def test_derivative_by_array():
    from sympy.abc import a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z

    bexpr = x*y**2*exp(z)*log(t)
    sexpr = sin(bexpr)
    cexpr = cos(bexpr)

    a = Array([sexpr])

    assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
    assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
    assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])

    assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
    assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
    assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
    assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
                                                                         [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])

    assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
    assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
    assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])

    assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
    assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
    assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
    assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
                                                                         [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])

def test_simple_3():
    assert Order(x) + x == Order(x)
    assert Order(x) + 2 == 2 + Order(x)
    assert Order(x) + x ** 2 == Order(x)
    assert Order(x) + 1 / x == 1 / x + Order(x)
    assert Order(1 / x) + 1 / x ** 2 == 1 / x ** 2 + Order(1 / x)
    assert Order(x) + exp(1 / x) == Order(x) + exp(1 / x)

def test_cot_rewrite():
    x = Symbol('x')
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert cot(x).rewrite(exp) == I*(pos_exp+neg_exp)/(pos_exp-neg_exp)
    assert cot(x).rewrite(sin) == 2*sin(2*x)/sin(x)**2
    assert cot(x).rewrite(cos) == -cos(x)/cos(x + S.Pi/2)
    assert cot(x).rewrite(tan) == 1/tan(x)

def test_solve_args():
    #implicit symbol to solve for
    assert set(int(tmp) for tmp in solve(x**2-4)) == set([2,-2])
    assert solve([x+y-3,x-y-5]) == {x: 4, y: -1}
    #no symbol to solve for
    assert solve(42) == []
    assert solve([1, 2]) is None
    #multiple symbols: take the first linear solution
    assert solve(x + y - 3, [x, y]) == [{x: 3 - y}]
    # unless it is an undetermined coefficients system
    assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
    # failing undetermined system
    assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b) == \
        [{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
    # failed single equation
    assert solve(1/(1/x - y + exp(y))) ==  []
    raises(NotImplementedError, 'solve(exp(x) + sin(x) + exp(y) + sin(y))')
    # failed system
    # --  when no symbols given, 1 fails
    assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}]
    #     both fail
    assert solve((exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}]
    # --  when symbols given
    solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)]
    #symbol is not a symbol or function
    raises(TypeError, "solve(x**2-pi, pi)")
    # no equations
    assert solve([], [x]) == []
    # overdetermined system
    # - nonlinear
    assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
    # - linear
    assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}

    def eq_sympy(input_dim, output_dim, ARD=False):
        """
        Latent force model covariance, exponentiated quadratic with multiple outputs. Derived from a diffusion equation with the initial spatial condition layed down by a Gaussian process with lengthscale given by shared_lengthscale.

        See IEEE Trans Pattern Anal Mach Intell. 2013 Nov;35(11):2693-705. doi: 10.1109/TPAMI.2013.86. Linear latent force models using Gaussian processes. Alvarez MA, Luengo D, Lawrence ND.

        :param input_dim: Dimensionality of the kernel
        :type input_dim: int
        :param output_dim: number of outputs in the covariance function.
        :type output_dim: int
        :param ARD: whether or not to user ARD (default False).
        :type ARD: bool

        """
        real_input_dim = input_dim
        if output_dim>1:
            real_input_dim -= 1
        X = sp.symbols('x_:' + str(real_input_dim))
        Z = sp.symbols('z_:' + str(real_input_dim))
        scale = sp.var('scale_i scale_j',positive=True)
        if ARD:
            lengthscales = [sp.var('lengthscale%i_i lengthscale%i_j' % i, positive=True) for i in range(real_input_dim)]
            shared_lengthscales = [sp.var('shared_lengthscale%i' % i, positive=True) for i in range(real_input_dim)]
            dist_string = ' + '.join(['(x_%i-z_%i)**2/(shared_lengthscale%i**2 + lengthscale%i_i**2 + lengthscale%i_j**2)' % (i, i, i) for i in range(real_input_dim)])
            dist = parse_expr(dist_string)
            f =  variance*sp.exp(-dist/2.)
        else:
            lengthscales = sp.var('lengthscale_i lengthscale_j',positive=True)
            shared_lengthscale = sp.var('shared_lengthscale',positive=True)
            dist_string = ' + '.join(['(x_%i-z_%i)**2' % (i, i) for i in range(real_input_dim)])
            dist = parse_expr(dist_string)
            f =  scale_i*scale_j*sp.exp(-dist/(2*(lengthscale_i**2 + lengthscale_j**2 + shared_lengthscale**2)))
        return kern(input_dim, [spkern(input_dim, f, output_dim=output_dim, name='eq_sympy')])

