from sympy.core.logic import FuzzyBool

from sympy.core import S, sympify, cacheit, pi, I, Rational
from sympy.core.add import Add
from sympy.core.function import Function, ArgumentIndexError, _coeff_isneg
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.elementary.exponential import exp, log, match_real_imag
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.integers import floor

from sympy.core.logic import fuzzy_or, fuzzy_and

def _rewrite_hyperbolics_as_exp(expr):
    expr = sympify(expr)
    return expr.xreplace({h: h.rewrite(exp)
        for h in expr.atoms(HyperbolicFunction)})


###############################################################################
########################### HYPERBOLIC FUNCTIONS ##############################
###############################################################################


class HyperbolicFunction(Function):
    """
    Base class for hyperbolic functions.

    See Also
    ========

    sinh, cosh, tanh, coth
    """

    unbranched = True


def _peeloff_ipi(arg):
    """
    Split ARG into two parts, a "rest" and a multiple of I*pi/2.
    This assumes ARG to be an Add.
    The multiple of I*pi returned in the second position is always a Rational.

    Examples
    ========

    >>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel
    >>> from sympy import pi, I
    >>> from sympy.abc import x, y
    >>> peel(x + I*pi/2)
    (x, I*pi/2)
    >>> peel(x + I*2*pi/3 + I*pi*y)
    (x + I*pi*y + I*pi/6, I*pi/2)
    """
    for a in Add.make_args(arg):
        if a == S.Pi*S.ImaginaryUnit:
            K = S.One
            break
        elif a.is_Mul:
            K, p = a.as_two_terms()
            if p == S.Pi*S.ImaginaryUnit and K.is_Rational:
                break
    else:
        return arg, S.Zero

    m1 = (K % S.Half)*S.Pi*S.ImaginaryUnit
    m2 = K*S.Pi*S.ImaginaryUnit - m1
    return arg - m2, m2


class sinh(HyperbolicFunction):
    r"""
    sinh(x) is the hyperbolic sine of x.

    The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$.

    Examples
    ========

    >>> from sympy import sinh
    >>> from sympy.abc import x
    >>> sinh(x)
    sinh(x)

    See Also
    ========

    cosh, tanh, asinh
    """

    def fdiff(self, argindex=1):
        """
        Returns the first derivative of this function.
        """
        if argindex == 1:
            return cosh(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return asinh

    @classmethod
    def eval(cls, arg):
        from sympy import sin

        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity
            elif arg is S.NegativeInfinity:
                return S.NegativeInfinity
            elif arg.is_zero:
                return S.Zero
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.NaN

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return S.ImaginaryUnit * sin(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

            if arg.is_Add:
                x, m = _peeloff_ipi(arg)
                if m:
                    return sinh(m)*cosh(x) + cosh(m)*sinh(x)

            if arg.is_zero:
                return S.Zero

            if arg.func == asinh:
                return arg.args[0]

            if arg.func == acosh:
                x = arg.args[0]
                return sqrt(x - 1) * sqrt(x + 1)

            if arg.func == atanh:
                x = arg.args[0]
                return x/sqrt(1 - x**2)

            if arg.func == acoth:
                x = arg.args[0]
                return 1/(sqrt(x - 1) * sqrt(x + 1))

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        """
        Returns the next term in the Taylor series expansion.
        """
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            if len(previous_terms) > 2:
                p = previous_terms[-2]
                return p * x**2 / (n*(n - 1))
            else:
                return x**(n) / factorial(n)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        """
        Returns this function as a complex coordinate.
        """
        from sympy import cos, sin
        if self.args[0].is_extended_real:
            if deep:
                hints['complex'] = False
                return (self.expand(deep, **hints), S.Zero)
            else:
                return (self, S.Zero)
        if deep:
            re, im = self.args[0].expand(deep, **hints).as_real_imag()
        else:
            re, im = self.args[0].as_real_imag()
        return (sinh(re)*cos(im), cosh(re)*sin(im))

    def _eval_expand_complex(self, deep=True, **hints):
        re_part, im_part = self.as_real_imag(deep=deep, **hints)
        return re_part + im_part*S.ImaginaryUnit

    def _eval_expand_trig(self, deep=True, **hints):
        if deep:
            arg = self.args[0].expand(deep, **hints)
        else:
            arg = self.args[0]
        x = None
        if arg.is_Add: # TODO, implement more if deep stuff here
            x, y = arg.as_two_terms()
        else:
            coeff, terms = arg.as_coeff_Mul(rational=True)
            if coeff is not S.One and coeff.is_Integer and terms is not S.One:
                x = terms
                y = (coeff - 1)*x
        if x is not None:
            return (sinh(x)*cosh(y) + sinh(y)*cosh(x)).expand(trig=True)
        return sinh(arg)

    def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
        return (exp(arg) - exp(-arg)) / 2

