def fourier_cos_seq(func, limits, n):
    """Returns the cos sequence in a Fourier series"""
    from sympy.integrals import integrate
    x, L = limits[0], limits[2] - limits[1]
    cos_term = cos(2*n*pi*x / L)
    formula = 2 * cos_term * integrate(func * cos_term, limits) / L
    a0 = formula.subs(n, S.Zero) / 2
    return a0, SeqFormula(2 * cos_term * integrate(func * cos_term, limits)
                          / L, (n, 1, oo))

    def bending_moment(self):
        """
        Returns a Singularity Function expression which represents
        the bending moment curve of the Beam object.

        Examples
        ========
        There is a beam of length 30 meters. A moment of magnitude 120 Nm is
        applied in the clockwise direction at the end of the beam. A pointload
        of magnitude 8 N is applied from the top of the beam at the starting
        point. There are two simple supports below the beam. One at the end
        and another one at a distance of 10 meters from the start. The
        deflection is restricted at both the supports.

        Using the sign convention of upward forces and clockwise moment
        being positive.

        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols
        >>> E, I = symbols('E, I')
        >>> R1, R2 = symbols('R1, R2')
        >>> b = Beam(30, E, I)
        >>> b.apply_load(-8, 0, -1)
        >>> b.apply_load(R1, 10, -1)
        >>> b.apply_load(R2, 30, -1)
        >>> b.apply_load(120, 30, -2)
        >>> b.bc_deflection = [(10, 0), (30, 0)]
        >>> b.solve_for_reaction_loads(R1, R2)
        >>> b.bending_moment()
        -8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)
        """
        x = self.variable
        return integrate(self.shear_force(), x)

def fourier_sin_seq(func, limits, n):
    """Returns the sin sequence in a Fourier series"""
    from sympy.integrals import integrate
    x, L = limits[0], limits[2] - limits[1]
    sin_term = sin(2*n*pi*x / L)
    return SeqFormula(2 * sin_term * integrate(func * sin_term, limits)
                      / L, (n, 1, oo))

def singularityintegrate(f, x):
    """
    This function handles the indefinite integrations of Singularity functions.
    The ``integrate`` function calls this function intenally whenever an
    instance of SingularityFunction is passed as argument.

    The idea for integration is the following:

    - If we are dealing with a SingularityFunction expression,
      i.e. ``SingularityFunction(x, a, n)``, we just return
      ``SingularityFunction(x, a, n + 1)/(n + 1)`` if ``n >= 0`` and
      ``SingularityFunction(x, a, n + 1)`` if ``n < 0``.

    - If the node is a multiplication or power node having a
      SingularityFunction term we rewrite the whole expression in terms of
      Heaviside and DiracDelta and then integrate the output. Lastly, we
      rewrite the output of integration back in terms of SingularityFunction.

    - If none of the above case arises, we return None.

    Examples
    ========
    >>> from sympy.integrals.singularityfunctions import singularityintegrate
    >>> from sympy import SingularityFunction, symbols, Function
    >>> x, a, n, y = symbols('x a n y')
    >>> f = Function('f')
    >>> singularityintegrate(SingularityFunction(x, a, 3), x)
    SingularityFunction(x, a, 4)/4
    >>> singularityintegrate(5*SingularityFunction(x, 5, -2), x)
    5*SingularityFunction(x, 5, -1)
    >>> singularityintegrate(6*SingularityFunction(x, 5, -1), x)
    6*SingularityFunction(x, 5, 0)
    >>> singularityintegrate(x*SingularityFunction(x, 0, -1), x)
    0
    >>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x)
    f(1)*SingularityFunction(x, 1, 0)

    """

    if not f.has(SingularityFunction):
        return None

    if f.func == SingularityFunction:
        x = sympify(f.args[0])
        a = sympify(f.args[1])
        n = sympify(f.args[2])
        if n.is_positive or n.is_zero:
            return SingularityFunction(x, a, n + 1)/(n + 1)
        elif n == -1 or n == -2:
            return SingularityFunction(x, a, n + 1)

    if f.is_Mul or f.is_Pow:

        expr = f.rewrite(DiracDelta)
        expr = integrate(expr, x)
        return expr.rewrite(SingularityFunction)
    return None

    def integrate(self, x=None, **kwargs):
        """Integrate Formal Power Series.