def test_evalf_bugs():
    assert NS(sin(1)+exp(-10**10),10) == NS(sin(1),10)
    assert NS(exp(10**10)+sin(1),10) == NS(exp(10**10),10)
    assert NS('log(1+1/10**50)',20) == '1.0000000000000000000e-50'
    assert NS('log(10**100,10)',10) == '100.0000000'
    assert NS('log(2)',10) == '0.6931471806'
    assert NS('(sin(x)-x)/x**3', 15, subs={x:'1/10**50'}) == '-0.166666666666667'
    assert NS(sin(1)+Rational(1,10**100)*I,15) == '0.841470984807897 + 1.00000000000000e-100*I'
    assert x.evalf() == x
    assert NS((1+I)**2*I, 6) == '-2.00000'
    d={n: (-1)**Rational(6,7), y: (-1)**Rational(4,7), x: (-1)**Rational(2,7)}
    assert NS((x*(1+y*(1 + n))).subs(d).evalf(),6) == '0.346011 + 0.433884*I'
    assert NS(((-I-sqrt(2)*I)**2).evalf()) == '-5.82842712474619'
    assert NS((1+I)**2*I,15) == '-2.00000000000000'
    #1659 (1/2):
    assert NS(pi.evalf(69) - pi) == '-4.43863937855894e-71'
    #1659 (2/2): With the bug present, this still only fails if the
    # terms are in the order given here. This is not generally the case,
    # because the order depends on the hashes of the terms.
    assert NS(20 - 5008329267844*n**25 - 477638700*n**37 - 19*n,
              subs={n:.01}) == '19.8100000000000'
    assert NS(((x - 1)*((1 - x))**1000).n()) == '(-x + 1.00000000000000)**1000*(x - 1.00000000000000)'
    assert NS((-x).n()) == '-x'
    assert NS((-2*x).n()) == '-2.00000000000000*x'
    assert NS((-2*x*y).n()) == '-2.00000000000000*x*y'

def test_polarify():
    from sympy import polar_lift, polarify
    x = Symbol('x')
    z = Symbol('z', polar=True)
    f = Function('f')
    ES = {}

    assert polarify(-1) == (polar_lift(-1), ES)
    assert polarify(1 + I) == (polar_lift(1 + I), ES)

    assert polarify(exp(x), subs=False) == exp(x)
    assert polarify(1 + x, subs=False) == 1 + x
    assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x

    assert polarify(x, lift=True) == polar_lift(x)
    assert polarify(z, lift=True) == z
    assert polarify(f(x), lift=True) == f(polar_lift(x))
    assert polarify(1 + x, lift=True) == polar_lift(1 + x)
    assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))

    newex, subs = polarify(f(x) + z)
    assert newex.subs(subs) == f(x) + z

    mu = Symbol("mu")
    sigma = Symbol("sigma", positive=True)

    # Make sure polarify(lift=True) doesn't try to lift the integration
    # variable
    assert polarify(
        Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
        (x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
        exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
        (2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))

def test_deriv_wrt_function():
    t = Symbol('t')
    xfunc = Function('x')
    yfunc = Function('y')
    x = xfunc(t)
    xd = diff(x, t)
    xdd = diff(xd, t)
    y = yfunc(t)
    yd = diff(y, t)
    ydd = diff(yd, t)

    assert diff(x, t) == xd
    assert diff(2 * x + 4, t) == 2 * xd
    assert diff(2 * x + 4 + y, t) == 2 * xd + yd
    assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
    assert diff(2 * x + 4 + y * x, x) == 2 + y
    assert (diff(4 * x**2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y +
            8 * x * xd)
    assert (diff(4 * x**2 + 3 * xd + x * y, t) ==  3 * xdd + x * yd + xd * y +
            8 * x * xd)
    assert diff(4 * x**2 + 3 * xd + x * y, xd) == 3
    assert diff(4 * x**2 + 3 * xd + x * y, xdd) == 0
    assert diff(sin(x), t) == xd * cos(x)
    assert diff(exp(x), t) == xd * exp(x)
    assert diff(sqrt(x), t) == xd / (2 * sqrt(x))

