def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial."""

    add_eq(f.as_expr(), 0)
    n = f.degree()
    
    a, b = f.nth(n), f.nth(0)
    alpha = (-cancel(b/a))**Rational(1, n)
    beta = (-cancel(b/a))
    tmp = str(Poly(f.as_expr()).factor_list()).split(',')[2][1:]
    TMP = tmp
    if (n > 1):
        TMP += '**' + str(n)
    add_eq(TMP, str(beta))
    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    roots, I = [], S.ImaginaryUnit

    for k in xrange(n):
        zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    roots = sorted(roots, key=default_sort_key)

    num = 1
    for i in roots:
        add_eq(tmp + str(num), str(i))
        num += 1
    add_comment('')
    return roots

def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial."""
    n = f.degree()

    a, b = f.nth(n), f.nth(0)
    alpha = (-cancel(b/a))**Rational(1, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    roots, I = [], S.ImaginaryUnit

    for k in xrange(n):
        zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    if all([ r.is_number for r in roots ]):
        reals, complexes = [], []

        for root in roots:
            if root.is_real:
                reals.append(root)
            else:
                complexes.append(root)

        roots = sorted(reals) + sorted(complexes, key=lambda r: (re(r), -im(r)))

    return roots

def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial."""

    add_comment("Solve the equation")
    add_eq(f.as_expr(), 0)
    n = f.degree()

    a, b = f.nth(n), f.nth(0)

    add_comment("Rewrite this equation as")
    add_eq(f.gen**n, -b/a)


    alpha = (-cancel(b/a))**Rational(1, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    roots, I = [], S.ImaginaryUnit

    add_comment("We have the following roots")
    for k in xrange(n):
        zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
        add_eq(f.gen, Mul(alpha, zeta, evaluate=False))
        roots.append((alpha*zeta).expand(power_base=False))

    roots = sorted(roots, key=default_sort_key)

    return roots

def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial. If the domain is ZZ
    then the roots will be sorted with negatives coming before positives.
    The ordering will be the same for any numerical coefficients as long as
    the assumptions tested are correct, otherwise the ordering will not be
    sorted (but will be canonical).
    """
    n = f.degree()

    a, b = f.nth(n), f.nth(0)
    base = -cancel(b/a)
    alpha = root(base, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    # define some parameters that will allow us to order the roots.
    # If the domain is ZZ this is guaranteed to return roots sorted
    # with reals before non-real roots and non-real sorted according
    # to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I
    neg = base.is_negative
    even = n % 2 == 0
    if neg:
        if even == True and (base + 1).is_positive:
            big = True
        else:
            big = False

    # get the indices in the right order so the computed
    # roots will be sorted when the domain is ZZ
    ks = []
    imax = n//2
    if even:
        ks.append(imax)
        imax -= 1
    if not neg:
        ks.append(0)
    for i in range(imax, 0, -1):
        if neg:
            ks.extend([i, -i])
        else:
            ks.extend([-i, i])
    if neg:
        ks.append(0)
        if big:
            for i in range(0, len(ks), 2):
                pair = ks[i: i + 2]
                pair = list(reversed(pair))

    # compute the roots
    roots, d = [], 2*I*pi/n
    for k in ks:
        zeta = exp(k*d).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    return roots

def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial."""
    n = f.degree()

    a, b = f.nth(n), f.nth(0)
    alpha = (-cancel(b/a))**Rational(1, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    roots, I = [], S.ImaginaryUnit

    for k in xrange(n):
        zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    return sorted(roots, key=default_sort_key)

    def is_coplanar(self, o):
        """ Returns True if `o` is coplanar with self, else False.

