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Python rationaltools.ratint函数代码示例

原作者: [db:作者] 来自: [db:来源] 收藏 邀请

本文整理汇总了Python中sympy.integrals.rationaltools.ratint函数的典型用法代码示例。如果您正苦于以下问题:Python ratint函数的具体用法?Python ratint怎么用?Python ratint使用的例子?那么恭喜您, 这里精选的函数代码示例或许可以为您提供帮助。



在下文中一共展示了ratint函数的10个代码示例,这些例子默认根据受欢迎程度排序。您可以为喜欢或者感觉有用的代码点赞,您的评价将有助于我们的系统推荐出更棒的Python代码示例。

示例1: test_ratint

def test_ratint():
    f = S(1)
    g = x + 1

    assert ratint(f / g, x) == log(x + 1)
    assert ratint((f,g), x) == log(x + 1)

    f = S(1)
    g = x**2 + 1

    assert ratint(f/g, x, real=None) == atan(x)
    assert ratint(f/g, x, real=True) == atan(x)

    assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2

    f = S(36)
    g = x**5-2*x**4-2*x**3+4*x**2+x-2

    assert ratint(f/g, x) == \
        -4*log(1 + x) + 4*log(-2 + x) - (6 + 12*x)/(1 - x**2)

    f = x**4-3*x**2+6
    g = x**6-5*x**4+5*x**2+4

    assert ratint(f/g, x) == \
        atan(x) + atan(x**3) + atan(x/2 - 3*x**S(3)/2 + S(1)/2*x**5)

    f = x**7-24*x**4-4*x**2+8*x-8
    g = x**8+6*x**6+12*x**4+8*x**2

    assert ratint(f/g, x) == \
        (4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)

    assert ratint((x**3*f)/(x*g), x) == \
        -(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
        5*2**(S(1)/2)*atan(x*2**(S(1)/2)/2) + S(1)/2*x**2 - 3*log(2 + x**2)

    f = x**5-x**4+4*x**3+x**2-x+5
    g = x**4-2*x**3+5*x**2-4*x+4

    assert ratint(f/g, x) == \
        x + S(1)/2*x**2 + S(1)/2*log(2-x+x**2) + (S(9)/7-4*x/7)/(2-x+x**2) + \
        13*7**(S(1)/2)*atan(-S(1)/7*7**(S(1)/2) + 2*x*7**(S(1)/2)/7)/49
开发者ID:gnulinooks,项目名称:sympy,代码行数:43,代码来源:test_rationaltools.py


示例2: _eval_integral


#.........这里部分代码省略.........
                parts.append(coeff*x)
                continue

            # g(x) = expr + O(x**n)
            order_term = g.getO()

            if order_term is not None:
                h = self._eval_integral(g.removeO(), x)

                if h is not None:
                    h_order_expr = self._eval_integral(order_term.expr, x)

                    if h_order_expr is not None:
                        h_order_term = order_term.func(h_order_expr, *order_term.variables)
                        parts.append(coeff*(h + h_order_term))
                        continue

                # NOTE: if there is O(x**n) and we fail to integrate then there is
                # no point in trying other methods because they will fail anyway.
                return None

            #               c
            # g(x) = (a*x+b)
            if g.is_Pow and not g.exp.has(x) and not meijerg:
                a = Wild('a', exclude=[x])
                b = Wild('b', exclude=[x])

                M = g.base.match(a*x + b)

                if M is not None:
                    if g.exp == -1:
                        h = C.log(g.base)
                    else:
                        h = g.base**(g.exp + 1) / (g.exp + 1)

                    parts.append(coeff * h / M[a])
                    continue

            #        poly(x)
            # g(x) = -------
            #        poly(x)
            if g.is_rational_function(x) and not meijerg:
                parts.append(coeff * ratint(g, x))
                continue

            if not meijerg:
                # g(x) = Mul(trig)
                h = trigintegrate(g, x)
                if h is not None:
                    parts.append(coeff * h)
                    continue

                # g(x) has at least a DiracDelta term
                h = deltaintegrate(g, x)
                if h is not None:
                    parts.append(coeff * h)
                    continue

            if not meijerg:
                # fall back to the more general algorithm
                try:
                    h = heurisch(g, x, hints=[])
                except PolynomialError:
                    # XXX: this exception means there is a bug in the
                    # implementation of heuristic Risch integration
                    # algorithm.
                    h = None
            else:
                h = None

            if meijerg is not False and h is None:
                # rewrite using G functions
                h = meijerint_indefinite(g, x)
                if h is not None:
                    parts.append(coeff * h)
                    continue