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        return (exp(arg) - exp(-arg)) / 2

    def _eval_rewrite_as_cosh(self, arg, **kwargs):
        return -S.ImaginaryUnit*cosh(arg + S.Pi*S.ImaginaryUnit/2)

    def _eval_rewrite_as_tanh(self, arg, **kwargs):
        tanh_half = tanh(S.Half*arg)
        return 2*tanh_half/(1 - tanh_half**2)

    def _eval_rewrite_as_coth(self, arg, **kwargs):
        coth_half = coth(S.Half*arg)
        return 2*coth_half/(coth_half**2 - 1)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.NaN:
            arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+')
        if arg0.is_zero:
            return arg
        elif arg0.is_finite:
            return self.func(arg0)
        else:
            return self

    def _eval_is_real(self):
        arg = self.args[0]
        if arg.is_real:
            return True

        # if `im` is of the form n*pi
        # else, check if it is a number
        re, im = arg.as_real_imag()
        return (im%pi).is_zero

    def _eval_is_extended_real(self):
        if self.args[0].is_extended_real:
            return True

    def _eval_is_positive(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_positive

    def _eval_is_negative(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_negative

    def _eval_is_finite(self):
        arg = self.args[0]
        return arg.is_finite

    def _eval_is_zero(self):
        arg = self.args[0]
        if arg.is_zero:
            return True


class cosh(HyperbolicFunction):
    r"""
    cosh(x) is the hyperbolic cosine of x.

    The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$.

    Examples
    ========

    >>> from sympy import cosh
    >>> from sympy.abc import x
    >>> cosh(x)
    cosh(x)

    See Also
    ========

    sinh, tanh, acosh
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return sinh(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy import cos
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity
            elif arg is S.NegativeInfinity:
                return S.Infinity
            elif arg.is_zero:
                return S.One
            elif arg.is_negative:
                return cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.NaN

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return cos(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return cls(-arg)

            if arg.is_Add:
                x, m = _peeloff_ipi(arg)
                if m:
                    return cosh(m)*cosh(x) + sinh(m)*sinh(x)

            if arg.is_zero:
                return S.One

            if arg.func == asinh:
                return sqrt(1 + arg.args[0]**2)

            if arg.func == acosh:
                return arg.args[0]

            if arg.func == atanh:
                return 1/sqrt(1 - arg.args[0]**2)

            if arg.func == acoth:
                x = arg.args[0]
                return x/(sqrt(x - 1) * sqrt(x + 1))

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)

            if len(previous_terms) > 2:
                p = previous_terms[-2]
                return p * x**2 / (n*(n - 1))
            else:
                return x**(n)/factorial(n)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        from sympy import cos, sin
        if self.args[0].is_extended_real:
            if deep:
                hints['complex'] = False
                return (self.expand(deep, **hints), S.Zero)
            else:
                return (self, S.Zero)
        if deep:
            re, im = self.args[0].expand(deep, **hints).as_real_imag()
        else:
            re, im = self.args[0].as_real_imag()

        return (cosh(re)*cos(im), sinh(re)*sin(im))

    def _eval_expand_complex(self, deep=True, **hints):
        re_part, im_part = self.as_real_imag(deep=deep, **hints)
        return re_part + im_part*S.ImaginaryUnit

    def _eval_expand_trig(self, deep=True, **hints):
        if deep:
            arg = self.args[0].expand(deep, **hints)
        else:
            arg = self.args[0]
        x = None
        if arg.is_Add: # TODO, implement more if deep stuff here
            x, y = arg.as_two_terms()
        else:
            coeff, terms = arg.as_coeff_Mul(rational=True)
            if coeff is not S.One and coeff.is_Integer and terms is not S.One:
                x = terms
                y = (coeff - 1)*x
        if x is not None:
            return (cosh(x)*cosh(y) + sinh(x)*sinh(y)).expand(trig=True)
        return cosh(arg)

    def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
        return (exp(arg) + exp(-arg)) / 2

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        return (exp(arg) + exp(-arg)) / 2

    def _eval_rewrite_as_sinh(self, arg, **kwargs):
        return -S.ImaginaryUnit*sinh(arg + S.Pi*S.ImaginaryUnit/2)

    def _eval_rewrite_as_tanh(self, arg, **kwargs):
        tanh_half = tanh(S.Half*arg)**2
        return (1 + tanh_half)/(1 - tanh_half)

    def _eval_rewrite_as_coth(self, arg, **kwargs):
        coth_half = coth(S.Half*arg)**2
        return (coth_half + 1)/(coth_half - 1)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.NaN:
            arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+')
        if arg0.is_zero:
            return S.One
        elif arg0.is_finite:
            return self.func(arg0)
        else:
            return self

    def _eval_is_real(self):
        arg = self.args[0]

