def test_nfloat():
    from sympy.core.basic import _aresame
    from sympy.polys.rootoftools import rootof

    x = Symbol("x")
    eq = x**(S(4)/3) + 4*x**(S(1)/3)/3
    assert _aresame(nfloat(eq), x**(S(4)/3) + (4.0/3)*x**(S(1)/3))
    assert _aresame(nfloat(eq, exponent=True), x**(4.0/3) + (4.0/3)*x**(1.0/3))
    eq = x**(S(4)/3) + 4*x**(x/3)/3
    assert _aresame(nfloat(eq), x**(S(4)/3) + (4.0/3)*x**(x/3))
    big = 12345678901234567890
    # specify precision to match value used in nfloat
    Float_big = Float(big, 15)
    assert _aresame(nfloat(big), Float_big)
    assert _aresame(nfloat(big*x), Float_big*x)
    assert _aresame(nfloat(x**big, exponent=True), x**Float_big)
    assert nfloat({x: sqrt(2)}) == {x: nfloat(sqrt(2))}
    assert nfloat({sqrt(2): x}) == {sqrt(2): x}
    assert nfloat(cos(x + sqrt(2))) == cos(x + nfloat(sqrt(2)))

    # issue 6342
    f = S('x*lamda + lamda**3*(x/2 + 1/2) + lamda**2 + 1/4')
    assert not any(a.free_symbols for a in solveset(f.subs(x, -0.139)))

    # issue 6632
    assert nfloat(-100000*sqrt(2500000001) + 5000000001) == \
        9.99999999800000e-11

    # issue 7122
    eq = cos(3*x**4 + y)*rootof(x**5 + 3*x**3 + 1, 0)
    assert str(nfloat(eq, exponent=False, n=1)) == '-0.7*cos(3.0*x**4 + y)'

def test_tan_rewrite():
    x = Symbol('x')
    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)

def test_sincos_rewrite():
    x = Symbol("x")
    y = Symbol("y")
    assert sin(pi/2-x) == cos(x)
    assert sin(pi-x) == sin(x)
    assert cos(pi/2-x) == sin(x)
    assert cos(pi-x) == -cos(x)

def test_cross_different_frames3():
    assert cross(A[1], C[1]) == sin(q3)*C[2]
    assert cross(A[1], C[2]) == -sin(q3)*C[1] + cos(q3)*C[3]
    assert cross(A[1], C[3]) == -cos(q3)*C[2]
    assert cross(C[1], A[1]) == -sin(q3)*C[2]
    assert cross(C[2], A[1]) == sin(q3)*C[1] - cos(q3)*C[3]
    assert cross(C[3], A[1]) == cos(q3)*C[2]

def test_cross_method():
    N = NewtonianReferenceFrame('N')
    q, qd = N.declare_coords('q', 3)
    q1, q2, q3 = q
    A = N.rotate('A', 1, q1)
    B = N.rotate('B', 2, q2)
    C = N.rotate('C', 3, q3)
    assert cross(N[1], N[1]) == Vector(0) == 0
    assert cross(N[1], N[2]) == N[3]
    assert N[1].cross(N[3]) == Vector({N[2]: -1})

    assert N[2].cross(N[1]) == Vector({N[3]: -1})
    assert N[2].cross(N[2]) == Vector(0)
    assert N[2].cross(N[3]) == N[1]

    assert N[3].cross(N[1]) == N[2]
    assert N[3].cross(N[2]) == Vector({N[1]: -1})
    assert N[3].cross(N[3]) == Vector(0)

    assert N[1].cross(A[1]) == Vector(0)
    assert N[1].cross(A[2]) == A[3]
    assert N[1].cross(A[3]) == Vector(-A[2])

    assert N[2].cross(A[1]) == Vector(-N[3])
    assert N[2].cross(A[2]) == Vector(sin(q1)*N[1])
    assert N[2].cross(A[3]) == Vector(cos(q1)*N[1])

    assert N[1].cross(B[1]) == Vector(sin(q2)*N[2])
    assert N[1].cross(B[2]) == N[3]
    assert N[1].cross(B[3]) == Vector(-cos(q2)*N[2])

def test_as_ordered_terms():
    f, g = symbols("f,g", cls=Function)

    assert x.as_ordered_terms() == [x]
    assert (sin(x) ** 2 * cos(x) + sin(x) * cos(x) ** 2 + 1).as_ordered_terms() == [
        sin(x) ** 2 * cos(x),
        sin(x) * cos(x) ** 2,
        1,
    ]

