from sympy.polys.domains import QQ, EX, RR
from sympy.polys.rings import ring
from sympy.polys.ring_series import (_invert_monoms, rs_integrate,
    rs_trunc, rs_mul, rs_square, rs_pow, _has_constant_term, rs_hadamard_exp,
    rs_series_from_list, rs_exp, rs_log, rs_newton, rs_series_inversion,
    rs_compose_add, rs_asin, rs_atan, rs_atanh, rs_tan, rs_cot, rs_sin, rs_cos,
    rs_cos_sin, rs_sinh, rs_cosh, rs_tanh, _tan1, rs_fun, rs_nth_root,
    rs_LambertW, rs_series_reversion, rs_is_puiseux, rs_series)
from sympy.utilities.pytest import raises
from sympy.core.compatibility import range
from sympy.core.symbol import symbols
from sympy.functions import (sin, cos, exp, tan, cot, atan, asin, atanh,
    tanh, log, sqrt)
from sympy.core.numbers import Rational
from sympy.core import expand, S

def is_close(a, b):
    tol = 10**(-10)
    assert abs(a - b) < tol

def test_ring_series1():
    R, x = ring('x', QQ)
    p = x**4 + 2*x**3 + 3*x + 4
    assert _invert_monoms(p) == 4*x**4 + 3*x**3 + 2*x + 1
    assert rs_hadamard_exp(p) == x**4/24 + x**3/3 + 3*x + 4
    R, x = ring('x', QQ)
    p = x**4 + 2*x**3 + 3*x + 4
    assert rs_integrate(p, x) == x**5/5 + x**4/2 + 3*x**2/2 + 4*x
    R, x, y = ring('x, y', QQ)
    p = x**2*y**2 + x + 1
    assert rs_integrate(p, x) == x**3*y**2/3 + x**2/2 + x
    assert rs_integrate(p, y) == x**2*y**3/3 + x*y + y

def test_trunc():
    R, x, y, t = ring('x, y, t', QQ)
    p = (y + t*x)**4
    p1 = rs_trunc(p, x, 3)
    assert p1 == y**4 + 4*y**3*t*x + 6*y**2*t**2*x**2

def test_mul_trunc():
    R, x, y, t = ring('x, y, t', QQ)
    p = 1 + t*x + t*y
    for i in range(2):
        p = rs_mul(p, p, t, 3)

    assert p == 6*x**2*t**2 + 12*x*y*t**2 + 6*y**2*t**2 + 4*x*t + 4*y*t + 1
    p = 1 + t*x + t*y + t**2*x*y
    p1 = rs_mul(p, p, t, 2)
    assert p1 == 1 + 2*t*x + 2*t*y
    R1, z = ring('z', QQ)
    def test1(p):
        p2 = rs_mul(p, z, x, 2)
    raises(ValueError, lambda: test1(p))

    p1 = 2 + 2*x + 3*x**2
    p2 = 3 + x**2
    assert rs_mul(p1, p2, x, 4) == 2*x**3 + 11*x**2 + 6*x + 6

def test_square_trunc():
    R, x, y, t = ring('x, y, t', QQ)
    p = (1 + t*x + t*y)*2
    p1 = rs_mul(p, p, x, 3)
    p2 = rs_square(p, x, 3)
    assert p1 == p2
    p = 1 + x + x**2 + x**3
    assert rs_square(p, x, 4) == 4*x**3 + 3*x**2 + 2*x + 1

def test_pow_trunc():
    R, x, y, z = ring('x, y, z', QQ)
    p0 = y + x*z
    p = p0**16
    for xx in (x, y, z):
        p1 = rs_trunc(p, xx, 8)
        p2 = rs_pow(p0, 16, xx, 8)
        assert p1 == p2