        # `cosh(x)` is real for real OR purely imaginary `x`
        if arg.is_real or arg.is_imaginary:
            return True

        # cosh(a+ib) = cos(b)*cosh(a) + i*sin(b)*sinh(a)
        # the imaginary part can be an expression like n*pi
        # if not, check if the imaginary part is a number
        re, im = arg.as_real_imag()
        return (im%pi).is_zero

    def _eval_is_positive(self):
        # cosh(x+I*y) = cos(y)*cosh(x) + I*sin(y)*sinh(x)
        # cosh(z) is positive iff it is real and the real part is positive.
        # So we need sin(y)*sinh(x) = 0 which gives x=0 or y=n*pi
        # Case 1 (y=n*pi): cosh(z) = (-1)**n * cosh(x) -> positive for n even
        # Case 2 (x=0): cosh(z) = cos(y) -> positive when cos(y) is positive
        z = self.args[0]

        x, y = z.as_real_imag()
        ymod = y % (2*pi)

        yzero = ymod.is_zero
        # shortcut if ymod is zero
        if yzero:
            return True

        xzero = x.is_zero
        # shortcut x is not zero
        if xzero is False:
            return yzero

        return fuzzy_or([
                # Case 1:
                yzero,
                # Case 2:
                fuzzy_and([
                    xzero,
                    fuzzy_or([ymod < pi/2, ymod > 3*pi/2])
                ])
            ])


    def _eval_is_nonnegative(self):
        z = self.args[0]

        x, y = z.as_real_imag()
        ymod = y % (2*pi)

        yzero = ymod.is_zero
        # shortcut if ymod is zero
        if yzero:
            return True

        xzero = x.is_zero
        # shortcut x is not zero
        if xzero is False:
            return yzero

        return fuzzy_or([
                # Case 1:
                yzero,
                # Case 2:
                fuzzy_and([
                    xzero,
                    fuzzy_or([ymod <= pi/2, ymod >= 3*pi/2])
                ])
            ])

    def _eval_is_finite(self):
        arg = self.args[0]
        return arg.is_finite


class tanh(HyperbolicFunction):
    r"""
    tanh(x) is the hyperbolic tangent of x.

    The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$.

    Examples
    ========

    >>> from sympy import tanh
    >>> from sympy.abc import x
    >>> tanh(x)
    tanh(x)

    See Also
    ========

    sinh, cosh, atanh
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return S.One - tanh(self.args[0])**2
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return atanh

    @classmethod
    def eval(cls, arg):
        from sympy import tan
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.One
            elif arg is S.NegativeInfinity:
                return S.NegativeOne
            elif arg.is_zero:
                return S.Zero
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.NaN

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                if _coeff_isneg(i_coeff):
                    return -S.ImaginaryUnit * tan(-i_coeff)
                return S.ImaginaryUnit * tan(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

            if arg.is_Add:
                x, m = _peeloff_ipi(arg)
                if m:
                    tanhm = tanh(m)
                    if tanhm is S.ComplexInfinity:
                        return coth(x)
                    else: # tanhm == 0
                        return tanh(x)

            if arg.is_zero:
                return S.Zero

            if arg.func == asinh:
                x = arg.args[0]
                return x/sqrt(1 + x**2)

            if arg.func == acosh:
                x = arg.args[0]
                return sqrt(x - 1) * sqrt(x + 1) / x

            if arg.func == atanh:
                return arg.args[0]

            if arg.func == acoth:
                return 1/arg.args[0]

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        from sympy import bernoulli
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            a = 2**(n + 1)

            B = bernoulli(n + 1)
            F = factorial(n + 1)

            return a*(a - 1) * B/F * x**n

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        from sympy import cos, sin
        if self.args[0].is_extended_real:
            if deep:
                hints['complex'] = False
                return (self.expand(deep, **hints), S.Zero)
            else:
                return (self, S.Zero)
        if deep:
            re, im = self.args[0].expand(deep, **hints).as_real_imag()
        else:
            re, im = self.args[0].as_real_imag()
        denom = sinh(re)**2 + cos(im)**2
        return (sinh(re)*cosh(re)/denom, sin(im)*cos(im)/denom)

    def _eval_expand_trig(self, **hints):
        arg = self.args[0]
        if arg.is_Add:
            from sympy import symmetric_poly
            n = len(arg.args)
            TX = [tanh(x, evaluate=False)._eval_expand_trig()
                for x in arg.args]
            p = [0, 0]  # [den, num]
            for i in range(n + 1):
                p[i % 2] += symmetric_poly(i, TX)
            return p[1]/p[0]
        elif arg.is_Mul:
            from sympy.functions.combinatorial.numbers import nC
            coeff, terms = arg.as_coeff_Mul()
            if coeff.is_Integer and coeff > 1:
                n = []
                d = []
                T = tanh(terms)
                for k in range(1, coeff + 1, 2):
                    n.append(nC(range(coeff), k)*T**k)
                for k in range(0, coeff + 1, 2):
                    d.append(nC(range(coeff), k)*T**k)
                return Add(*n)/Add(*d)
        return tanh(arg)

    def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
        neg_exp, pos_exp = exp(-arg), exp(arg)
        return (pos_exp - neg_exp)/(pos_exp + neg_exp)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        neg_exp, pos_exp = exp(-arg), exp(arg)
        return (pos_exp - neg_exp)/(pos_exp + neg_exp)

    def _eval_rewrite_as_sinh(self, arg, **kwargs):
        return S.ImaginaryUnit*sinh(arg)/sinh(S.Pi*S.ImaginaryUnit/2 - arg)

    def _eval_rewrite_as_cosh(self, arg, **kwargs):
        return S.ImaginaryUnit*cosh(S.Pi*S.ImaginaryUnit/2 - arg)/cosh(arg)

    def _eval_rewrite_as_coth(self, arg, **kwargs):
        return 1/coth(arg)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return arg
        else:
            return self.func(arg)

    def _eval_is_real(self):
        arg = self.args[0]
        if arg.is_real:
            return True

        re, im = arg.as_real_imag()