    args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
    expr = Add(*args)

    assert expr.as_ordered_terms() == args

    assert (1 + 4 * sqrt(3) * pi * x).as_ordered_terms() == [4 * pi * x * sqrt(3), 1]

    assert (2 + 3 * I).as_ordered_terms() == [2, 3 * I]
    assert (-2 + 3 * I).as_ordered_terms() == [-2, 3 * I]
    assert (2 - 3 * I).as_ordered_terms() == [2, -3 * I]
    assert (-2 - 3 * I).as_ordered_terms() == [-2, -3 * I]

    assert (4 + 3 * I).as_ordered_terms() == [4, 3 * I]
    assert (-4 + 3 * I).as_ordered_terms() == [-4, 3 * I]
    assert (4 - 3 * I).as_ordered_terms() == [4, -3 * I]
    assert (-4 - 3 * I).as_ordered_terms() == [-4, -3 * I]

    f = x ** 2 * y ** 2 + x * y ** 4 + y + 2

    assert f.as_ordered_terms(order="lex") == [x ** 2 * y ** 2, x * y ** 4, y, 2]
    assert f.as_ordered_terms(order="grlex") == [x * y ** 4, x ** 2 * y ** 2, y, 2]
    assert f.as_ordered_terms(order="rev-lex") == [2, y, x * y ** 4, x ** 2 * y ** 2]
    assert f.as_ordered_terms(order="rev-grlex") == [2, y, x ** 2 * y ** 2, x * y ** 4]

def plotgrid_and_save(name):
    tmp_file = TmpFileManager.tmp_file

    x = Symbol('x')
    y = Symbol('y')
    z = Symbol('z')
    p1 = plot(x)
    p2 = plot_parametric((sin(x), cos(x)), (x, sin(x)), show=False)
    p3 = plot_parametric(cos(x), sin(x), adaptive=False, nb_of_points=500, show=False)
    p4 = plot3d_parametric_line(sin(x), cos(x), x, show=False)
    # symmetric grid
    p = PlotGrid(2, 2, p1, p2, p3, p4)
    p.save(tmp_file('%s_grid1' % name))
    p._backend.close()

    # grid size greater than the number of subplots
    p = PlotGrid(3, 4, p1, p2, p3, p4)
    p.save(tmp_file('%s_grid2' % name))
    p._backend.close()

    p5 = plot(cos(x),(x, -pi, pi), show=False)
    p5[0].line_color = lambda a: a
    p6 = plot(Piecewise((1, x > 0), (0, True)), (x, -1, 1), show=False)
    p7 = plot_contour((x**2 + y**2, (x, -5, 5), (y, -5, 5)), (x**3 + y**3, (x, -3, 3), (y, -3, 3)), show=False)
    # unsymmetric grid (subplots in one line)
    p = PlotGrid(1, 3, p5, p6, p7)
    p.save(tmp_file('%s_grid3' % name))
    p._backend.close()

def test_matrices():
    M = Matrix(2, 2, lambda i, j: (i + j + 1)*sin((i + j + 1)*x))

    assert integrate(M, x) == Matrix([
        [-cos(x), -cos(2*x)],
        [-cos(2*x), -cos(3*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 arbitrary_point(self, t=None):
        """ Returns an arbitrary point on the Plane; varying `t` from 0 to 2*pi
        will move the point in a circle of radius 1 about p1 of the Plane.

        Examples
        ========

        >>> from sympy.geometry.plane import Plane
        >>> from sympy.abc import t
        >>> p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0))
        >>> p.arbitrary_point(t)
        Point3D(0, cos(t), sin(t))
        >>> _.distance(p.p1).simplify()
        1

        Returns
        =======

        Point3D

        """
        from sympy import cos, sin
        t = t or Dummy('t')
        x, y, z = self.normal_vector
        a, b, c = self.p1.args
        if x == y == 0:
            return Point3D(a + cos(t), b + sin(t), c)
        elif x == z == 0:
            return Point3D(a + cos(t), b, c + sin(t))
        elif y == z == 0:
            return Point3D(a, b + cos(t), c + sin(t))
        m = Dummy()
        p = self.projection(Point3D(self.p1.x + cos(t), self.p1.y + sin(t), 0)*m)
        return p.xreplace({m: solve(p.distance(self.p1) - 1, m)[0]})