        Examples
        ========

        >>> from sympy import fps, sin, integrate
        >>> from sympy.abc import x
        >>> f = fps(sin(x))
        >>> f.integrate(x).truncate()
        -1 + x**2/2 - x**4/24 + O(x**6)
        >>> integrate(f, (x, 0, 1))
        -cos(1) + 1
        """
        from sympy.integrals import integrate

        if x is None:
            x = self.x
        elif iterable(x):
            return integrate(self.function, x)

        f = integrate(self.function, x)
        ind = integrate(self.ind, x)
        ind += (f - ind).limit(x, 0)  # constant of integration

        pow_xk = self._get_pow_x(self.xk.formula)
        ak = self.ak
        k = ak.variables[0]
        if ak.formula.has(x):
            form = []
            for e, c in ak.formula.args:
                temp = S.Zero
                for t in Add.make_args(e):
                    pow_x = self._get_pow_x(t)
                    temp += t / (pow_xk + pow_x + 1)
                form.append((temp, c))
            form = Piecewise(*form)
            ak = sequence(form.subs(k, k - 1), (k, ak.start + 1, ak.stop))
        else:
            ak = sequence((ak.formula / (pow_xk + 1)).subs(k, k - 1),
                          (k, ak.start + 1, ak.stop))

        return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))

def _laplace_transform(f, t, s, simplify=True):
    """ The backend function for laplace transforms. """
    from sympy import (re, Max, exp, pi, Abs, Min, periodic_argument as arg,
                       cos, Wild, symbols)
    F = integrate(exp(-s*t) * f, (t, 0, oo))

    if not F.has(Integral):
        return _simplify(F, simplify), -oo, True

    if not F.is_Piecewise:
        raise IntegralTransformError('Laplace', f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError('Laplace', f, 'integral in unexpected form')

    a = -oo
    aux = True
    conds = conjuncts(to_cnf(cond))
    u = Dummy('u', real=True)
    p, q, w1, w2, w3 = symbols('p q w1 w2 w3', cls=Wild, exclude=[s])
    for c in conds:
        a_ = oo
        aux_ = []
        for d in disjuncts(c):
            m = d.match(abs(arg((s + w3)**p*q, w1)) < w2)
            if m:
                if m[q] > 0 and m[w2]/m[p] == pi/2:
                    d = re(s + m[w3]) > 0
            m = d.match(0 < cos(abs(arg(s, q)))*abs(s) - p)
            if m:
                d = re(s) > m[p]
            d_ = d.replace(re, lambda x: x.expand().as_real_imag()[0]).subs(re(s), t)
            if not d.is_Relational or (d.rel_op != '<' and d.rel_op != '<=') \
               or d_.has(s) or not d_.has(t):
                aux_ += [d]
                continue
            soln = _solve_inequality(d_, t)
            if not soln.is_Relational or \
               (soln.rel_op != '<' and soln.rel_op != '<='):
                aux_ += [d]
                continue
            if soln.lhs == t:
                raise IntegralTransformError('Laplace', f,
                                     'convergence not in half-plane?')
            else:
                a_ = Min(soln.lhs, a_)
        if a_ != oo:
            a = Max(a_, a)
        else:
            aux = And(aux, Or(*aux_))

    return _simplify(F, simplify), a, aux

    def length(self):
        """The curve length.

        Examples
        ========

        >>> from sympy.geometry.curve import Curve
        >>> from sympy import cos, sin
        >>> from sympy.abc import t
        >>> Curve((t, t), (t, 0, 1)).length
        sqrt(2)
        """
        integrand = sqrt(sum(diff(func, self.limits[0])**2 for func in self.functions))
        return integrate(integrand, self.limits)

    def slope(self):
        """
        Returns a Singularity Function expression which represents
        the slope the elastic curve of the Beam object.