        # if denom = 0, tanh(arg) = zoo
        if re == 0 and im % pi == pi/2:
            return None

        # check if im is of the form n*pi/2 to make sin(2*im) = 0
        # if not, im could be a number, return False in that case
        return (im % (pi/2)).is_zero

    def _eval_is_extended_real(self):
        if self.args[0].is_extended_real:
            return True

    def _eval_is_positive(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_positive

    def _eval_is_negative(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_negative

    def _eval_is_finite(self):
        from sympy import sinh, cos
        arg = self.args[0]

        re, im = arg.as_real_imag()
        denom = cos(im)**2 + sinh(re)**2
        if denom == 0:
            return False
        elif denom.is_number:
            return True
        if arg.is_extended_real:
            return True

    def _eval_is_zero(self):
        arg = self.args[0]
        if arg.is_zero:
            return True


class coth(HyperbolicFunction):
    r"""
    coth(x) is the hyperbolic cotangent of x.

    The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$.

    Examples
    ========

    >>> from sympy import coth
    >>> from sympy.abc import x
    >>> coth(x)
    coth(x)

    See Also
    ========

    sinh, cosh, acoth
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -1/sinh(self.args[0])**2
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return acoth

    @classmethod
    def eval(cls, arg):
        from sympy import cot
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.One
            elif arg is S.NegativeInfinity:
                return S.NegativeOne
            elif arg.is_zero:
                return S.ComplexInfinity
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.NaN

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                if _coeff_isneg(i_coeff):
                    return S.ImaginaryUnit * cot(-i_coeff)
                return -S.ImaginaryUnit * cot(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

            if arg.is_Add:
                x, m = _peeloff_ipi(arg)
                if m:
                    cothm = coth(m)
                    if cothm is S.ComplexInfinity:
                        return coth(x)
                    else: # cothm == 0
                        return tanh(x)

            if arg.is_zero:
                return S.ComplexInfinity

            if arg.func == asinh:
                x = arg.args[0]
                return sqrt(1 + x**2)/x

            if arg.func == acosh:
                x = arg.args[0]
                return x/(sqrt(x - 1) * sqrt(x + 1))

            if arg.func == atanh:
                return 1/arg.args[0]

            if arg.func == acoth:
                return arg.args[0]

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        from sympy import bernoulli
        if n == 0:
            return 1 / sympify(x)
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            B = bernoulli(n + 1)
            F = factorial(n + 1)

            return 2**(n + 1) * B/F * x**n

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        from sympy import cos, sin
        if self.args[0].is_extended_real:
            if deep:
                hints['complex'] = False
                return (self.expand(deep, **hints), S.Zero)
            else:
                return (self, S.Zero)
        if deep:
            re, im = self.args[0].expand(deep, **hints).as_real_imag()
        else:
            re, im = self.args[0].as_real_imag()
        denom = sinh(re)**2 + sin(im)**2
        return (sinh(re)*cosh(re)/denom, -sin(im)*cos(im)/denom)

    def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
        neg_exp, pos_exp = exp(-arg), exp(arg)
        return (pos_exp + neg_exp)/(pos_exp - neg_exp)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        neg_exp, pos_exp = exp(-arg), exp(arg)
        return (pos_exp + neg_exp)/(pos_exp - neg_exp)

    def _eval_rewrite_as_sinh(self, arg, **kwargs):
        return -S.ImaginaryUnit*sinh(S.Pi*S.ImaginaryUnit/2 - arg)/sinh(arg)

    def _eval_rewrite_as_cosh(self, arg, **kwargs):
        return -S.ImaginaryUnit*cosh(arg)/cosh(S.Pi*S.ImaginaryUnit/2 - arg)

    def _eval_rewrite_as_tanh(self, arg, **kwargs):
        return 1/tanh(arg)

    def _eval_is_positive(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_positive

    def _eval_is_negative(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_negative

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return 1/arg
        else:
            return self.func(arg)

    def _eval_expand_trig(self, **hints):
        arg = self.args[0]
        if arg.is_Add:
            from sympy import symmetric_poly
            CX = [coth(x, evaluate=False)._eval_expand_trig() for x in arg.args]
            p = [[], []]
            n = len(arg.args)
            for i in range(n, -1, -1):
                p[(n - i) % 2].append(symmetric_poly(i, CX))
            return Add(*p[0])/Add(*p[1])
        elif arg.is_Mul:
            from sympy import binomial
            coeff, x = arg.as_coeff_Mul(rational=True)
            if coeff.is_Integer and coeff > 1:
                c = coth(x, evaluate=False)
                p = [[], []]
                for i in range(coeff, -1, -1):
                    p[(coeff - i) % 2].append(binomial(coeff, i)*c**i)
                return Add(*p[0])/Add(*p[1])
        return coth(arg)


class ReciprocalHyperbolicFunction(HyperbolicFunction):
    """Base class for reciprocal functions of hyperbolic functions. """