            # if we failed maybe it was because we had
            # a product that could have been expanded,
            # so let's try an expansion of the whole
            # thing before giving up; we don't try this
            # out the outset because there are things
            # that cannot be solved unless they are
            # NOT expanded e.g., x**x*(1+log(x)). There
            # should probably be a checker somewhere in this
            # routine to look for such cases and try to do
            # collection on the expressions if they are already
            # in an expanded form
            if not h and len(args) == 1:
                f = f.expand(mul=True, deep=False)
                if f.is_Add:
                    return self._eval_integral(f, x, meijerg)


            if h is not None:
                parts.append(coeff * h)
            else:
                return None

        return Add(*parts)
开发者ID:manoj2378,项目名称:sympy,代码行数:101,代码来源:integrals.py


示例3: _eval_integral


#.........这里部分代码省略.........
                            h_order_expr, *order_term.variables)
                        parts.append(coeff*(h + h_order_term))
                        continue

                # NOTE: if there is O(x**n) and we fail to integrate then there is
                # no point in trying other methods because they will fail anyway.
                return None

            #               c
            # g(x) = (a*x+b)
            if g.is_Pow and not g.exp.has(x) and not meijerg:
                a = Wild('a', exclude=[x])
                b = Wild('b', exclude=[x])

                M = g.base.match(a*x + b)

                if M is not None:
                    if g.exp == -1:
                        h = C.log(g.base)
                    elif conds != 'piecewise':
                        h = g.base**(g.exp + 1) / (g.exp + 1)
                    else:
                        h1 = C.log(g.base)
                        h2 = g.base**(g.exp + 1) / (g.exp + 1)
                        h = Piecewise((h1, Eq(g.exp, -1)), (h2, True))

                    parts.append(coeff * h / M[a])
                    continue

            #        poly(x)
            # g(x) = -------
            #        poly(x)
            if g.is_rational_function(x) and not meijerg:
                parts.append(coeff * ratint(g, x))
                continue

            if not meijerg:
                # g(x) = Mul(trig)
                h = trigintegrate(g, x, conds=conds)
                if h is not None:
                    parts.append(coeff * h)
                    continue

                # g(x) has at least a DiracDelta term
                h = deltaintegrate(g, x)
                if h is not None:
                    parts.append(coeff * h)
                    continue

                # Try risch again.
                if risch is not False:
                    try:
                        h, i = risch_integrate(g, x, separate_integral=True, conds=conds)
                    except NotImplementedError:
                        h = None
                    else:
                        if i:
                            h = h + i.doit(risch=False)

                        parts.append(coeff*h)
                        continue

                # fall back to heurisch
                try:
                    if conds == 'piecewise':
                        h = heurisch_wrapper(g, x, hints=[])
开发者ID:hrashk,项目名称:sympy,代码行数:67,代码来源:integrals.py


示例4: test_ratint

def test_ratint():
    assert ratint(S(0), x) == 0
    assert ratint(S(7), x) == 7*x

    assert ratint(x, x) == x**2/2
    assert ratint(2*x, x) == x**2
    assert ratint(-2*x, x) == -x**2

    assert ratint(8*x**7 + 2*x + 1, x) == x**8 + x**2 + x

    f = S(1)
    g = x + 1

    assert ratint(f / g, x) == log(x + 1)
    assert ratint((f, g), x) == log(x + 1)

    f = x**3 - x
    g = x - 1

    assert ratint(f/g, x) == x**3/3 + x**2/2

    f = x
    g = (x - a)*(x + a)

    assert ratint(f/g, x) == log(x**2 - a**2)/2

    f = S(1)
    g = x**2 + 1

    assert ratint(f/g, x, real=None) == atan(x)
    assert ratint(f/g, x, real=True) == atan(x)

    assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2

    f = S(36)
    g = x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2

    assert ratint(f/g, x) == \
        -4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1)

    f = x**4 - 3*x**2 + 6
    g = x**6 - 5*x**4 + 5*x**2 + 4

    assert ratint(f/g, x) == \
        atan(x) + atan(x**3) + atan(x/2 - 3*x**S(3)/2 + S(1)/2*x**5)

    f = x**7 - 24*x**4 - 4*x**2 + 8*x - 8
    g = x**8 + 6*x**6 + 12*x**4 + 8*x**2

    assert ratint(f/g, x) == \
        (4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)

    assert ratint((x**3*f)/(x*g), x) == \
        -(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
        5*sqrt(2)*atan(x*sqrt(2)/2) + S(1)/2*x**2 - 3*log(2 + x**2)