        Examples
        ========
        There is a beam of length 30 meters. A moment of magnitude 120 Nm is
        applied in the clockwise direction at the end of the beam. A pointload
        of magnitude 8 N is applied from the top of the beam at the starting
        point. There are two simple supports below the beam. One at the end
        and another one at a distance of 10 meters from the start. The
        deflection is restricted at both the supports.

        Using the sign convention of upward forces and clockwise moment
        being positive.

        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols
        >>> E, I = symbols('E, I')
        >>> R1, R2 = symbols('R1, R2')
        >>> b = Beam(30, E, I)
        >>> b.apply_load(-8, 0, -1)
        >>> b.apply_load(R1, 10, -1)
        >>> b.apply_load(R2, 30, -1)
        >>> b.apply_load(120, 30, -2)
        >>> b.bc_deflection = [(10, 0), (30, 0)]
        >>> b.solve_for_reaction_loads(R1, R2)
        >>> b.slope()
        (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
            + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
        """
        x = self.variable
        E = self.elastic_modulus
        I = self.second_moment
        if not self._boundary_conditions['slope']:
            return diff(self.deflection(), x)

        C3 = Symbol('C3')
        slope_curve = integrate(self.bending_moment(), x) + C3

        bc_eqs = []
        for position, value in self._boundary_conditions['slope']:
            eqs = slope_curve.subs(x, position) - value
            bc_eqs.append(eqs)

        constants = list(linsolve(bc_eqs, C3))
        slope_curve = slope_curve.subs({C3: constants[0][0]})
        return S(1)/(E*I)*slope_curve

def test_trim():
    f = Function("f")

    assert trim((f(x) ** 2 + f(x)) / f(x)) == 1 + f(x)
    assert trim((sin(x) ** 2 + sin(x)) / sin(x)) == 1 + sin(x)

    assert trim((f(x) + y * f(x)) / f(x)) == 1 + y

    expr = integrate(1 / (x ** 3 + 1), x)

    assert trim(together(expr.diff(x))) == 1 / (x ** 3 + 1)
    assert cancel(together(expr.diff(x))) == 1 / (x ** 3 + 1)

    expr = together(expr.subs(x, sin(x)).diff(x))

    assert trim(expr) == cos(x) / (1 + sin(x) ** 3)

    assert trim((2 * (1 / n - cos(n * pi) / n)) / pi) == 1 / pi / n * (2 - 2 * cos(pi * n))

    assert trim(sin((f(x) ** 2 + f(x)) / f(x))) == sin(1 + f(x))

    assert trim(exp(x) * sin(x) / 2 + cos(x) * exp(x)) == exp(x) * (sin(x) + 2 * cos(x)) / 2

def _fourier_transform(f, x, k, a, b, name, simplify=True):
    """
    Compute a general fourier-type transform
        F(k) = a int_-oo^oo exp(b*I*x*k) f(x) dx.

    For suitable choice of a and b, this reduces to the standard fourier
    and inverse fourier transforms.
    """
    from sympy import exp, I, oo
    F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo))

    if not F.has(Integral):
        return _simplify(F, simplify), True

    if not F.is_Piecewise:
        raise IntegralTransformError(name, f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(name, f, 'integral in unexpected form')

    return _simplify(F, simplify), cond

    def piecewise_integrate(self, x, **kwargs):
        """Return the Piecewise with each expression being
        replaced with its antiderivative. To obtain a continuous
        antiderivative, use the `integrate` function or method.

