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from sympy.core import Ne, Rational, Symbol
from sympy.functions import sin, cos, tan, csc, sec, cot, log, Piecewise
from sympy.integrals.trigonometry import trigintegrate
x = Symbol('x')
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(
(-cos(y*x)/y, Ne(y, 0)), (0, True))
assert trigintegrate(cos(y*x), x) == Piecewise(
(sin(y*x)/y, Ne(y, 0)), (x, True))
assert trigintegrate(sin(y*x)**2, x) == Piecewise(
((x*y/2 - sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (0, True))
assert trigintegrate(sin(y*x)*cos(y*x), x) == Piecewise(
(sin(x*y)**2/(2*y), Ne(y, 0)), (0, True))
assert trigintegrate(cos(y*x)**2, x) == Piecewise(
((x*y/2 + sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (x, 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
# both n, m are odd and -ve, and not necessarily equal
assert trigintegrate(sin(x)**-1*cos(x)**-1, x) == \
-log(sin(x)**2 - 1)/2 + log(sin(x))
def test_trigintegrate_even():
assert trigintegrate(sin(x)**2, x) == x/2 - cos(x)*sin(x)/2
assert trigintegrate(cos(x)**2, x) == x/2 + cos(x)*sin(x)/2
assert trigintegrate(sin(3*x)**2, x) == x/2 - cos(3*x)*sin(3*x)/6
assert trigintegrate(cos(3*x)**2, x) == x/2 + cos(3*x)*sin(3*x)/6
assert trigintegrate(sin(x)**2 * cos(x)**2, x) == \
x/8 - sin(2*x)*cos(2*x)/16
assert trigintegrate(sin(x)**4 * cos(x)**2, x) == \
x/16 - sin(x) *cos(x)/16 - sin(x)**3*cos(x)/24 + \
sin(x)**5*cos(x)/6
assert trigintegrate(sin(x)**2 * cos(x)**4, x) == \
x/16 + cos(x) *sin(x)/16 + cos(x)**3*sin(x)/24 - \
cos(x)**5*sin(x)/6
assert trigintegrate(sin(x)**(-4), x) == -2*cos(x)/(3*sin(x)) \
- cos(x)/(3*sin(x)**3)
assert trigintegrate(cos(x)**(-6), x) == sin(x)/(5*cos(x)**5) \
+ 4*sin(x)/(15*cos(x)**3) + 8*sin(x)/(15*cos(x))
def test_trigintegrate_mixed():
assert trigintegrate(sin(x)*sec(x), x) == -log(cos(x))
assert trigintegrate(sin(x)*csc(x), x) == x
assert trigintegrate(sin(x)*cot(x), x) == sin(x)
assert trigintegrate(cos(x)*sec(x), x) == x
assert trigintegrate(cos(x)*csc(x), x) == log(sin(x))
assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
- log(cos(x) + 1)/2 + cos(x)
assert trigintegrate(cot(x)*cos(x)**2, x) == log(sin(x)) - sin(x)**2/2
def test_trigintegrate_symbolic():
n = Symbol('n', integer=True)
assert trigintegrate(cos(x)**n, x) is None
assert trigintegrate(sin(x)**n, x) is None
assert trigintegrate(cot(x)**n, x) is None
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