    p = 1 + x
    p1 = rs_pow(p, 3, x, 2)
    assert p1 == 1 + 3*x
    assert rs_pow(p, 0, x, 2) == 1
    assert rs_pow(p, -2, x, 2) == 1 - 2*x
    p = x + y
    assert rs_pow(p, 3, y, 3) == x**3 + 3*x**2*y + 3*x*y**2
    assert rs_pow(1 + x, Rational(2, 3), x, 4) == 4*x**3/81 - x**2/9 + 2*x/3 + 1

def test_has_constant_term():
    R, x, y, z = ring('x, y, z', QQ)
    p = y + x*z
    assert _has_constant_term(p, x)
    p = x + x**4
    assert not _has_constant_term(p, x)
    p = 1 + x + x**4
    assert _has_constant_term(p, x)
    p = x + y + x*z

def test_inversion():
    R, x = ring('x', QQ)
    p = 2 + x + 2*x**2
    n = 5
    p1 = rs_series_inversion(p, x, n)
    assert rs_trunc(p*p1, x, n) == 1
    R, x, y = ring('x, y', QQ)
    p = 2 + x + 2*x**2 + y*x + x**2*y
    p1 = rs_series_inversion(p, x, n)
    assert rs_trunc(p*p1, x, n) == 1

    R, x, y = ring('x, y', QQ)
    p = 1 + x + y
    def test2(p):
        p1 = rs_series_inversion(p, x, 4)
    raises(NotImplementedError, lambda: test2(p))
    p = R.zero
    def test3(p):
        p1 = rs_series_inversion(p, x, 3)
    raises(ZeroDivisionError, lambda: test3(p))


def test_series_reversion():
    R, x, y = ring('x, y', QQ)

    p = rs_tan(x, x, 10)
    assert rs_series_reversion(p, x, 8, y) == rs_atan(y, y, 8)

    p = rs_sin(x, x, 10)
    assert rs_series_reversion(p, x, 8, y) == 5*y**7/112 + 3*y**5/40 + \
        y**3/6 + y

def test_series_from_list():
    R, x = ring('x', QQ)
    p = 1 + 2*x + x**2 + 3*x**3
    c = [1, 2, 0, 4, 4]
    r = rs_series_from_list(p, c, x, 5)
    pc = R.from_list(list(reversed(c)))
    r1 = rs_trunc(pc.compose(x, p), x, 5)
    assert r == r1
    R, x, y = ring('x, y', QQ)
    c = [1, 3, 5, 7]
    p1 = rs_series_from_list(x + y, c, x, 3, concur=0)
    p2 = rs_trunc((1 + 3*(x+y) + 5*(x+y)**2 + 7*(x+y)**3), x, 3)
    assert p1 == p2

    R, x = ring('x', QQ)
    h = 25
    p = rs_exp(x, x, h) - 1
    p1 = rs_series_from_list(p, c, x, h)
    p2 = 0
    for i, cx in enumerate(c):
        p2 += cx*rs_pow(p, i, x, h)
    assert p1 == p2

def test_log():
    R, x = ring('x', QQ)
    p = 1 + x
    p1 = rs_log(p, x, 4)/x**2
    assert p1 == S(1)/3*x - S(1)/2 + x**(-1)
    p = 1 + x +2*x**2/3
    p1 = rs_log(p, x, 9)
    assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
      7*x**4/36 - x**3/3 + x**2/6 + x
    p2 = rs_series_inversion(p, x, 9)
    p3 = rs_log(p2, x, 9)
    assert p3 == -p1

    R, x, y = ring('x, y', QQ)
    p = 1 + x + 2*y*x**2
    p1 = rs_log(p, x, 6)
    assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
                  x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
        - EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
    assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
        EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
        EX(1/a)*x + EX(log(a))

    p = x + x**2 + 3
    assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + S(19281291595)/9920232)

def test_exp():
    R, x = ring('x', QQ)
    p = x + x**4
    for h in [10, 30]:
        q = rs_series_inversion(1 + p, x, h) - 1
        p1 = rs_exp(q, x, h)
        q1 = rs_log(p1, x, h)
        assert q1 == q
    p1 = rs_exp(p, x, 30)
    assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
    prec = 21
    p = rs_log(1 + x, x, prec)
    p1 = rs_exp(p, x, prec)
    assert p1 == x + 1