        Examples
        ========

        >>> from sympy import Piecewise
        >>> from sympy.abc import x
        >>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
        >>> p.piecewise_integrate(x)
        Piecewise((0, x < 0), (x, x < 1), (2*x, True))

        Note that this does not give a continuous function, e.g.
        at x = 1 the 3rd condition applies and the antiderivative
        there is 2*x so the value of the antiderivative is 2:

        >>> anti = _
        >>> anti.subs(x, 1)
        2

        The continuous derivative accounts for the integral *up to*
        the point of interest, however:

        >>> p.integrate(x)
        Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
        >>> _.subs(x, 1)
        1

        See Also
        ========
        Piecewise._eval_integral
        """
        from sympy.integrals import integrate
        return self.func(*[(integrate(e, x, **kwargs), c) for e, c in self.args])

 def integrate(self, *args, **kwargs):
     from sympy.integrals import integrate
     return integrate(self, *args, **kwargs)

def _rsolve_hypergeometric(f, x, P, Q, k, m):
    """Recursive wrapper to rsolve_hypergeometric.

    Returns a Tuple of (formula, series independent terms,
    maximum power of x in independent terms) if successful
    otherwise ``None``.

    See :func:`rsolve_hypergeometric` for details.
    """
    from sympy.polys import lcm, roots
    from sympy.integrals import integrate

    # tranformation - c
    proots, qroots = roots(P, k), roots(Q, k)
    all_roots = dict(proots)
    all_roots.update(qroots)
    scale = lcm([r.as_numer_denom()[1] for r, t in all_roots.items()
                 if r.is_rational])
    f, P, Q, m = _transformation_c(f, x, P, Q, k, m, scale)

    # transformation - a
    qroots = roots(Q, k)
    if qroots:
        k_min = Min(*qroots.keys())
    else:
        k_min = S.Zero
    shift = k_min + m
    f, P, Q, m = _transformation_a(f, x, P, Q, k, m, shift)

    l = (x*f).limit(x, 0)
    if not isinstance(l, Limit) and l != 0:  # Ideally should only be l != 0
        return None

    qroots = roots(Q, k)
    if qroots:
        k_max = Max(*qroots.keys())
    else:
        k_max = S.Zero

    ind, mp = S.Zero, -oo
    for i in range(k_max + m + 1):
        r = f.diff(x, i).limit(x, 0) / factorial(i)
        if r.is_finite is False:
            old_f = f
            f, P, Q, m = _transformation_a(f, x, P, Q, k, m, i)
            f, P, Q, m = _transformation_e(f, x, P, Q, k, m)
            sol, ind, mp = _rsolve_hypergeometric(f, x, P, Q, k, m)
            sol = _apply_integrate(sol, x, k)
            sol = _apply_shift(sol, i)
            ind = integrate(ind, x)
            ind += (old_f - ind).limit(x, 0)  # constant of integration
            mp += 1
            return sol, ind, mp
        elif r:
            ind += r*x**(i + shift)
            pow_x = Rational((i + shift), scale)
            if pow_x > mp:
                mp = pow_x  # maximum power of x
    ind = ind.subs(x, x**(1/scale))

    sol = _compute_formula(f, x, P, Q, k, m, k_max)
    sol = _apply_shift(sol, shift)
    sol = _apply_scale(sol, scale)

    return sol, ind, mp

def rational_algorithm(f, x, k, order=4, full=False):
    """Rational algorithm for computing
    formula of coefficients of Formal Power Series
    of a function.

    Applicable when f(x) or some derivative of f(x)
    is a rational function in x.

    :func:`rational_algorithm` uses :func:`apart` function for partial fraction
    decomposition. :func:`apart` by default uses 'undetermined coefficients
    method'. By setting ``full=True``, 'Bronstein's algorithm' can be used
    instead.

    Looks for derivative of a function up to 4'th order (by default).
    This can be overriden using order option.