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_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_dyadic():
    d1 = A.x | A.x
    d2 = A.y | A.y
    d3 = A.x | A.y
    assert d1 * 0 == 0
    assert d1 != 0
    assert d1 * 2 == 2 * A.x | A.x
    assert d1 / 2. == 0.5 * d1
    assert d1 & (0 * d1) == 0
    assert d1 & d2 == 0
    assert d1 & A.x == A.x
    assert d1 ^ A.x == 0
    assert d1 ^ A.y == A.x | A.z
    assert d1 ^ A.z == - A.x | A.y
    assert d2 ^ A.x == - A.y | A.z
    assert A.x ^ d1 == 0
    assert A.y ^ d1 == - A.z | A.x
    assert A.z ^ d1 == A.y | A.x
    assert A.x & d1 == A.x
    assert A.y & d1 == 0
    assert A.y & d2 == A.y
    assert d1 & d3 == A.x | A.y
    assert d3 & d1 == 0
    assert d1.dt(A) == 0
    q = dynamicsymbols('q')
    qd = dynamicsymbols('q', 1)
    B = A.orientnew('B', 'Axis', [q, A.z])
    assert d1.express(B) == d1.express(B, B)
    assert d1.express(B) == ((cos(q)**2) * (B.x | B.x) + (-sin(q) * cos(q)) *
            (B.x | B.y) + (-sin(q) * cos(q)) * (B.y | B.x) + (sin(q)**2) *
            (B.y | B.y))
    assert d1.express(B, A) == (cos(q)) * (B.x | A.x) + (-sin(q)) * (B.y | A.x)
    assert d1.express(A, B) == (cos(q)) * (A.x | B.x) + (-sin(q)) * (A.x | B.y)
    assert d1.dt(B) == (-qd) * (A.y | A.x) + (-qd) * (A.x | A.y)

def test_solve_sqrt_3():
    R = Symbol("R")
    eq = sqrt(2) * R * sqrt(1 / (R + 1)) + (R + 1) * (sqrt(2) * sqrt(1 / (R + 1)) - 1)
    sol = solveset_complex(eq, R)

    assert sol == FiniteSet(
        *[
            S(5) / 3 + 4 * sqrt(10) * cos(atan(3 * sqrt(111) / 251) / 3) / 3,
            -sqrt(10) * cos(atan(3 * sqrt(111) / 251) / 3) / 3
            + 40 * re(1 / ((-S(1) / 2 - sqrt(3) * I / 2) * (S(251) / 27 + sqrt(111) * I / 9) ** (S(1) / 3))) / 9
            + sqrt(30) * sin(atan(3 * sqrt(111) / 251) / 3) / 3
            + S(5) / 3
            + I
            * (
                -sqrt(30) * cos(atan(3 * sqrt(111) / 251) / 3) / 3
                - sqrt(10) * sin(atan(3 * sqrt(111) / 251) / 3) / 3
                + 40 * im(1 / ((-S(1) / 2 - sqrt(3) * I / 2) * (S(251) / 27 + sqrt(111) * I / 9) ** (S(1) / 3))) / 9
            ),
        ]
    )

    # the number of real roots will depend on the value of m: for m=1 there are 4
    # and for m=-1 there are none.
    eq = -sqrt((m - q) ** 2 + (-m / (2 * q) + S(1) / 2) ** 2) + sqrt(
        (-m ** 2 / 2 - sqrt(4 * m ** 4 - 4 * m ** 2 + 8 * m + 1) / 4 - S(1) / 4) ** 2
        + (m ** 2 / 2 - m - sqrt(4 * m ** 4 - 4 * m ** 2 + 8 * m + 1) / 4 - S(1) / 4) ** 2
    )
    raises(NotImplementedError, lambda: solveset_real(eq, q))

def test_as_real_imag():
    n = pi**1000
    # the special code for working out the real
    # and complex parts of a power with Integer exponent
    # should not run if there is no imaginary part, hence
    # this should not hang
    assert n.as_real_imag() == (n, 0)

    # issue 6261
    x = Symbol('x')
    assert sqrt(x).as_real_imag() == \
        ((re(x)**2 + im(x)**2)**(S(1)/4)*cos(atan2(im(x), re(x))/2),
     (re(x)**2 + im(x)**2)**(S(1)/4)*sin(atan2(im(x), re(x))/2))