    #To be defined in class
    _reciprocal_of = None
    _is_even = None  # type: FuzzyBool
    _is_odd = None  # type: FuzzyBool

    @classmethod
    def eval(cls, arg):
        if arg.could_extract_minus_sign():
            if cls._is_even:
                return cls(-arg)
            if cls._is_odd:
                return -cls(-arg)

        t = cls._reciprocal_of.eval(arg)
        if hasattr(arg, 'inverse') and arg.inverse() == cls:
            return arg.args[0]
        return 1/t if t is not None else t

    def _call_reciprocal(self, method_name, *args, **kwargs):
        # Calls method_name on _reciprocal_of
        o = self._reciprocal_of(self.args[0])
        return getattr(o, method_name)(*args, **kwargs)

    def _calculate_reciprocal(self, method_name, *args, **kwargs):
        # If calling method_name on _reciprocal_of returns a value != None
        # then return the reciprocal of that value
        t = self._call_reciprocal(method_name, *args, **kwargs)
        return 1/t if t is not None else t

    def _rewrite_reciprocal(self, method_name, arg):
        # Special handling for rewrite functions. If reciprocal rewrite returns
        # unmodified expression, then return None
        t = self._call_reciprocal(method_name, arg)
        if t is not None and t != self._reciprocal_of(arg):
            return 1/t

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)

    def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_tractable", arg)

    def _eval_rewrite_as_tanh(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_tanh", arg)

    def _eval_rewrite_as_coth(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_coth", arg)

    def as_real_imag(self, deep = True, **hints):
        return (1 / self._reciprocal_of(self.args[0])).as_real_imag(deep, **hints)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def _eval_expand_complex(self, deep=True, **hints):
        re_part, im_part = self.as_real_imag(deep=True, **hints)
        return re_part + S.ImaginaryUnit*im_part

    def _eval_expand_trig(self, **hints):
        return self._calculate_reciprocal("_eval_expand_trig", **hints)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)

    def _eval_is_extended_real(self):
        return self._reciprocal_of(self.args[0]).is_extended_real

    def _eval_is_finite(self):
        return (1/self._reciprocal_of(self.args[0])).is_finite


class csch(ReciprocalHyperbolicFunction):
    r"""
    csch(x) is the hyperbolic cosecant of x.

    The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$

    Examples
    ========

    >>> from sympy import csch
    >>> from sympy.abc import x
    >>> csch(x)
    csch(x)

    See Also
    ========

    sinh, cosh, tanh, sech, asinh, acosh
    """

    _reciprocal_of = sinh
    _is_odd = True

    def fdiff(self, argindex=1):
        """
        Returns the first derivative of this function
        """
        if argindex == 1:
            return -coth(self.args[0]) * csch(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        """
        Returns the next term in the Taylor series expansion
        """
        from sympy import bernoulli
        if n == 0:
            return 1/sympify(x)
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            B = bernoulli(n + 1)
            F = factorial(n + 1)

            return 2 * (1 - 2**n) * B/F * x**n

    def _eval_rewrite_as_cosh(self, arg, **kwargs):
        return S.ImaginaryUnit / cosh(arg + S.ImaginaryUnit * S.Pi / 2)

    def _eval_is_positive(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_positive

    def _eval_is_negative(self):
        if self.args[0].is_extended_real:
            return self.args[0].is_negative


class sech(ReciprocalHyperbolicFunction):
    r"""
    sech(x) is the hyperbolic secant of x.

    The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$

    Examples
    ========

    >>> from sympy import sech
    >>> from sympy.abc import x
    >>> sech(x)
    sech(x)

    See Also
    ========

    sinh, cosh, tanh, coth, csch, asinh, acosh
    """

    _reciprocal_of = cosh
    _is_even = True

    def fdiff(self, argindex=1):
        if argindex == 1:
            return - tanh(self.args[0])*sech(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        from sympy.functions.combinatorial.numbers import euler
        if n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)
            return euler(n) / factorial(n) * x**(n)

    def _eval_rewrite_as_sinh(self, arg, **kwargs):
        return S.ImaginaryUnit / sinh(arg + S.ImaginaryUnit * S.Pi /2)

    def _eval_is_positive(self):
        if self.args[0].is_extended_real:
            return True


###############################################################################
############################# HYPERBOLIC INVERSES #############################
###############################################################################

class InverseHyperbolicFunction(Function):
    """Base class for inverse hyperbolic functions."""

    pass


class asinh(InverseHyperbolicFunction):
    """
    asinh(x) is the inverse hyperbolic sine of x.

    The inverse hyperbolic sine function.

