re < r2) and (i1 < im and im < i2))
def _sin_pow_integrate(n, x):
if n > 0:
if n == 1:
# Recursion break
return -cos(x)
# n > 0
# / /
# | |
# | n -1 n-1 n - 1 | n-2
# | sin (x) dx = ______ cos (x) sin (x) + _______ | sin (x) dx
# | |
# | n n |
# / /
#
return (Rational(-1, n) * cos(x) * sin(x)**(n - 1) +
Rational(n - 1, n) * _sin_pow_integrate(n - 2, x))
if n < 0:
if n == -1:
# Make sure this does not come back here again.
# Recursion breaks here or at n==0.
return trigintegrate(1 / sin(x), x)
# n < 0
# / /
# | |
# | n 1 n+1 n + 2 | n+2
# | sin (x) dx = _______ cos (x) sin (x) + _______ | sin (x) dx
# | |
# | n + 1 n + 1 |
# / /
#
return (Rational(1, n + 1) * cos(x) * sin(x)**(n + 1) +
Rational(n + 2, n + 1) * _sin_pow_integrate(n + 2, x))
else:
# n == 0
# Recursion break.
return x
def test_1_over_x_and_sqrt():
# 1.0 and 0.5 would do something different in regular StrPrinter,
# but these are exact in IEEE floating point so no different here.
assert mcode(1 / x) == '1./x'
assert mcode(x**-1) == mcode(x**-1.0) == '1./x'
assert mcode(1 / sqrt(x)) == '1./sqrt(x)'
assert mcode(x**Rational(-1, 2)) == mcode(x**-0.5) == '1./sqrt(x)'
assert mcode(sqrt(x)) == mcode(x**0.5) == 'sqrt(x)'
assert mcode(1 / pi) == '1/pi'
assert mcode(pi**-1) == mcode(pi**-1.0) == '1/pi'
assert mcode(pi**-0.5) == '1/sqrt(pi)'
def test_ccode_Pow():
assert ccode(x**3) == "pow(x, 3)"
assert ccode(x**(y**3)) == "pow(x, pow(y, 3))"
g = implemented_function('g', Lambda(x, 2 * x))
assert ccode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"pow(3.5*2*x, -x + pow(y, x))/(pow(x, 2) + y)"
assert ccode(x**-1.0) == '1.0/x'
assert ccode(x**Rational(2, 3)) == 'pow(x, 2.0L/3.0L)'
_cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"),
(lambda base, exp: not exp.is_integer, "pow")]
assert ccode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)'
assert ccode(x**3.2, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 3.2)'
def test_MatrixSymbol():
n = Symbol('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert mcode(A * B) == "A*B"
assert mcode(B * A) == "B*A"
assert mcode(2 * A * B) == "2*A*B"
assert mcode(B * 2 * A) == "2*B*A"
assert mcode(A * (B + 3 * Identity(n))) == "A*(B + 3*eye(n))"
assert mcode(A**(x**2)) == "A^(x.^2)"
assert mcode(A**3) == "A^3"
assert mcode(A**Rational(1, 2)) == "A^(1/2)"
def test_neg_rational():
r = Rational(-3, 4)
assert r.is_positive is False
assert r.is_nonpositive is True
assert r.is_negative is True
assert r.is_nonnegative is False
r = Rational(-1, 4)
assert r.is_nonpositive is True
assert r.is_positive is False
assert r.is_negative is True
assert r.is_nonnegative is False
r = Rational(-5, 4)
assert r.is_negative is True
assert r.is_positive is False
assert r.is_nonpositive is True
assert r.is_nonnegative is False
r = Rational(-5, 3)
assert r.is_nonnegative is False
assert r.is_positive is False
assert r.is_negative is True
assert r.is_nonpositive is True
def test_riemann_cyclic():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i, j, k, l, m, n, p, q = tensor_indices('i,j,k,l,m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
t = R(i, j, k, l) + R(i, l, j, k) + R(i, k, l, j) - \
R(i, j, l, k) - R(i, l, k, j) - R(i, k, j, l)
t2 = t * R(-i, -j, -k, -l)
t3 = riemann_cyclic(t2)
assert t3 == 0
t = R(i, j, k, l) * (R(-i, -j, -k, -l) - 2 * R(-i, -k, -j, -l))
t1 = riemann_cyclic(t)
assert t1 == 0
t = R(i, j, k, l)
t1 = riemann_cyclic(t)
assert t1 == -Rational(1, 3) * R(i, l, j, k) + Rational(1, 3) * R(
i, k, j, l) + Rational(2, 3) * R(i, j, k, l)
t = R(i, j, k, l) * R(-k, -l, m,
n) * (R(-m, -n, -i, -j) + 2 * R(-m, -j, -n, -i))
t1 = riemann_cyclic(t)
assert t1 == 0
def test_ccode_Rational():
assert ccode(Rational(3, 7)) == "3.0L/7.0L"
assert ccode(Rational(18, 9)) == "2"
assert ccode(Rational(3, -7)) == "-3.0L/7.0L"
assert ccode(Rational(-3, -7)) == "3.0L/7.0L"
assert ccode(x + Rational(3, 7)) == "x + 3.0L/7.0L"
assert ccode(Rational(3, 7) * x) == "(3.0L/7.0L)*x"
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == cbrt((1 + I)**(12*I*f))
assert cbrt((1 + I)**(I*(1 + 7*f))).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, Rational(5, 2))) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**(2/Integer(3))
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == Rational(1, 2)
assert sqrt(r**(4/Integer(3))) != r**(2/Integer(3))
assert sqrt((p + I)**(4/Integer(3))) == (p + I)**(2/Integer(3))
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**Rational(5, 2)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def _cos_pow_integrate(n, x):
if n > 0:
if n == 1:
# Recursion break.
return sin(x)
# n > 0
# / /
# | |
# | n 1 n-1 n - 1 | n-2
# | sin (x) dx = ______ sin (x) cos (x) + _______ | cos (x) dx
# | |
# | n n |
# / /
#
return (Rational(1, n) * sin(x) * cos(x)**(n - 1) +
Rational(n - 1, n) * _cos_pow_integrate(n - 2, x))
if n < 0:
if n == -1:
# Recursion break
return trigintegrate(1 / cos(x), x)
# n < 0
# / /
# | |
# | n -1 n+1 n + 2 | n+2
# | cos (x) dx = _______ sin (x) cos (x) + _______ | cos (x) dx
# | |
# | n + 1 n + 1 |
# / /
#
return (Rational(-1, n + 1) * sin(x) * cos(x)**(n + 1) +
Rational(n + 2, n + 1) * _cos_pow_integrate(n + 2, x))
else:
# n == 0
# Recursion Break.
return x
def test_Rational():
assert mcode(Rational(3, 7)) == "3/7"
assert mcode(Rational(18, 9)) == "2"
assert mcode(Rational(3, -7)) == "-3/7"
assert mcode(Rational(-3, -7)) == "3/7"
assert mcode(x + Rational(3, 7)) == "x + 3/7"
assert mcode(Rational(3, 7)*x) == "(3/7)*x"
def test_pos_rational():
r = Rational(3, 4)
assert r.is_commutative is True
assert r.is_integer is False
assert r.is_rational is True
assert r.is_algebraic is True
assert r.is_transcendental is False
assert r.is_extended_real is True
assert r.is_complex is True
assert r.is_noninteger is True
assert r.is_irrational is False
assert r.is_imaginary is False
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonpositive is False
assert r.is_nonnegative is True
assert r.is_even is False
assert r.is_odd is False
assert r.is_finite is True
assert r.is_infinite is False
assert r.is_comparable is True
assert r.is_prime is False
assert r.is_composite is False
r = Rational(1, 4)
assert r.is_nonpositive is False
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonnegative is True
r = Rational(5, 4)
assert r.is_negative is False
assert r.is_positive is True
assert r.is_nonpositive is False
assert r.is_nonnegative is True
r = Rational(5, 3)
assert r.is_nonnegative is True
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonpositive is False
def test_MatPow():
A = MatrixSymbol('A', n, n)
AA = MatPow(A, 2)
assert AA.exp == 2
assert AA.base == A
assert (A**n).exp == n
assert A**0 == Identity(n)
assert A**1 == A
assert A**2 == AA
assert A**-1 == Inverse(A)
assert A**Rational(1, 2) == sqrt(A)
pytest.raises(ShapeError, lambda: MatrixSymbol('B', 3, 2)**2)
def test_int_to_Integer():
# XXX: Warning, don't test with == here. 0.5 == Rational(1, 2) is True!
app = init_ipython_session(auto_int_to_Integer=True)
app.run_cell("from diofant import Integer")
app.run_cell("a = 1")
assert isinstance(app.user_ns['a'], Integer)
app.run_cell("a = 1/2")
assert isinstance(app.user_ns['a'], Rational)
app.run_cell("a = 1")
assert isinstance(app.user_ns['a'], Integer)
app.run_cell("a = int(1)")
assert isinstance(app.user_ns['a'], int)
app.run_cell("a = (1/\n2)")
assert app.user_ns['a'] == Rational(1, 2)
def random_point(self, seed=None):
""" Returns a random point on the Plane.