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> o = (0, 0, 0)
        >>> p = Plane(o, (1, 1, 1))
        >>> p2 = Plane(o, (2, 2, 2))
        >>> p == p2
        False
        >>> p.is_coplanar(p2)
        True
        """
        if isinstance(o, Plane):
            x, y, z = map(Dummy, 'xyz')
            return not cancel(self.equation(x, y, z)/o.equation(x, y, z)).has(x, y, z)
        if isinstance(o, Point3D):
            return o in self
        elif isinstance(o, LinearEntity3D):
            return all(i in self for i in self)
        elif isinstance(o, GeometryEntity):  # XXX should only be handling 2D objects now
            return all(i == 0 for i in self.normal_vector[:2])

def _quintic_simplify(expr):
    expr = powsimp(expr)
    expr = cancel(expr)
    return together(expr)

def integral_steps(integrand, symbol, **options):
    """Returns the steps needed to compute an integral.

    This function attempts to mirror what a student would do by hand as
    closely as possible.

    SymPy Gamma uses this to provide a step-by-step explanation of an
    integral. The code it uses to format the results of this function can be
    found at
    https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.

    Examples
    ========

    >>> from sympy import exp, sin, cos
    >>> from sympy.integrals.manualintegrate import integral_steps
    >>> from sympy.abc import x
    >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    URule(u_var=_u, u_func=exp(x), constant=1,
        substep=ArctanRule(context=1/(_u**2 + 1), symbol=_u),
        context=exp(x)/(exp(2*x) + 1), symbol=x)
    >>> print(repr(integral_steps(sin(x), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
    >>> print(repr(integral_steps((x**2 + 3)**2 , x))) \
    # doctest: +NORMALIZE_WHITESPACE
    RewriteRule(rewritten=x**4 + 6*x**2 + 9,
    substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
        ConstantTimesRule(constant=6, other=x**2,
            substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
                context=6*x**2, symbol=x),
        ConstantRule(constant=9, context=9, symbol=x)],
    context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)


    Returns
    =======
    rule : namedtuple
        The first step; most rules have substeps that must also be
        considered. These substeps can be evaluated using ``manualintegrate``
        to obtain a result.

    """
    integrand_ = integrand
    if integrand.is_Mul:
        num, den = integrand.as_numer_denom()
        if num.is_Pow and den.is_Pow and num.args[1] == den.args[1]:
            p = den.args[1]
            e = cancel(num.args[0] / den.args[0])
            n, d = e.as_numer_denom()
            if n < 0:
                integrand = pow(-n, p) / pow(-d, p)
            else:
                integrand = pow(n, p) / pow(d, p)
            add_comment("Evaluate the integral")
            add_exp(sympy.Integral(integrand_, symbol))
            add_comment("Rewrite the integrand")
            add_eq(integrand_, integrand)
    cachekey = (integrand, symbol)
    if cachekey in _integral_cache:
        if _integral_cache[cachekey] is None:
            # cyclic integral! null_safe will eliminate that path
            return None
        else:
            return _integral_cache[cachekey]
    else:
        _integral_cache[cachekey] = None

    integral = IntegralInfo(integrand, symbol)

    def key(integral):
        integrand = integral.integrand

        if isinstance(integrand, TrigonometricFunction):
            return TrigonometricFunction
        elif isinstance(integrand, sympy.Derivative):
            return sympy.Derivative
        elif symbol not in integrand.free_symbols:
            return sympy.Number
        else:
            for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.log,
                        sympy.Add, sympy.Mul, sympy.atan):
                if isinstance(integrand, cls):
                    return cls

    def integral_is_subclass(*klasses):
        def _integral_is_subclass(integral):
            k = key(integral)
            return k and issubclass(k, klasses)
        return _integral_is_subclass

    result = do_one(
        null_safe(switch(key, {
            sympy.Pow: do_one(null_safe(power_rule), null_safe(arctan_rule), null_safe(cos_m2_rule),null_safe(sin_m2_rule), null_safe(sqrt_rule)),
            sympy.Symbol: power_rule,
            sympy.exp: exp_rule,
            sympy.Add: add_rule,
            sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule)),
            sympy.Derivative: derivative_rule,
#.........这里部分代码省略.........