    # issue 3853
    a, b = symbols('a,b', real=True)
    assert ((1 + sqrt(a + b*I))/2).as_real_imag() == \
           (
               (a**2 + b**2)**Rational(
                   1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2),
               (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2)

    assert sqrt(a**2).as_real_imag() == (sqrt(a**2), 0)
    i = symbols('i', imaginary=True)
    assert sqrt(i**2).as_real_imag() == (0, abs(i))

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_equals():
    assert (-3 - sqrt(5) + (-sqrt(10) / 2 - sqrt(2) / 2) ** 2).equals(0)
    assert (x ** 2 - 1).equals((x + 1) * (x - 1))
    assert (cos(x) ** 2 + sin(x) ** 2).equals(1)
    assert (a * cos(x) ** 2 + a * sin(x) ** 2).equals(a)
    r = sqrt(2)
    assert (-1 / (r + r * x) + 1 / r / (1 + x)).equals(0)
    assert factorial(x + 1).equals((x + 1) * factorial(x))
    assert sqrt(3).equals(2 * sqrt(3)) is False
    assert (sqrt(5) * sqrt(3)).equals(sqrt(3)) is False
    assert (sqrt(5) + sqrt(3)).equals(0) is False
    assert (sqrt(5) + pi).equals(0) is False
    assert meter.equals(0) is False
    assert (3 * meter ** 2).equals(0) is False

    # from integrate(x*sqrt(1+2*x), x);
    # diff is zero, but differentiation does not show it
    i = 2 * sqrt(2) * x ** (S(5) / 2) * (1 + 1 / (2 * x)) ** (S(5) / 2) / 5 + 2 * sqrt(2) * x ** (S(3) / 2) * (
        1 + 1 / (2 * x)
    ) ** (S(5) / 2) / (-6 - 3 / x)
    ans = sqrt(2 * x + 1) * (6 * x ** 2 + x - 1) / 15
    diff = i - ans
    assert diff.equals(0) is not False  # should be True, but now it's None
    # XXX TODO add a force=True option to equals to posify both
    # self and other before beginning comparisions
    p = Symbol("p", positive=True)
    assert diff.subs(x, p).equals(0) is True

def trig_rule(integral):
    integrand, symbol = integral
    if isinstance(integrand, sympy.sin) or isinstance(integrand, sympy.cos):
        arg = integrand.args[0]

        if not isinstance(arg, sympy.Symbol):
            return  # perhaps a substitution can deal with it

        if isinstance(integrand, sympy.sin):
            func = "sin"
        else:
            func = "cos"

        return TrigRule(func, arg, integrand, symbol)

    if integrand == sympy.sec(symbol) ** 2:
        return TrigRule("sec**2", symbol, integrand, symbol)
    elif integrand == sympy.csc(symbol) ** 2:
        return TrigRule("csc**2", symbol, integrand, symbol)

    if isinstance(integrand, sympy.tan):
        rewritten = sympy.sin(*integrand.args) / sympy.cos(*integrand.args)
    elif isinstance(integrand, sympy.cot):
        rewritten = sympy.cos(*integrand.args) / sympy.sin(*integrand.args)
    elif isinstance(integrand, sympy.sec):
        arg = integrand.args[0]
        rewritten = (sympy.sec(arg) ** 2 + sympy.tan(arg) * sympy.sec(arg)) / (sympy.sec(arg) + sympy.tan(arg))
    elif isinstance(integrand, sympy.csc):
        arg = integrand.args[0]
        rewritten = (sympy.csc(arg) ** 2 + sympy.cot(arg) * sympy.csc(arg)) / (sympy.csc(arg) + sympy.cot(arg))
    else:
        return

    return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand, symbol)

def test_image_interval():
    from sympy.core.numbers import Rational
    x = Symbol('x', real=True)
    a = Symbol('a', real=True)
    assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2)
    assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \
        Interval(-4, 2, True, False)
    assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
        Interval(0, 4, False, True)
    assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4)
    assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
        Interval(0, 4, False, True)
    assert imageset(x, x**2, Interval(-2, 1, True, True)) == \
        Interval(0, 4, False, True)
    assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1)
    assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \
        Interval(-35, 0)  # Multiple Maxima
    assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \
        + Interval(2, oo)  # Single Infinite discontinuity
    assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \
        Interval(Rational(3, 2), oo, False)  # Multiple Infinite discontinuities

    # Test for Python lambda
    assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2)

    assert imageset(Lambda(x, a*x), Interval(0, 1)) == \
            ImageSet(Lambda(x, a*x), Interval(0, 1))

    assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \
            ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1))