    Returns
    =======

    formula : Expr
    ind : Expr
        Independent terms.
    order : int

    Examples
    ========

    >>> from sympy import log, atan, I
    >>> from sympy.series.formal import rational_algorithm as ra
    >>> from sympy.abc import x, k

    >>> ra(1 / (1 - x), x, k)
    (1, 0, 0)
    >>> ra(log(1 + x), x, k)
    (-(-1)**(-k)/k, 0, 1)

    >>> ra(atan(x), x, k, full=True)
    ((-I*(-I)**(-k)/2 + I*I**(-k)/2)/k, 0, 1)

    Notes
    =====

    By setting ``full=True``, range of admissible functions to be solved using
    ``rational_algorithm`` can be increased. This option should be used
    carefully as it can signifcantly slow down the computation as ``doit`` is
    performed on the :class:`RootSum` object returned by the ``apart`` function.
    Use ``full=False`` whenever possible.

    See Also
    ========

    sympy.polys.partfrac.apart

    References
    ==========

    .. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf
    .. [2] Power Series in Computer Algebra - Wolfram Koepf
    """
    from sympy.polys import RootSum, apart
    from sympy.integrals import integrate

    diff = f
    ds = []  # list of diff

    for i in range(order + 1):
        if i:
            diff = diff.diff(x)

        if diff.is_rational_function(x):
            coeff, sep = S.Zero, S.Zero

            terms = apart(diff, x, full=full)
            if terms.has(RootSum):
                terms = terms.doit()

            for t in Add.make_args(terms):
                num, den = t.as_numer_denom()
                if not den.has(x):
                    sep += t
                else:
                    if isinstance(den, Mul):
                        # m*(n*x - a)**j -> (n*x - a)**j
                        ind = den.as_independent(x)
                        den = ind[1]
                        num /= ind[0]

                    # (n*x - a)**j -> (x - b)
                    den, j = den.as_base_exp()
                    a, xterm = den.as_coeff_add(x)

                    # term -> m/x**n
                    if not a:
                        sep += t
                        continue

                    xc = xterm[0].coeff(x)
                    a /= -xc
                    num /= xc**j

#.........这里部分代码省略.........

def deltaintegrate(f, x):
    """
    deltaintegrate(f, x)

    The idea for integration is the following:

    - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
      we try to simplify it.

      If we could simplify it, then we integrate the resulting expression.
      We already know we can integrate a simplified expression, because only
      simple DiracDelta expressions are involved.

      If we couldn't simplify it, there are two cases:

      1) The expression is a simple expression: we return the integral,
         taking care if we are dealing with a Derivative or with a proper
         DiracDelta.

      2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do
         nothing at all.

    - If the node is a multiplication node having a DiracDelta term:

      First we expand it.

      If the expansion did work, then we try to integrate the expansion.

      If not, we try to extract a simple DiracDelta term, then we have two
      cases:

      1) We have a simple DiracDelta term, so we return the integral.

      2) We didn't have a simple term, but we do have an expression with
         simplified DiracDelta terms, so we integrate this expression.

    Examples
    ========

        >>> from sympy.abc import x, y, z
        >>> from sympy.integrals.deltafunctions import deltaintegrate
        >>> from sympy import sin, cos, DiracDelta, Heaviside
        >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
        sin(1)*cos(1)*Heaviside(x - 1)
        >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
        z**2*DiracDelta(x - z)*Heaviside(y - z)

    See Also
    ========

    sympy.functions.special.delta_functions.DiracDelta
    sympy.integrals.integrals.Integral
    """
    if not f.has(DiracDelta):
        return None

    from sympy.integrals import Integral, integrate
    from sympy.solvers import solve

    # g(x) = DiracDelta(h(x))
    if f.func == DiracDelta:
        h = f.expand(diracdelta=True, wrt=x)
        if h == f:  # can't simplify the expression
            # FIXME: the second term tells whether is DeltaDirac or Derivative
            # For integrating derivatives of DiracDelta we need the chain rule
            if f.is_simple(x):
                if len(f.args) <= 1 or f.args[1] == 0:
                    return Heaviside(f.args[0])
                else:
                    return DiracDelta(f.args[0], f.args[1] - 1) / f.args[0].as_poly().LC()
        else:  # let's try to integrate the simplified expression
            fh = integrate(h, x)
            return fh
    elif f.is_Mul or f.is_Pow:  # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e
        g = f.expand()
        if f != g:  # the expansion worked
            fh = integrate(g, x)
            if fh is not None and not isinstance(fh, Integral):
                return fh
        else:
            # no expansion performed, try to extract a simple DiracDelta term
            deltaterm, rest_mult = change_mul(f, x)