    Examples
    ========

    >>> from sympy import asinh
    >>> from sympy.abc import x
    >>> asinh(x).diff(x)
    1/sqrt(x**2 + 1)
    >>> asinh(1)
    log(1 + sqrt(2))

    See Also
    ========

    acosh, atanh, sinh
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/sqrt(self.args[0]**2 + 1)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy import asin
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity
            elif arg is S.NegativeInfinity:
                return S.NegativeInfinity
            elif arg.is_zero:
                return S.Zero
            elif arg is S.One:
                return log(sqrt(2) + 1)
            elif arg is S.NegativeOne:
                return log(sqrt(2) - 1)
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.ComplexInfinity

            if arg.is_zero:
                return S.Zero

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return S.ImaginaryUnit * asin(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

        if isinstance(arg, sinh) and arg.args[0].is_number:
            z = arg.args[0]
            if z.is_real:
                return z
            r, i = match_real_imag(z)
            if r is not None and i is not None:
                f = floor((i + pi/2)/pi)
                m = z - I*pi*f
                even = f.is_even
                if even is True:
                    return m
                elif even is False:
                    return -m

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) >= 2 and n > 2:
                p = previous_terms[-2]
                return -p * (n - 2)**2/(n*(n - 1)) * x**2
            else:
                k = (n - 1) // 2
                R = RisingFactorial(S.Half, k)
                F = factorial(k)
                return (-1)**k * R / F * x**n / n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return arg
        else:
            return self.func(arg)

    def _eval_rewrite_as_log(self, x, **kwargs):
        return log(x + sqrt(x**2 + 1))

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return sinh

    def _eval_is_zero(self):
        arg = self.args[0]
        if arg.is_zero:
            return True


class acosh(InverseHyperbolicFunction):
    """
    acosh(x) is the inverse hyperbolic cosine of x.

    The inverse hyperbolic cosine function.

    Examples
    ========

    >>> from sympy import acosh
    >>> from sympy.abc import x
    >>> acosh(x).diff(x)
    1/sqrt(x**2 - 1)
    >>> acosh(1)
    0

    See Also
    ========

    asinh, atanh, cosh
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/sqrt(self.args[0]**2 - 1)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity
            elif arg is S.NegativeInfinity:
                return S.Infinity
            elif arg.is_zero:
                return S.Pi*S.ImaginaryUnit / 2
            elif arg is S.One:
                return S.Zero
            elif arg is S.NegativeOne:
                return S.Pi*S.ImaginaryUnit

        if arg.is_number:
            cst_table = {
                S.ImaginaryUnit: log(S.ImaginaryUnit*(1 + sqrt(2))),
                -S.ImaginaryUnit: log(-S.ImaginaryUnit*(1 + sqrt(2))),
                S.Half: S.Pi/3,
                Rational(-1, 2): S.Pi*Rational(2, 3),
                sqrt(2)/2: S.Pi/4,
                -sqrt(2)/2: S.Pi*Rational(3, 4),
                1/sqrt(2): S.Pi/4,
                -1/sqrt(2): S.Pi*Rational(3, 4),
                sqrt(3)/2: S.Pi/6,
                -sqrt(3)/2: S.Pi*Rational(5, 6),
                (sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(5, 12),
                -(sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(7, 12),
                sqrt(2 + sqrt(2))/2: S.Pi/8,
                -sqrt(2 + sqrt(2))/2: S.Pi*Rational(7, 8),
                sqrt(2 - sqrt(2))/2: S.Pi*Rational(3, 8),
                -sqrt(2 - sqrt(2))/2: S.Pi*Rational(5, 8),
                (1 + sqrt(3))/(2*sqrt(2)): S.Pi/12,
                -(1 + sqrt(3))/(2*sqrt(2)): S.Pi*Rational(11, 12),
                (sqrt(5) + 1)/4: S.Pi/5,
                -(sqrt(5) + 1)/4: S.Pi*Rational(4, 5)
            }

            if arg in cst_table:
                if arg.is_extended_real:
                    return cst_table[arg]*S.ImaginaryUnit
                return cst_table[arg]

        if arg is S.ComplexInfinity:
            return S.ComplexInfinity
        if arg == S.ImaginaryUnit*S.Infinity:
            return S.Infinity + S.ImaginaryUnit*S.Pi/2
        if arg == -S.ImaginaryUnit*S.Infinity:
            return S.Infinity - S.ImaginaryUnit*S.Pi/2

        if arg.is_zero:
            return S.Pi*S.ImaginaryUnit*S.Half

        if isinstance(arg, cosh) and arg.args[0].is_number:
            z = arg.args[0]
            if z.is_real:
                from sympy.functions.elementary.complexes import Abs
                return Abs(z)
            r, i = match_real_imag(z)
            if r is not None and i is not None:
                f = floor(i/pi)
                m = z - I*pi*f
                even = f.is_even
                if even is True:
                    if r.is_nonnegative:
                        return m
                    elif r.is_negative:
                        return -m
                elif even is False:
                    m -= I*pi
                    if r.is_nonpositive:
                        return -m
                    elif r.is_positive:
                        return m

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return S.Pi*S.ImaginaryUnit / 2
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) >= 2 and n > 2:
                p = previous_terms[-2]
                return p * (n - 2)**2/(n*(n - 1)) * x**2
            else:
                k = (n - 1) // 2
                R = RisingFactorial(S.Half, k)
                F = factorial(k)
                return -R / F * S.ImaginaryUnit * x**n / n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return S.ImaginaryUnit*S.Pi/2
        else:
            return self.func(arg)

    def _eval_rewrite_as_log(self, x, **kwargs):
        return log(x + sqrt(x + 1) * sqrt(x - 1))

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return cosh


class atanh(InverseHyperbolicFunction):
    """
    atanh(x) is the inverse hyperbolic tangent of x.