def test_ode_solutions():
    # only a few examples here, the rest will be tested in the actual dsolve tests
    assert ode_renumber(constantsimp(C1*exp(2*x)+exp(x)*(C2+C3), x, 3), 'C', 1, 3) == \
        ode_renumber(C1*exp(x)+C2*exp(2*x), 'C', 1, 2)
    assert ode_renumber(constantsimp(Eq(f(x),I*C1*sinh(x/3) + C2*cosh(x/3)), x, 2),
        'C', 1, 2) == ode_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3)), 'C', 1, 2)
    assert ode_renumber(constantsimp(Eq(f(x),acos((-C1)/cos(x))), x, 1), 'C', 1, 1) == \
        Eq(f(x),acos(C1/cos(x)))
    assert ode_renumber(constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), x, 1),
        'C', 1, 1) ==  Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
    assert ode_renumber(constantsimp(Eq(log(x*2**Rational(1,2)*(1/x)**Rational(1,2)*f(x)\
        **Rational(1,2)/C1) + x**2/(2*f(x)**2), 0), x, 1), 'C', 1, 1) == \
        Eq(log(C1*x*(1/x)**Rational(1,2)*f(x)**Rational(1,2)) + x**2/(2*f(x)**2), 0)
    assert ode_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) - \
        cos(f(x)/x)*exp(-f(x)/x)/2, 0), x, 1), 'C', 1, 1) == \
        Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*exp(-f(x)/x)/2, 0)
    u2 = Symbol('u2')
    _a = Symbol('_a')
    assert ode_renumber(constantsimp(Eq(-Integral(-1/((1 - u2**2)**Rational(1,2)*u2), \
        (u2, _a, x/f(x))) + log(f(x)/C1), 0), x, 1), 'C', 1, 1) == \
        Eq(-Integral(-1/(u2*(1 - u2**2)**Rational(1,2)), (u2, _a, x/f(x))) + \
        log(C1*f(x)), 0)
    assert map(lambda i: ode_renumber(constantsimp(i, x, 1), 'C', 1, 1),
        [Eq(f(x), (-C1*x + x**2)**Rational(1,2)), Eq(f(x), -(-C1*x +
        x**2)**Rational(1,2))]) == [Eq(f(x), (C1*x + x**2)**Rational(1,2)),
        Eq(f(x), -(C1*x + x**2)**Rational(1,2))]

def test_integrate_linearterm_pow():
    # check integrate((a*x+b)^c, x)  --  issue 3499
    y = Symbol('y', positive=True)
    # TODO: Remove conds='none' below, let the assumption take care of it.
    assert integrate(x**y, x, conds='none') == x**(y + 1)/(y + 1)
    assert integrate((exp(y)*x + 1/y)**(1 + sin(y)), x, conds='none') == \
        exp(-y)*(exp(y)*x + 1/y)**(2 + sin(y)) / (2 + sin(y))

def test_transform():
    a = Integral(x**2 + 1, (x, -1, 2))
    fx = x
    fy = 3*y + 1
    assert a.doit() == a.transform(fx, fy).doit()
    assert a.transform(fx, fy).transform(fy, fx) == a
    fx = 3*x + 1
    fy = y
    assert a.transform(fx, fy).transform(fy, fx) == a
    a = Integral(sin(1/x), (x, 0, 1))
    assert a.transform(x, 1/y) == Integral(sin(y)/y**2, (y, 1, oo))
    assert a.transform(x, 1/y).transform(y, 1/x) == a
    a = Integral(exp(-x**2), (x, -oo, oo))
    assert a.transform(x, 2*y) == Integral(2*exp(-4*y**2), (y, -oo, oo))
    # < 3 arg limit handled properly
    assert Integral(x, x).transform(x, a*y).doit() == \
        Integral(y*a**2, y).doit()
    _3 = S(3)
    assert Integral(x, (x, 0, -_3)).transform(x, 1/y).doit() == \
        Integral(-1/x**3, (x, -oo, -1/_3)).doit()
    assert Integral(x, (x, 0, _3)).transform(x, 1/y) == \
        Integral(y**(-3), (y, 1/_3, oo))
    # issue 8400
    i = Integral(x + y, (x, 1, 2), (y, 1, 2))
    assert i.transform(x, (x + 2*y, x)).doit() == \
        i.transform(x, (x + 2*z, x)).doit() == 3