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[exp(a), a])
    assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
        exp(a)*x**2/2 + exp(a)*x + exp(a)
    assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
            exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
            exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)

    R, x, y = ring('x, y', EX)
    assert rs_exp(x + a, x, 5) ==  EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
        EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
    assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
        EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
        EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
        EX(exp(a))*x + EX(exp(a))

def test_newton():
    R, x = ring('x', QQ)
    p = x**2 - 2
    r = rs_newton(p, x, 4)
    f = [1, 0, -2]
    assert r == 8*x**4 + 4*x**2 + 2

def test_compose_add():
    R, x = ring('x', QQ)
    p1 = x**3 - 1
    p2 = x**2 - 2
    assert rs_compose_add(p1, p2) == x**6 - 6*x**4 - 2*x**3 + 12*x**2 - 12*x - 7

def test_fun():
    R, x, y = ring('x, y', QQ)
    p = x*y + x**2*y**3 + x**5*y
    assert rs_fun(p, rs_tan, x, 10) == rs_tan(p, x, 10)
    assert rs_fun(p, _tan1, x, 10) == _tan1(p, x, 10)

def test_nth_root():
    R, x, y = ring('x, y', QQ)
    r1 = rs_nth_root(1 + x**2*y, 4, x, 10)
    assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \
        7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1
    assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \
        5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \
        x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1
    assert rs_nth_root(8*x, 3, x, 3) == 2*x**QQ(1, 3)
    assert rs_nth_root(8*x + x**2 + x**3, 3, x, 3) == x**QQ(4,3)/12 + 2*x**QQ(1,3)
    r = rs_nth_root(8*x + x**2*y + x**3, 3, x, 4)
    assert r == -x**QQ(7,3)*y**2/288 + x**QQ(7,3)/12 + x**QQ(4,3)*y/12 + 2*x**QQ(1,3)

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_nth_root(x + a, 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \
        EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3))
    assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\
        x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \
        EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \
        EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5))

def test_atan():
    R, x, y = ring('x, y', QQ)
    assert rs_atan(x, x, 9) == -x**7/7 + x**5/5 - x**3/3 + x
    assert rs_atan(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 - x**8*y**9 + \
        2*x**7*y**9 - x**7*y**7/7 - x**6*y**9/3 + x**6*y**7 - x**5*y**7 + \
        x**5*y**5/5 - x**4*y**5 - x**3*y**3/3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_atan(x + a, x, 5) == -EX((a**3 - a)/(a**8 + 4*a**6 + 6*a**4 + \
        4*a**2 + 1))*x**4 + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + \
        9*a**2 + 3))*x**3 - EX(a/(a**4 + 2*a**2 + 1))*x**2 + \
        EX(1/(a**2 + 1))*x + EX(atan(a))
    assert rs_atan(x + x**2*y + a, x, 4) == -EX(2*a/(a**4 + 2*a**2 + 1)) \
        *x**3*y + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + 9*a**2 + 3))*x**3 + \
        EX(1/(a**2 + 1))*x**2*y - EX(a/(a**4 + 2*a**2 + 1))*x**2 + EX(1/(a**2 \
        + 1))*x + EX(atan(a))

def test_asin():
    R, x, y = ring('x, y', QQ)
    assert rs_asin(x + x*y, x, 5) == x**3*y**3/6 + x**3*y**2/2 + x**3*y/2 + \
        x**3/6 + x*y + x
    assert rs_asin(x*y + x**2*y**3, x, 6) == x**5*y**7/2 + 3*x**5*y**5/40 + \
        x**4*y**5/2 + x**3*y**3/6 + x**2*y**3 + x*y

def test_tan():
    R, x, y = ring('x, y', QQ)
    assert rs_tan(x, x, 9)/x**5 == \
        S(17)/315*x**2 + S(2)/15 + S(1)/3*x**(-2) + x**(-4)
    assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
        4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
        x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[tan(a), a])
    assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**S(3)/3 +
        2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + S(1)/3)*x**3 + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
    assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
        (tan(a)**4 + S(4)/3*tan(a)**2 + S(1)/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)