    The inverse hyperbolic tangent function.

    Examples
    ========

    >>> from sympy import atanh
    >>> from sympy.abc import x
    >>> atanh(x).diff(x)
    1/(1 - x**2)

    See Also
    ========

    asinh, acosh, tanh
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/(1 - self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy import atan
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg.is_zero:
                return S.Zero
            elif arg is S.One:
                return S.Infinity
            elif arg is S.NegativeOne:
                return S.NegativeInfinity
            elif arg is S.Infinity:
                return -S.ImaginaryUnit * atan(arg)
            elif arg is S.NegativeInfinity:
                return S.ImaginaryUnit * atan(-arg)
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                from sympy.calculus.util import AccumBounds
                return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return S.ImaginaryUnit * atan(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

        if arg.is_zero:
            return S.Zero

        if isinstance(arg, tanh) and arg.args[0].is_number:
            z = arg.args[0]
            if z.is_real:
                return z
            r, i = match_real_imag(z)
            if r is not None and i is not None:
                f = floor(2*i/pi)
                even = f.is_even
                m = z - I*f*pi/2
                if even is True:
                    return m
                elif even is False:
                    return m - I*pi/2

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            return x**n / n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return arg
        else:
            return self.func(arg)

    def _eval_rewrite_as_log(self, x, **kwargs):
        return (log(1 + x) - log(1 - x)) / 2

    def _eval_is_zero(self):
        arg = self.args[0]
        if arg.is_zero:
            return True


    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return tanh


class acoth(InverseHyperbolicFunction):
    """
    acoth(x) is the inverse hyperbolic cotangent of x.

    The inverse hyperbolic cotangent function.

    Examples
    ========

    >>> from sympy import acoth
    >>> from sympy.abc import x
    >>> acoth(x).diff(x)
    1/(1 - x**2)

    See Also
    ========

    asinh, acosh, coth
    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/(1 - self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy import acot
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Zero
            elif arg is S.NegativeInfinity:
                return S.Zero
            elif arg.is_zero:
                return S.Pi*S.ImaginaryUnit / 2
            elif arg is S.One:
                return S.Infinity
            elif arg is S.NegativeOne:
                return S.NegativeInfinity
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.Zero

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return -S.ImaginaryUnit * acot(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

        if arg.is_zero:
            return S.Pi*S.ImaginaryUnit*S.Half

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return S.Pi*S.ImaginaryUnit / 2
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            return x**n / n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy import Order
        arg = self.args[0].as_leading_term(x)

        if x in arg.free_symbols and Order(1, x).contains(arg):
            return S.ImaginaryUnit*S.Pi/2
        else:
            return self.func(arg)

    def _eval_rewrite_as_log(self, x, **kwargs):
        return (log(1 + 1/x) - log(1 - 1/x)) / 2

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return coth


class asech(InverseHyperbolicFunction):
    """
    asech(x) is the inverse hyperbolic secant of x.

    The inverse hyperbolic secant function.

    Examples
    ========

    >>> from sympy import asech, sqrt, S
    >>> from sympy.abc import x
    >>> asech(x).diff(x)
    -1/(x*sqrt(1 - x**2))
    >>> asech(1).diff(x)
    0
    >>> asech(1)
    0
    >>> asech(S(2))
    I*pi/3
    >>> asech(-sqrt(2))
    3*I*pi/4
    >>> asech((sqrt(6) - sqrt(2)))
    I*pi/12

    See Also
    ========

    asinh, atanh, cosh, acoth

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
    .. [2] http://dlmf.nist.gov/4.37
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/

    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            z = self.args[0]
            return -1/(z*sqrt(1 - z**2))
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Pi*S.ImaginaryUnit / 2
            elif arg is S.NegativeInfinity:
                return S.Pi*S.ImaginaryUnit / 2
            elif arg.is_zero:
                return S.Infinity
            elif arg is S.One:
                return S.Zero
            elif arg is S.NegativeOne:
                return S.Pi*S.ImaginaryUnit