def test_issue_8368():
    assert integrate(exp(-s*x)*cosh(x), (x, 0, oo)) == \
        Piecewise(
            (   pi*Piecewise(
                    (   -s/(pi*(-s**2 + 1)),
                        Abs(s**2) < 1),
                    (   1/(pi*s*(1 - 1/s**2)),
                        Abs(s**(-2)) < 1),
                    (   meijerg(
                            ((S(1)/2,), (0, 0)),
                            ((0, S(1)/2), (0,)),
                            polar_lift(s)**2),
                        True)
                ),
                And(
                    Abs(periodic_argument(polar_lift(s)**2, oo)) < pi,
                    cos(Abs(periodic_argument(polar_lift(s)**2, oo))/2)*sqrt(Abs(s**2)) - 1 > 0,
                    Ne(s**2, 1))
            ),
            (
                Integral(exp(-s*x)*cosh(x), (x, 0, oo)),
                True))
    assert integrate(exp(-s*x)*sinh(x), (x, 0, oo)) == \
        Piecewise(
            (   -1/(s + 1)/2 - 1/(-s + 1)/2,
                And(
                    Ne(1/s, 1),
                    Abs(periodic_argument(s, oo)) < pi/2,
                    Abs(periodic_argument(s, oo)) <= pi/2,
                    cos(Abs(periodic_argument(s, oo)))*Abs(s) - 1 > 0)),
            (   Integral(exp(-s*x)*sinh(x), (x, 0, oo)),
                True))

def test_tan_rewrite():
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
    assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
    assert tan(x).rewrite(cos) == -cos(x + S.Pi/2)/cos(x)
    assert tan(x).rewrite(cot) == 1/cot(x)
    assert tan(sinh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
    assert tan(cosh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
    assert tan(tanh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
    assert tan(coth(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
    assert tan(sin(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
    assert tan(cos(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
    assert tan(tan(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
    assert tan(cot(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
    assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
    assert 0 == (cos(pi/15)*tan(pi/15) - sin(pi/15)).rewrite(pow)
    assert tan(pi/19).rewrite(pow) == tan(pi/19)
    assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)

def test_convolution_fft():
    assert all(convolution_fft([], x, dps=y) == [] for x in ([], [1]) for y in (None, 3))
    assert convolution_fft([1, 2, 3], [4, 5, 6]) == [4, 13, 28, 27, 18]
    assert convolution_fft([1], [5, 6, 7]) == [5, 6, 7]
    assert convolution_fft([1, 3], [5, 6, 7]) == [5, 21, 25, 21]

    assert convolution_fft([1 + 2*I], [2 + 3*I]) == [-4 + 7*I]
    assert convolution_fft([1 + 2*I, 3 + 4*I, 5 + S(3)/5*I], [S(2)/5 + S(4)/7*I]) == \
            [-S(26)/35 + 48*I/35, -S(38)/35 + 116*I/35, S(58)/35 + 542*I/175]

    assert convolution_fft([S(3)/4, S(5)/6], [S(7)/8, S(1)/3, S(2)/5]) == \
                                    [S(21)/32, S(47)/48, S(26)/45, S(1)/3]

    assert convolution_fft([S(1)/9, S(2)/3, S(3)/5], [S(2)/5, S(3)/7, S(4)/9]) == \
                                [S(2)/45, S(11)/35, S(8152)/14175, S(523)/945, S(4)/15]

    assert convolution_fft([pi, E, sqrt(2)], [sqrt(3), 1/pi, 1/E]) == \
                    [sqrt(3)*pi, 1 + sqrt(3)*E, E/pi + pi*exp(-1) + sqrt(6),
                                            sqrt(2)/pi + 1, sqrt(2)*exp(-1)]

    assert convolution_fft([2321, 33123], [5321, 6321, 71323]) == \
                        [12350041, 190918524, 374911166, 2362431729]

    assert convolution_fft([312313, 31278232], [32139631, 319631]) == \
                        [10037624576503, 1005370659728895, 9997492572392]

    raises(TypeError, lambda: convolution_fft(x, y))
    raises(ValueError, lambda: convolution_fft([x, y], [y, x]))

def test_pde_separate_add():
    x, y, z, t = symbols("x,y,z,t")
    F, T, X, Y, Z, u = map(Function, 'FTXYZu')

    eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t)))
    res = pde_separate_add(eq, u(x, t), [X(x), T(t)])
    assert res == [D(X(x), x)*exp(-X(x)), D(T(t), t)*exp(T(t))]