    R, x, y = ring('x, y', EX)
    assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
        2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
    assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
        2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
        EX(tan(a)**2 + 1)*x + EX(tan(a))

    p = x + x**2 + 5
    assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
        668083460499)

def test_cot():
    R, x, y = ring('x, y', QQ)
    assert rs_cot(x**6 + x**7, x, 8) == x**(-6) - x**(-5) + x**(-4) - \
        x**(-3) + x**(-2) - x**(-1) + 1 - x + x**2 - x**3 + x**4 - x**5 + \
        2*x**6/3 - 4*x**7/3
    assert rs_cot(x + x**2*y, x, 5) == -x**4*y**5 - x**4*y/15 + x**3*y**4 - \
        x**3/45 - x**2*y**3 - x**2*y/3 + x*y**2 - x/3 - y + x**(-1)

def test_sin():
    R, x, y = ring('x, y', QQ)
    assert rs_sin(x, x, 9)/x**5 == \
        -S(1)/5040*x**2 + S(1)/120 - S(1)/6*x**(-2) + x**(-4)
    assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
        x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
        x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
        x**3*y**3/6 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
    assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
        sin(a)*x**2/2 + cos(a)*x + sin(a)
    assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
        cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
        cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)

    R, x, y = ring('x, y', EX)
    assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
        EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
    assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
        EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
        EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
        EX(cos(a))*x + EX(sin(a))

def test_cos():
    R, x, y = ring('x, y', QQ)
    assert rs_cos(x, x, 9)/x**5 == \
        S(1)/40320*x**3 - S(1)/720*x + S(1)/24*x**(-1) - S(1)/2*x**(-3) + x**(-5)
    assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
        x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
        x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
        x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
    assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
        cos(a)*x**2/2 - sin(a)*x + cos(a)
    assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
        sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
        sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)

    R, x, y = ring('x, y', EX)
    assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
        EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
    assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
        EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
        EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
        EX(sin(a))*x + EX(cos(a))

def test_cos_sin():
    R, x, y = ring('x, y', QQ)
    cos, sin = rs_cos_sin(x, x, 9)
    assert cos == rs_cos(x, x, 9)
    assert sin == rs_sin(x, x, 9)
    cos, sin = rs_cos_sin(x + x*y, x, 5)
    assert cos == rs_cos(x + x*y, x, 5)
    assert sin == rs_sin(x + x*y, x, 5)

def test_atanh():
    R, x, y = ring('x, y', QQ)
    assert rs_atanh(x, x, 9)/x**5 == S(1)/7*x**2 + S(1)/5 + S(1)/3*x**(-2) + x**(-4)
    assert rs_atanh(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 + x**8*y**9 + \
        2*x**7*y**9 + x**7*y**7/7 + x**6*y**9/3 + x**6*y**7 + x**5*y**7 + \
        x**5*y**5/5 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_atanh(x + a, x, 5) == EX((a**3 + a)/(a**8 - 4*a**6 + 6*a**4 - \
        4*a**2 + 1))*x**4 - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + \
        9*a**2 - 3))*x**3 + EX(a/(a**4 - 2*a**2 + 1))*x**2 - EX(1/(a**2 - \
        1))*x + EX(atanh(a))
    assert rs_atanh(x + x**2*y + a, x, 4) == EX(2*a/(a**4 - 2*a**2 + \
        1))*x**3*y - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + 9*a**2 - 3))*x**3 - \
        EX(1/(a**2 - 1))*x**2*y + EX(a/(a**4 - 2*a**2 + 1))*x**2 - \
        EX(1/(a**2 - 1))*x + EX(atanh(a))

    p = x + x**2 + 5
    assert rs_atanh(p, x, 10).compose(x, 10) == EX(-S(733442653682135)/5079158784 \
        + atanh(5))

