def test_gruntz_eval_special_slow():
    sskip()
    assert gruntz(gamma(x+1)/sqrt(2*pi)
                  - exp(-x)*(x**(x+S(1)/2) + x**(x-S(1)/2)/12), x, oo) == oo
    assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0
    # XXX This sometimes fails!!!
    assert gruntz(exp(gamma(x-exp(-x))*exp(1/x)) - exp(gamma(x)), x, oo) == oo

def test_beta():
    x, y = Symbol('x'), Symbol('y')

    assert isinstance(beta(x, y), beta)

    assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y)
    assert expand_func(beta(x, y) - beta(y, x)) == 0  # Symmetric
    assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify()

    assert diff(beta(x, y), x) == beta(x, y)*(digamma(x) - digamma(x + y))
    assert diff(beta(x, y), y) == beta(x, y)*(digamma(y) - digamma(x + y))

def test_gruntz_eval_special():
    # Gruntz, p. 126
    assert gruntz(exp(x) * (sin(1 / x + exp(-x)) - sin(1 / x + exp(-x ** 2))), x, oo) == 1
    assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x ** 2), x, oo) == -2 / sqrt(pi)
    assert gruntz(exp(exp(x)) * (exp(sin(1 / x + exp(-exp(x)))) - exp(sin(1 / x))), x, oo) == 1
    assert gruntz(exp(x) * (gamma(x + exp(-x)) - gamma(x)), x, oo) == oo
    assert gruntz(exp(exp(digamma(digamma(x)))) / x, x, oo) == exp(-S(1) / 2)
    assert gruntz(exp(exp(digamma(log(x)))) / x, x, oo) == exp(-S(1) / 2)
    assert gruntz(digamma(digamma(digamma(x))), x, oo) == oo
    assert gruntz(loggamma(loggamma(x)), x, oo) == oo
    assert gruntz(((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x) - cos(1 / x)) * x * log(x), x, oo) == -S(1) / 2
    assert gruntz(x * (gamma(x - 1 / gamma(x)) - gamma(x) + log(x)), x, oo) == S(1) / 2
    assert gruntz((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x), x, oo) == 1

def test_leading_term():
    from sympy import digamma

    assert O(1 / digamma(1 / x)) == O(1 / log(x))

def test_gruntz_eval_special_slow():
    _sskip()
    assert gruntz(gamma(x + 1)/sqrt(2*pi)
                  - exp(-x)*(x**(x + S(1)/2) + x**(x - S(1)/2)/12), x, oo) == oo
    assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0