def test_hankel_transform():
    from sympy import sinh, cosh, gamma, sqrt, exp

    r = Symbol("r")
    k = Symbol("k")
    nu = Symbol("nu")
    m = Symbol("m")
    a = symbols("a")

    assert hankel_transform(1/r, r, k, 0) == 1/k
    assert inverse_hankel_transform(1/k, k, r, 0) == 1/r

    assert hankel_transform(
        1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
    assert inverse_hankel_transform(
        2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)

    assert hankel_transform(1/r**m, r, k, nu) == (
        2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
    assert inverse_hankel_transform(2**(-m + 1)*k**(
        m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)

    assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
                                                     3)/2)*gamma(nu + S(3)/2)/sqrt(pi)
    assert inverse_hankel_transform(
        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(3)/2)*gamma(
        nu + S(3)/2)/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)

def test_ode_solutions():
    # only a few examples here, the rest will be tested in the actual dsolve tests
    assert constant_renumber(constantsimp(C1*exp(2*x)+exp(x)*(C2+C3), x, 3), 'C', 1, 3) == \
        constant_renumber(C1*exp(x)+C2*exp(2*x), 'C', 1, 2)
    assert constant_renumber(constantsimp(Eq(f(x),I*C1*sinh(x/3) + C2*cosh(x/3)), x, 2),
        'C', 1, 2) == constant_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3)), 'C', 1, 2)
    assert constant_renumber(constantsimp(Eq(f(x),acos((-C1)/cos(x))), x, 1), 'C', 1, 1) == \
        Eq(f(x),acos(C1/cos(x)))
    assert constant_renumber(constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), x, 1),
        'C', 1, 1) ==  Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
    assert constant_renumber(constantsimp(Eq(log(x*sqrt(2)*sqrt(1/x)*sqrt(f(x))\
        /C1) + x**2/(2*f(x)**2), 0), x, 1), 'C', 1, 1) == \
        Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
    assert constant_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) - \
        cos(f(x)/x)*exp(-f(x)/x)/2, 0), x, 1), 'C', 1, 1) == \
        Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*exp(-f(x)/x)/2, 0)
    u2 = Symbol('u2')
    _a = Symbol('_a')
    assert constant_renumber(constantsimp(Eq(-Integral(-1/(sqrt(1 - u2**2)*u2), \
        (u2, _a, x/f(x))) + log(f(x)/C1), 0), x, 1), 'C', 1, 1) == \
        Eq(-Integral(-1/(u2*sqrt(1 - u2**2)), (u2, _a, x/f(x))) + \
        log(C1*f(x)), 0)
    assert [constant_renumber(constantsimp(i, x, 1), 'C', 1, 1) for i in
        [Eq(f(x), sqrt(-C1*x + x**2)), Eq(f(x), -sqrt(-C1*x +
        x**2))]] == [Eq(f(x), sqrt(C1*x + x**2)),
        Eq(f(x), -sqrt(C1*x + x**2))]

def test_branching():
    from sympy import exp_polar, polar_lift, Symbol, I, exp
    assert besselj(polar_lift(k), x) == besselj(k, x)
    assert besseli(polar_lift(k), x) == besseli(k, x)

    n = Symbol('n', integer=True)
    assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
    assert besselj(n, polar_lift(x)) == besselj(n, x)
    assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
    assert besseli(n, polar_lift(x)) == besseli(n, x)

    def tn(func, s):
        from random import uniform
        c = uniform(1, 5)
        expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
        eps = 1e-15
        expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
        return abs(expr.n() - expr2.n()).n() < 1e-10

    nu = Symbol('nu')
    assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
    assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
    assert tn(besselj, 2)
    assert tn(besselj, pi)
    assert tn(besselj, I)
    assert tn(besseli, 2)
    assert tn(besseli, pi)
    assert tn(besseli, I)

def test_fisher_z():
    d1 = Symbol("d1", positive=True)
    d2 = Symbol("d2", positive=True)

    X = FisherZ("x", d1, d2)
    assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2)*(d1*exp(2*x) + d2)
                             **(-d1/2 - d2/2)*exp(d1*x)/beta(d1/2, d2/2))

def test_issue_1364():
    assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
    assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
    assert solve(x**x) == []
    assert solve(x**x - 2) == [exp(LambertW(log(2)))]
    assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
    assert solve((a/x + exp(x/2)).diff(x), x) == [4*LambertW(sqrt(2)*sqrt(a)/4)]