def test_trigintegrate_mixed():
assert trigintegrate(sin(x) * sec(x), x) == -log(sin(x)**2 - 1) / 2
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(cos(x)**2 - 1) / 2
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)
def test_sech():
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) == nan
assert sech(zoo) == nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi * I) == sec(pi)
assert sech(-pi * I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi * I / 2) == zoo
assert sech(-pi * I / 2) == zoo
assert sech((-3 * 10**73 + 1) * pi * I / 2) == zoo
assert sech((7 * 10**103 + 1) * pi * I / 2) == zoo
assert sech(pi * I) == -1
assert sech(-pi * I) == -1
assert sech(5 * pi * I) == -1
assert sech(8 * pi * I) == 1
assert sech(pi * I / 3) == 2
assert sech(-2 * pi * I / 3) == -2
assert sech(pi * I / 4) == sqrt(2)
assert sech(-pi * I / 4) == sqrt(2)
assert sech(5 * pi * I / 4) == -sqrt(2)
assert sech(-5 * pi * I / 4) == -sqrt(2)
assert sech(pi * I / 6) == 2 / sqrt(3)
assert sech(-pi * I / 6) == 2 / sqrt(3)
assert sech(7 * pi * I / 6) == -2 / sqrt(3)
assert sech(-5 * pi * I / 6) == -2 / sqrt(3)
assert sech(pi * I / 105) == 1 / cos(pi / 105)
assert sech(-pi * I / 105) == 1 / cos(pi / 105)
assert sech(x * I) == 1 / cos(x)
assert sech(k * pi * I) == 1 / cos(k * pi)
assert sech(17 * k * pi * I) == 1 / cos(17 * k * pi)
assert sech(n).is_extended_real is True
assert csch(n).is_finite is None
pytest.raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_sech():
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) == nan
assert sech(zoo) == nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi*I) == sec(pi)
assert sech(-pi*I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi*I/2) == zoo
assert sech(-pi*I/2) == zoo
assert sech((-3*10**73 + 1)*pi*I/2) == zoo
assert sech((7*10**103 + 1)*pi*I/2) == zoo
assert sech(pi*I) == -1
assert sech(-pi*I) == -1
assert sech(5*pi*I) == -1
assert sech(8*pi*I) == 1
assert sech(pi*I/3) == 2
assert sech(-2*pi*I/3) == -2
assert sech(pi*I/4) == sqrt(2)
assert sech(-pi*I/4) == sqrt(2)
assert sech(5*pi*I/4) == -sqrt(2)
assert sech(-5*pi*I/4) == -sqrt(2)
assert sech(pi*I/6) == 2/sqrt(3)
assert sech(-pi*I/6) == 2/sqrt(3)
assert sech(7*pi*I/6) == -2/sqrt(3)
assert sech(-5*pi*I/6) == -2/sqrt(3)
assert sech(pi*I/105) == 1/cos(pi/105)
assert sech(-pi*I/105) == 1/cos(pi/105)
assert sech(x*I) == 1/cos(x)
assert sech(k*pi*I) == 1/cos(k*pi)
assert sech(17*k*pi*I) == 1/cos(17*k*pi)
assert sech(n).is_extended_real is True
assert csch(n).is_finite is None
pytest.raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_inverses():
pytest.raises(AttributeError, lambda: sin(x).inverse())
pytest.raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
pytest.raises(AttributeError, lambda: csc(x).inverse())
pytest.raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_cos_rewrite():
assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert (cos(sinh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: sinh(3)}).evalf())
assert (cos(cosh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cosh(3)}).evalf())
assert (cos(tanh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: tanh(3)}).evalf())
assert (cos(coth(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: coth(3)}).evalf())
assert (cos(sin(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: sin(3)}).evalf())
assert (cos(cos(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cos(3)}).evalf())
assert (cos(tan(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: tan(3)}).evalf())
assert (cos(cot(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cot(3)}).evalf())
assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
assert cos(x).rewrite(Pow) == cos(x)
assert cos(x).rewrite(sec) == 1/sec(x)
def test_sec():
x = symbols('x', extended_real=True)
z = symbols('z')
assert sec.nargs == FiniteSet(1)
assert sec(0) == 1
assert sec(pi) == -1
assert sec(pi/2) == zoo
assert sec(-pi/2) == zoo
assert sec(pi/6) == 2*sqrt(3)/3
assert sec(pi/3) == 2
assert sec(5*pi/2) == zoo
assert sec(9*pi/7) == -sec(2*pi/7)
assert sec(3*pi/4) == -sqrt(2) # issue sympy/sympy#8421
assert sec(I) == sech(1)
assert sec(x*I) == sech(x)
assert sec(-x) == sec(x)
assert sec(asec(x)) == x
assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2)
assert sec(x).rewrite(sin) == sec(x)
assert sec(x).rewrite(cos) == 1/cos(x)
assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1)
assert sec(x).rewrite(sqrt) == sec(x)
assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1)
assert sec(z).conjugate() == sec(conjugate(z))
assert (sec(z).as_real_imag() ==
(cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2),
sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2)))
assert sec(x).expand(trig=True) == 1/cos(x)
assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1)
assert sec(a).is_algebraic is None
assert sec(na).is_algebraic is False
assert sec(x).as_leading_term() == sec(x)
assert sec(0).is_finite
assert sec(x).is_finite is None
assert sec(pi/2).is_finite is False
assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6)
# https://github.com/sympy/sympy/issues/7166
assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6)
# https://github.com/sympy/sympy/issues/7167
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**Rational(3, 2)/12 +
(x - 3*pi/2)**Rational(7, 2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2)))
assert sec(x).diff(x) == tan(x)*sec(x)
# Taylor Term checks
assert sec(z).taylor_term(4, z) == 5*z**4/24
assert sec(z).taylor_term(6, z) == 61*z**6/720
assert sec(z).taylor_term(5, z) == 0
pytest.raises(ArgumentIndexError, lambda: sec(x).fdiff(2))