def test_sinh():
    R, x, y = ring('x, y', QQ)
    assert rs_sinh(x, x, 9)/x**5 == S(1)/5040*x**2 + S(1)/120 + S(1)/6*x**(-2) + x**(-4)
    assert rs_sinh(x*y + x**2*y**3, x, 9) == x**8*y**11/12 + \
        x**8*y**9/720 + x**7*y**9/12 + x**7*y**7/5040 + x**6*y**9/6 + \
        x**6*y**7/24 + x**5*y**7/2 + x**5*y**5/120 + x**4*y**5/2 + \
        x**3*y**3/6 + x**2*y**3 + x*y

def test_cosh():
    R, x, y = ring('x, y', QQ)
    assert rs_cosh(x, x, 9)/x**5 == S(1)/40320*x**3 + S(1)/720*x + S(1)/24*x**(-1) + \
        S(1)/2*x**(-3) + x**(-5)
    assert rs_cosh(x*y + x**2*y**3, x, 9) == x**8*y**12/24 + \
        x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 + \
        x**7*y**8/120 + x**6*y**8/4 + x**6*y**6/720 + x**5*y**6/6 + \
        x**4*y**6/2 + x**4*y**4/24 + x**3*y**4 + x**2*y**2/2 + 1

def test_tanh():
    R, x, y = ring('x, y', QQ)
    assert rs_tanh(x, x, 9)/x**5 == -S(17)/315*x**2 + S(2)/15 - S(1)/3*x**(-2) + x**(-4)
    assert rs_tanh(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 - \
        17*x**8*y**9/45 + 4*x**7*y**9/3 - 17*x**7*y**7/315 - x**6*y**9/3 + \
        2*x**6*y**7/3 - x**5*y**7 + 2*x**5*y**5/15 - x**4*y**5 - \
        x**3*y**3/3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_tanh(x + a, x, 5) == EX(tanh(a)**5 - 5*tanh(a)**3/3 +
        2*tanh(a)/3)*x**4 + EX(-tanh(a)**4 + 4*tanh(a)**2/3 - QQ(1, 3))*x**3 + \
        EX(tanh(a)**3 - tanh(a))*x**2 + EX(-tanh(a)**2 + 1)*x + EX(tanh(a))

    p = rs_tanh(x + x**2*y + a, x, 4)
    assert (p.compose(x, 10)).compose(y, 5) == EX(-1000*tanh(a)**4 + \
        10100*tanh(a)**3 + 2470*tanh(a)**2/3 - 10099*tanh(a) + QQ(530, 3))

def test_RR():
    rs_funcs = [rs_sin, rs_cos, rs_tan, rs_cot, rs_atan, rs_tanh]
    sympy_funcs = [sin, cos, tan, cot, atan, tanh]
    R, x, y = ring('x, y', RR)
    a = symbols('a')
    for rs_func, sympy_func in zip(rs_funcs, sympy_funcs):
        p = rs_func(2 + x, x, 5).compose(x, 5)
        q = sympy_func(2 + a).series(a, 0, 5).removeO()
        is_close(p.as_expr(), q.subs(a, 5).n())

    p = rs_nth_root(2 + x, 5, x, 5).compose(x, 5)
    q = ((2 + a)**QQ(1, 5)).series(a, 0, 5).removeO()
    is_close(p.as_expr(), q.subs(a, 5).n())

def test_is_regular():
    R, x, y = ring('x, y', QQ)
    p = 1 + 2*x + x**2 + 3*x**3
    assert not rs_is_puiseux(p, x)

    p = x + x**QQ(1,5)*y
    assert rs_is_puiseux(p, x)
    assert not rs_is_puiseux(p, y)

    p = x + x**2*y**QQ(1,5)*y
    assert not rs_is_puiseux(p, x)

def test_puiseux():
    R, x, y = ring('x, y', QQ)
    p = x**QQ(2,5) + x**QQ(2,3) + x