        if arg.is_number:
            cst_table = {
                S.ImaginaryUnit: - (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
                -S.ImaginaryUnit: (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
                (sqrt(6) - sqrt(2)): S.Pi / 12,
                (sqrt(2) - sqrt(6)): 11*S.Pi / 12,
                sqrt(2 - 2/sqrt(5)): S.Pi / 10,
                -sqrt(2 - 2/sqrt(5)): 9*S.Pi / 10,
                2 / sqrt(2 + sqrt(2)): S.Pi / 8,
                -2 / sqrt(2 + sqrt(2)): 7*S.Pi / 8,
                2 / sqrt(3): S.Pi / 6,
                -2 / sqrt(3): 5*S.Pi / 6,
                (sqrt(5) - 1): S.Pi / 5,
                (1 - sqrt(5)): 4*S.Pi / 5,
                sqrt(2): S.Pi / 4,
                -sqrt(2): 3*S.Pi / 4,
                sqrt(2 + 2/sqrt(5)): 3*S.Pi / 10,
                -sqrt(2 + 2/sqrt(5)): 7*S.Pi / 10,
                S(2): S.Pi / 3,
                -S(2): 2*S.Pi / 3,
                sqrt(2*(2 + sqrt(2))): 3*S.Pi / 8,
                -sqrt(2*(2 + sqrt(2))): 5*S.Pi / 8,
                (1 + sqrt(5)): 2*S.Pi / 5,
                (-1 - sqrt(5)): 3*S.Pi / 5,
                (sqrt(6) + sqrt(2)): 5*S.Pi / 12,
                (-sqrt(6) - sqrt(2)): 7*S.Pi / 12,
            }

            if arg in cst_table:
                if arg.is_extended_real:
                    return cst_table[arg]*S.ImaginaryUnit
                return cst_table[arg]

        if arg is S.ComplexInfinity:
            from sympy.calculus.util import AccumBounds
            return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)

        if arg.is_zero:
            return S.Infinity

    @staticmethod
    @cacheit
    def expansion_term(n, x, *previous_terms):
        if n == 0:
            return log(2 / x)
        elif n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) > 2 and n > 2:
                p = previous_terms[-2]
                return p * (n - 1)**2 // (n // 2)**2 * x**2 / 4
            else:
                k = n // 2
                R = RisingFactorial(S.Half , k) *  n
                F = factorial(k) * n // 2 * n // 2
                return -1 * R / F * x**n / 4

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return sech

    def _eval_rewrite_as_log(self, arg, **kwargs):
        return log(1/arg + sqrt(1/arg - 1) * sqrt(1/arg + 1))


class acsch(InverseHyperbolicFunction):
    """
    acsch(x) is the inverse hyperbolic cosecant of x.

    The inverse hyperbolic cosecant function.

    Examples
    ========

    >>> from sympy import acsch, sqrt, S
    >>> from sympy.abc import x
    >>> acsch(x).diff(x)
    -1/(x**2*sqrt(1 + x**(-2)))
    >>> acsch(1).diff(x)
    0
    >>> acsch(1)
    log(1 + sqrt(2))
    >>> acsch(S.ImaginaryUnit)
    -I*pi/2
    >>> acsch(-2*S.ImaginaryUnit)
    I*pi/6
    >>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2)))
    -5*I*pi/12

    See Also
    ========

    asinh

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
    .. [2] http://dlmf.nist.gov/4.37
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/

    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            z = self.args[0]
            return -1/(z**2*sqrt(1 + 1/z**2))
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Zero
            elif arg is S.NegativeInfinity:
                return S.Zero
            elif arg.is_zero:
                return S.ComplexInfinity
            elif arg is S.One:
                return log(1 + sqrt(2))
            elif arg is S.NegativeOne:
                return - log(1 + sqrt(2))

        if arg.is_number:
            cst_table = {
                S.ImaginaryUnit: -S.Pi / 2,
                S.ImaginaryUnit*(sqrt(2) + sqrt(6)): -S.Pi / 12,
                S.ImaginaryUnit*(1 + sqrt(5)): -S.Pi / 10,
                S.ImaginaryUnit*2 / sqrt(2 - sqrt(2)): -S.Pi / 8,
                S.ImaginaryUnit*2: -S.Pi / 6,
                S.ImaginaryUnit*sqrt(2 + 2/sqrt(5)): -S.Pi / 5,
                S.ImaginaryUnit*sqrt(2): -S.Pi / 4,
                S.ImaginaryUnit*(sqrt(5)-1): -3*S.Pi / 10,
                S.ImaginaryUnit*2 / sqrt(3): -S.Pi / 3,
                S.ImaginaryUnit*2 / sqrt(2 + sqrt(2)): -3*S.Pi / 8,
                S.ImaginaryUnit*sqrt(2 - 2/sqrt(5)): -2*S.Pi / 5,
                S.ImaginaryUnit*(sqrt(6) - sqrt(2)): -5*S.Pi / 12,
                S(2): -S.ImaginaryUnit*log((1+sqrt(5))/2),
            }

            if arg in cst_table:
                return cst_table[arg]*S.ImaginaryUnit

        if arg is S.ComplexInfinity:
            return S.Zero

        if arg.is_zero:
            return S.ComplexInfinity

        if _coeff_isneg(arg):
            return -cls(-arg)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return csch

    def _eval_rewrite_as_log(self, arg, **kwargs):
        return log(1/arg + sqrt(1/arg**2 + 1))