            if not deltaterm:
                if rest_mult:
                    fh = integrate(rest_mult, x)
                    return fh
            else:
                deltaterm = deltaterm.expand(diracdelta=True, wrt=x)
                if deltaterm.is_Mul:  # Take out any extracted factors
                    deltaterm, rest_mult_2 = change_mul(deltaterm, x)
                    rest_mult = rest_mult * rest_mult_2
                point = solve(deltaterm.args[0], x)[0]

                # Return the largest hyperreal term left after
                # repeated integration by parts.  For example,
                #
                #   integrate(y*DiracDelta(x, 1),x) == y*DiracDelta(x,0),  not 0
                #
                # This is so Integral(y*DiracDelta(x).diff(x),x).doit()
#.........这里部分代码省略.........

    def deflection(self):
        """
        Returns a Singularity Function expression which represents
        the elastic curve or deflection of the Beam object.

        Examples
        ========
        There is a beam of length 30 meters. A moment of magnitude 120 Nm is
        applied in the clockwise direction at the end of the beam. A pointload
        of magnitude 8 N is applied from the top of the beam at the starting
        point. There are two simple supports below the beam. One at the end
        and another one at a distance of 10 meters from the start. The
        deflection is restricted at both the supports.

        Using the sign convention of upward forces and clockwise moment
        being positive.

        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols
        >>> E, I = symbols('E, I')
        >>> R1, R2 = symbols('R1, R2')
        >>> b = Beam(30, E, I)
        >>> b.apply_load(-8, 0, -1)
        >>> b.apply_load(R1, 10, -1)
        >>> b.apply_load(R2, 30, -1)
        >>> b.apply_load(120, 30, -2)
        >>> b.bc_deflection = [(10, 0), (30, 0)]
        >>> b.solve_for_reaction_loads(R1, R2)
        >>> b.deflection()
        (4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
            + 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
        """
        x = self.variable
        E = self.elastic_modulus
        I = self.second_moment
        if self._composite_type == "hinge":
            return self._hinge_beam_deflection
        if not self._boundary_conditions['deflection'] and not self._boundary_conditions['slope']:
            if self._composite_type == "fixed":
                args = I.args
                conditions = []
                prev_def = 0
                prev_end = 0
                for i in range(len(args)):
                    if i != 0:
                        prev_end = args[i-1][1].args[1]
                    deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
                    conditions.append(((prev_def + deflection_value), args[i][1]))
                    prev_def = deflection_value.subs(x, args[i][1].args[1])
                return Piecewise(*conditions)
            return S(1)/(E*I)*integrate(integrate(self.bending_moment(), x), x)
        elif not self._boundary_conditions['deflection']:
            return integrate(self.slope(), x)
        elif not self._boundary_conditions['slope'] and self._boundary_conditions['deflection']:
            if self._composite_type == "fixed":
                args = I.args
                conditions = []
                prev_def = 0
                prev_end = 0
                for i in range(len(args)):
                    if i != 0:
                        prev_end = args[i-1][1].args[1]
                    deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
                    conditions.append(((prev_def + deflection_value), args[i][1]))
                    prev_def = deflection_value.subs(x, args[i][1].args[1])
                return Piecewise(*conditions)
            C3 = Symbol('C3')
            C4 = Symbol('C4')
            slope_curve = integrate(self.bending_moment(), x) + C3
            deflection_curve = integrate(slope_curve, x) + C4
            bc_eqs = []
            for position, value in self._boundary_conditions['deflection']:
                eqs = deflection_curve.subs(x, position) - value
                bc_eqs.append(eqs)
            constants = list(linsolve(bc_eqs, (C3, C4)))
            deflection_curve = deflection_curve.subs({C3: constants[0][0], C4: constants[0][1]})
            return S(1)/(E*I)*deflection_curve

        if self._composite_type == "fixed":
            args = I.args
            conditions = []
            prev_def = 0
            prev_end = 0
            for i in range(len(args)):
                if i != 0:
                    prev_end = args[i-1][1].args[1]
                deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
                conditions.append(((prev_def + deflection_value), args[i][1]))
                prev_def = deflection_value.subs(x, args[i][1].args[1])
            return Piecewise(*conditions)

