def test_expint():
    from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei

    aneg = Symbol("a", negative=True)
    u = Symbol("u", polar=True)

    assert mellin_transform(E1(x), x, s) == (gamma(s) / s, (0, oo), True)
    assert inverse_mellin_transform(gamma(s) / s, s, x, (0, oo)).rewrite(expint).expand() == E1(x)
    assert mellin_transform(expint(a, x), x, s) == (gamma(s) / (a + s - 1), (Max(1 - re(a), 0), oo), True)
    # XXX IMT has hickups with complicated strips ...
    assert simplify(
        unpolarify(
            inverse_mellin_transform(gamma(s) / (aneg + s - 1), s, x, (1 - aneg, oo)).rewrite(expint).expand(func=True)
        )
    ) == expint(aneg, x)

    assert mellin_transform(Si(x), x, s) == (
        -2 ** s * sqrt(pi) * gamma(s / 2 + S(1) / 2) / (2 * s * gamma(-s / 2 + 1)),
        (-1, 0),
        True,
    )
    assert inverse_mellin_transform(
        -2 ** s * sqrt(pi) * gamma((s + 1) / 2) / (2 * s * gamma(-s / 2 + 1)), s, x, (-1, 0)
    ) == Si(x)

    assert mellin_transform(Ci(sqrt(x)), x, s) == (
        -2 ** (2 * s - 1) * sqrt(pi) * gamma(s) / (s * gamma(-s + S(1) / 2)),
        (0, 1),
        True,
    )
    assert inverse_mellin_transform(
        -4 ** s * sqrt(pi) * gamma(s) / (2 * s * gamma(-s + S(1) / 2)), s, u, (0, 1)
    ).expand() == Ci(sqrt(u))

    # TODO LT of Si, Shi, Chi is a mess ...
    assert laplace_transform(Ci(x), x, s) == (-log(1 + s ** 2) / 2 / s, 0, True)
    assert laplace_transform(expint(a, x), x, s) == (lerchphi(s * polar_lift(-1), 1, a), 0, S(0) < re(a))
    assert laplace_transform(expint(1, x), x, s) == (log(s + 1) / s, 0, True)
    assert laplace_transform(expint(2, x), x, s) == ((s - log(s + 1)) / s ** 2, 0, True)

    assert inverse_laplace_transform(-log(1 + s ** 2) / 2 / s, s, u).expand() == Heaviside(u) * Ci(u)
    assert inverse_laplace_transform(log(s + 1) / s, s, x).rewrite(expint) == Heaviside(x) * E1(x)
    assert (
        inverse_laplace_transform((s - log(s + 1)) / s ** 2, s, x).rewrite(expint).expand()
        == (expint(2, x) * Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
    )

def test_as_integral():
    from sympy import Function, Integral
    f = Function('f')
    assert mellin_transform(f(x), x, s).rewrite('Integral') == \
        Integral(x**(s - 1)*f(x), (x, 0, oo))
    assert fourier_transform(f(x), x, s).rewrite('Integral') == \
        Integral(f(x)*exp(-2*I*pi*s*x), (x, -oo, oo))
    assert laplace_transform(f(x), x, s).rewrite('Integral') == \
        Integral(f(x)*exp(-s*x), (x, 0, oo))
    assert str(inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
        == "Integral(x**(-s)*f(s), (s, _c - oo*I, _c + oo*I))"
    assert str(inverse_laplace_transform(f(s), s, x).rewrite('Integral')) == \
        "Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I))"
    assert inverse_fourier_transform(f(s), s, x).rewrite('Integral') == \
        Integral(f(s)*exp(2*I*pi*s*x), (s, -oo, oo))

def test_issue_8882():
    # This is the original test.
    # from sympy import diff, Integral, integrate
    # r = Symbol('r')
    # psi = 1/r*sin(r)*exp(-(a0*r))
    # h = -1/2*diff(psi, r, r) - 1/r*psi
    # f = 4*pi*psi*h*r**2
    # assert integrate(f, (r, -oo, 3), meijerg=True).has(Integral) == True

    # To save time, only the critical part is included.
    F = -a**(-s + 1)*(4 + 1/a**2)**(-s/2)*sqrt(1/a**2)*exp(-s*I*pi)* \
        sin(s*atan(sqrt(1/a**2)/2))*gamma(s)
    raises(IntegralTransformError, lambda:
        inverse_mellin_transform(F, s, x, (-1, oo),
        **{'as_meijerg': True, 'needeval': True}))