    r = rs_series_inversion(p, x, 1)
    r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \
        2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \
        + x**QQ(-2,5)
    assert r == r1

    r = rs_nth_root(1 + p, 3, x, 1)
    assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1

    r = rs_log(1 + p, x, 1)
    assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5)

    r = rs_LambertW(p, x, 1)
    assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5)

    p1 = x + x**QQ(1,5)*y
    r = rs_exp(p1, x, 1)
    assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \
        x**QQ(1,5)*y + 1

    r = rs_atan(p, x, 2)
    assert r ==  -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
        x + x**QQ(2,3) + x**QQ(2,5)

    r = rs_atan(p1, x, 2)
    assert r ==  x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \
        x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y

    r = rs_asin(p, x, 2)
    assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)

    r = rs_cot(p, x, 1)
    assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \
        2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \
        x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5)

    r = rs_cos_sin(p, x, 2)
    assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \
        x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1
    assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \
        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)

    r = rs_atanh(p, x, 2)
    assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \
        x**QQ(2,3) + x**QQ(2,5)

    r = rs_sinh(p, x, 2)
    assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)

    r = rs_cosh(p, x, 2)
    assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \
        x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1

    r = rs_tanh(p, x, 2)
    assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
        x + x**QQ(2,3) + x**QQ(2,5)

def test1():
    R, x = ring('x', QQ)
    r = rs_sin(x, x, 15)*x**(-5)
    assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \
        QQ(1,120) - x**-2/6 + x**-4

    p = rs_sin(x, x, 10)
    r = rs_nth_root(p, 2, x, 10)
    assert  r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \
        x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2)

    p = rs_sin(x, x, 10)
    r = rs_nth_root(p, 7, x, 10)
    r = rs_pow(r, 5, x, 10)
    assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \
        11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7)

    r = rs_exp(x**QQ(1,2), x, 10)
    assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \
        x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \
        x**QQ(15,2)/1307674368000 + x**7/87178291200 + \
        x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \
        x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \
        x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \
        x**QQ(1,2) + 1

def test_puiseux2():
    R, y = ring('y', QQ)
    S, x = ring('x', R)

    p = x + x**QQ(1,5)*y
    r = rs_atan(p, x, 3)
    assert r == (y**13/13 + y**8 + 2*y**3)*x**QQ(13,5) - (y**11/11 + y**6 +
        y)*x**QQ(11,5) + (y**9/9 + y**4)*x**QQ(9,5) - (y**7/7 +
        y**2)*x**QQ(7,5) + (y**5/5 + 1)*x - y**3*x**QQ(3,5)/3 + y*x**QQ(1,5)

def test_rs_series():
    x, a, b, c = symbols('x, a, b, c')

    assert rs_series(a, a, 5).as_expr() == a
    assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
        5)).removeO()
    assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
        cos(a)).series(a, 0, 5)).removeO()
    assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
        cos(a)).series(a, 0, 5)).removeO()

    p = (sin(a) - a)*(cos(a**2) + a**4/2)
    assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
        10).removeO())

    p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
        5).removeO())

    p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
        5).removeO())

    p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
    assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
        10).removeO())

    p = sin(a + b + c)
    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
        5).removeO())

    p = tan(sin(a**2 + 4) + b + c)
    assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
        6).removeO())

    p = a**QQ(2,5) + a**QQ(2,3) + a

    r = rs_series(tan(p), a, 2)
    assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
        a + a**QQ(2,3) + a**QQ(2,5)

    r = rs_series(exp(p), a, 1)
    assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1

    r = rs_series(sin(p), a, 2)
    assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
        a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)

    r = rs_series(cos(p), a, 2)
    assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
        a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1

    assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0,
            5).removeO()

    assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \
                    x**2/2 + x
    assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \
                    8*x**2 + 4*x
    assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \
                    x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + \
                    x**2/2 + x
    assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \
                    x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - \
                    x**2*a**4/2 + x*a**2