        C4 = Symbol('C4')
        deflection_curve = integrate((E*I)*self.slope(), x) + C4

        bc_eqs = []
        for position, value in self._boundary_conditions['deflection']:
            eqs = deflection_curve.subs(x, position) - value
            bc_eqs.append(eqs)

        constants = list(linsolve(bc_eqs, C4))
#.........这里部分代码省略.........

 def integrate(self, *args, **kwargs):
     """See the integrate function in sympy.integrals"""
     from sympy.integrals import integrate
     return integrate(self, *args, **kwargs)

    def solve_for_reaction_loads(self, *reactions):
        """
        Solves for the reaction forces.

        Examples
        ========
        There is a beam of length 30 meters. A moment of magnitude 120 Nm is
        applied in the clockwise direction at the end of the beam. A pointload
        of magnitude 8 N is applied from the top of the beam at the starting
        point. There are two simple supports below the beam. One at the end
        and another one at a distance of 10 meters from the start. The
        deflection is restricted at both the supports.

        Using the sign convention of upward forces and clockwise moment
        being positive.

        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols, linsolve, limit
        >>> E, I = symbols('E, I')
        >>> R1, R2 = symbols('R1, R2')
        >>> b = Beam(30, E, I)
        >>> b.apply_load(-8, 0, -1)
        >>> b.apply_load(R1, 10, -1)  # Reaction force at x = 10
        >>> b.apply_load(R2, 30, -1)  # Reaction force at x = 30
        >>> b.apply_load(120, 30, -2)
        >>> b.bc_deflection = [(10, 0), (30, 0)]
        >>> b.load
        R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1)
            - 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2)
        >>> b.solve_for_reaction_loads(R1, R2)
        >>> b.reaction_loads
        {R1: 6, R2: 2}
        >>> b.load
        -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
            + 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
        """
        if self._composite_type == "hinge":
            return self._solve_hinge_beams(*reactions)

        x = self.variable
        l = self.length
        C3 = Symbol('C3')
        C4 = Symbol('C4')

        shear_curve = limit(self.shear_force(), x, l)
        moment_curve = limit(self.bending_moment(), x, l)

        slope_eqs = []
        deflection_eqs = []

        slope_curve = integrate(self.bending_moment(), x) + C3
        for position, value in self._boundary_conditions['slope']:
            eqs = slope_curve.subs(x, position) - value
            slope_eqs.append(eqs)

        deflection_curve = integrate(slope_curve, x) + C4
        for position, value in self._boundary_conditions['deflection']:
            eqs = deflection_curve.subs(x, position) - value
            deflection_eqs.append(eqs)

        solution = list((linsolve([shear_curve, moment_curve] + slope_eqs
                            + deflection_eqs, (C3, C4) + reactions).args)[0])
        solution = solution[2:]

        self._reaction_loads = dict(zip(reactions, solution))
        self._load = self._load.subs(self._reaction_loads)

    def _solve_hinge_beams(self, *reactions):
        """Method to find integration constants and reactional variables in a
        composite beam connected via hinge.
        This method resolves the composite Beam into its sub-beams and then
        equations of shear force, bending moment, slope and deflection are
        evaluated for both of them separately. These equations are then solved
        for unknown reactions and integration constants using the boundary
        conditions applied on the Beam. Equal deflection of both sub-beams
        at the hinge joint gives us another equation to solve the system.