Returns
=======
Point3D
"""
import random
if seed is not None:
rng = random.Random(seed)
else:
rng = random
t = Dummy('t')
return self.arbitrary_point(t).subs(t, Rational(rng.random()))
def test_rational():
a = Rational(1, 5)
r = sqrt(5)/5
assert sqrt(a) == r
assert 2*sqrt(a) == 2*r
r = a*a**Rational(1, 2)
assert a**Rational(3, 2) == r
assert 2*a**Rational(3, 2) == 2*r
r = a**5*a**Rational(2, 3)
assert a**Rational(17, 3) == r
assert 2 * a**Rational(17, 3) == 2*r
def eval(cls, n, sym=None):
if n.is_Number:
if n.is_Integer and n.is_nonnegative:
if n is S.Zero:
return S.One
elif n is S.One:
if sym is None:
return -S.Half
else:
return sym - S.Half
# Bernoulli numbers
elif sym is None:
if n.is_odd:
return S.Zero
n = int(n)
# Use mpmath for enormous Bernoulli numbers
if n > 500:
p, q = bernfrac(n)
return Rational(int(p), int(q))
case = n % 6
highest_cached = cls._highest[case]
if n <= highest_cached:
return cls._cache[n]
# To avoid excessive recursion when, say, bernoulli(1000) is
# requested, calculate and cache the entire sequence ... B_988,
# B_994, B_1000 in increasing order
for i in range(highest_cached + 6, n + 6, 6):
b = cls._calc_bernoulli(i)
cls._cache[i] = b
cls._highest[case] = i
return b
# Bernoulli polynomials
else:
n, result = int(n), []
for k in range(n + 1):
result.append(binomial(n, k) * cls(k) * sym**(n - k))
return Add(*result)
else:
raise ValueError("Bernoulli numbers are defined only"
" for nonnegative integer indices.")
if sym is None:
if n.is_odd and (n - 1).is_positive:
return S.Zero
def test_trigintegrate_odd():
assert trigintegrate(Rational(1), x) == x
assert trigintegrate(x, x) is None
assert trigintegrate(x**2, x) is None
assert trigintegrate(sin(x), x) == -cos(x)
assert trigintegrate(cos(x), x) == sin(x)
assert trigintegrate(sin(3 * x), x) == -cos(3 * x) / 3
assert trigintegrate(cos(3 * x), x) == sin(3 * x) / 3
y = Symbol('y')
assert trigintegrate(sin(y*x), x) == \
Piecewise((0, Eq(y, 0)), (-cos(y*x)/y, True))
assert trigintegrate(cos(y*x), x) == \
Piecewise((x, Eq(y, 0)), (sin(y*x)/y, True))
assert trigintegrate(sin(y*x)**2, x) == \
Piecewise((0, Eq(y, 0)), ((x*y/2 - sin(x*y)*cos(x*y)/2)/y, True))
assert trigintegrate(sin(y*x)*cos(y*x), x) == \
Piecewise((0, Eq(y, 0)), (sin(x*y)**2/(2*y), True))
assert trigintegrate(cos(y*x)**2, x) == \
Piecewise((x, Eq(y, 0)), ((x*y/2 + sin(x*y)*cos(x*y)/2)/y, True))
y = Symbol('y', positive=True)
# TODO: remove conds='none' below. For this to work we would have to rule
# out (e.g. by trying solve) the condition y = 0, incompatible with
# y.is_positive being True.
assert trigintegrate(sin(y * x), x, conds='none') == -cos(y * x) / y
assert trigintegrate(cos(y * x), x, conds='none') == sin(y * x) / y
assert trigintegrate(sin(x) * cos(x), x) == sin(x)**2 / 2
assert trigintegrate(sin(x) * cos(x)**2, x) == -cos(x)**3 / 3
assert trigintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
# check if it selects right function to substitute,
# so the result is kept simple
assert trigintegrate(sin(x)**7 * cos(x), x) == sin(x)**8 / 8
assert trigintegrate(sin(x) * cos(x)**7, x) == -cos(x)**8 / 8
assert trigintegrate(sin(x)**7 * cos(x)**3, x) == \
-sin(x)**10/10 + sin(x)**8/8
assert trigintegrate(sin(x)**3 * cos(x)**7, x) == \
cos(x)**10/10 - cos(x)**8/8