        Examples
        ========
        A combined beam, with constant fkexural rigidity E*I, is formed by joining
        a Beam of length 2*l to the right of another Beam of length l. The whole beam
        is fixed at both of its both end. A point load of magnitude P is also applied
        from the top at a distance of 2*l from starting point.

        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols
        >>> E, I = symbols('E, I')
        >>> l=symbols('l', positive=True)
        >>> b1=Beam(l ,E,I)
        >>> b2=Beam(2*l ,E,I)
        >>> b=b1.join(b2,"hinge")
        >>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
        >>> b.apply_load(A1,0,-1)
        >>> b.apply_load(M1,0,-2)
        >>> b.apply_load(P,2*l,-1)
        >>> b.apply_load(A2,3*l,-1)
        >>> b.apply_load(M2,3*l,-2)
        >>> b.bc_slope=[(0,0), (3*l, 0)]
        >>> b.bc_deflection=[(0,0), (3*l, 0)]
        >>> b.solve_for_reaction_loads(M1, A1, M2, A2)
        >>> b.reaction_loads
        {A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
        >>> b.slope()
        Piecewise(((5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36
        + 5*P*SingularityFunction(x, l, 2)/36)/(E*I), l >= x), ((P*l**2/18 - 4*P*l*SingularityFunction(-l +
        x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
        - 13*P*SingularityFunction(-l + x, 2*l, 2)/36)/(E*I), x < 3*l))
        >>> b.deflection()
        Piecewise(((5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108
        + 5*P*SingularityFunction(x, l, 3)/108)/(E*I), l >= x), ((5*P*l**3/54 + P*l**2*(-l + x)/18
        - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108
        + P*SingularityFunction(-l + x, l, 3)/6 - 13*P*SingularityFunction(-l + x, 2*l, 3)/108)/(E*I), x < 3*l))
        """
        x = self.variable
        l = self._hinge_position
        E = self._elastic_modulus
        I = self._second_moment

        if isinstance(I, Piecewise):
            I1 = I.args[0][0]
            I2 = I.args[1][0]
        else:
            I1 = I2 = I

        load_1 = 0       # Load equation on first segment of composite beam
        load_2 = 0       # Load equation on second segment of composite beam

        # Distributing load on both segments
        for load in self.applied_loads:
            if load[1] < l:
                load_1 += load[0]*SingularityFunction(x, load[1], load[2])
                if load[2] == 0:
                    load_1 -= load[0]*SingularityFunction(x, load[3], load[2])
                elif load[2] > 0:
                    load_1 -= load[0]*SingularityFunction(x, load[3], load[2]) + load[0]*SingularityFunction(x, load[3], 0)
            elif load[1] == l:
                load_1 += load[0]*SingularityFunction(x, load[1], load[2])
                load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
            elif load[1] > l:
                load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
                if load[2] == 0:
                    load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2])
                elif load[2] > 0:
                    load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2]) + load[0]*SingularityFunction(x, load[3] - l, 0)

        h = Symbol('h')     # Force due to hinge
        load_1 += h*SingularityFunction(x, l, -1)
        load_2 -= h*SingularityFunction(x, 0, -1)

        eq = []
        shear_1 = integrate(load_1, x)
        shear_curve_1 = limit(shear_1, x, l)
        eq.append(shear_curve_1)
        bending_1 = integrate(shear_1, x)
        moment_curve_1 = limit(bending_1, x, l)
        eq.append(moment_curve_1)

        shear_2 = integrate(load_2, x)
        shear_curve_2 = limit(shear_2, x, self.length - l)
        eq.append(shear_curve_2)
        bending_2 = integrate(shear_2, x)
        moment_curve_2 = limit(bending_2, x, self.length - l)
        eq.append(moment_curve_2)

        C1 = Symbol('C1')
        C2 = Symbol('C2')
        C3 = Symbol('C3')
#.........这里部分代码省略.........

 def _eval_integral(self,x):
     from sympy.integrals import integrate
     return  Piecewise(*[(integrate(e, x), c) for e, c in self.args])