    f = x**5 - x**4 + 4*x**3 + x**2 - x + 5
    g = x**4 - 2*x**3 + 5*x**2 - 4*x + 4

    assert ratint(f/g, x) == \
        x + S(1)/2*x**2 + S(1)/2*log(2 - x + x**2) - (4*x - 9)/(14 - 7*x + 7*x**2) + \
        13*sqrt(7)*atan(-S(1)/7*sqrt(7) + 2*x*sqrt(7)/7)/49

    assert ratint(1/(x**2 + x + 1), x) == \
        2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3

    assert ratint(1/(x**3 + 1), x) == \
        -log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3)
             /3 + 2*x*sqrt(3)/3)/3

    assert ratint(1/(x**2 + x + 1), x, real=False) == \
        -I*3**half*log(half + x - half*I*3**half)/3 + \
        I*3**half*log(half + x + half*I*3**half)/3

    assert ratint(1/(x**3 + 1), x, real=False) == log(1 + x)/3 + \
        (-S(1)/6 + I*3**half/6)*log(-half + x + I*3**half/2) + \
        (-S(1)/6 - I*3**half/6)*log(-half + x - I*3**half/2)

    # issue 4991
    assert ratint(1/(x*(a + b*x)**3), x) == \
        (3*a + 2*b*x)/(2*a**4 + 4*a**3*b*x + 2*a**2*b**2*x**2) + (
            log(x) - log(a/b + x))/a**3

    assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2
    assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2

    assert ratint((x/4 - 4/(1 - x)).diff(x), x) == x/4 + 4/(x - 1)

    ans = atan(x)
    assert ratint(1/(x**2 + 1), x, symbol=x) == ans
    assert ratint(1/(x**2 + 1), x, symbol='x') == ans
    assert ratint(1/(x**2 + 1), x, symbol=a) == ans
开发者ID:AdrianPotter,项目名称:sympy,代码行数:92,代码来源:test_rationaltools.py


示例5: test_issue_5817

def test_issue_5817():
    a, b, c = symbols('a,b,c', positive=True)

    assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
        sqrt(a)*atan(sqrt(
            b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))
开发者ID:AdrianPotter,项目名称:sympy,代码行数:6,代码来源:test_rationaltools.py


示例6: test_issue_5249

def test_issue_5249():
    assert ratint(
        1/(x**2 + a**2), x) == (-I*log(-I*a + x)/2 + I*log(I*a + x)/2)/a
开发者ID:AdrianPotter,项目名称:sympy,代码行数:3,代码来源:test_rationaltools.py


示例7: test_issue_5414

def test_issue_5414():
    assert ratint(1/(x**2 + 16), x) == atan(x/4)/4
开发者ID:AdrianPotter,项目名称:sympy,代码行数:2,代码来源:test_rationaltools.py


示例8: _eval_integral

    def _eval_integral(self, f, x):
        """Calculate the anti-derivative to the function f(x).

        This is a powerful function that should in theory be able to integrate
        everything that can be integrated. If you find something, that it
        doesn't, it is easy to implement it.

        (1) Simple heuristics (based on pattern matching and integral table):

         - most frequently used functions (e.g. polynomials)
         - functions non-integrable by any of the following algorithms (e.g.
           exp(-x**2))

        (2) Integration of rational functions:

         (a) using apart() - apart() is full partial fraction decomposition
         procedure based on Bronstein-Salvy algorithm. It gives formal
         decomposition with no polynomial factorization at all (so it's fast
         and gives the most general results). However it needs much better
         implementation of RootsOf class (if fact any implementation).
         (b) using Trager's algorithm - possibly faster than (a) but needs
         implementation :)

        (3) Whichever implementation of pmInt (Mateusz, Kirill's or a
        combination of both).

          - this way we can handle efficiently huge class of elementary and
            special functions

        (4) Recursive Risch algorithm as described in Bronstein's integration
        tutorial.

          - this way we can handle those integrable functions for which (3)
            fails

        (5) Powerful heuristics based mostly on user defined rules.

         - handle complicated, rarely used cases
        """

        # if it is a poly(x) then let the polynomial integrate itself (fast)
        #
        # It is important to make this check first, otherwise the other code
        # will return a sympy expression instead of a Polynomial.
        #
        # see Polynomial for details.
        if isinstance(f, Poly):
            return f.integrate(x)

        # Piecewise antiderivatives need to call special integrate.
        if f.func is Piecewise:
            return f._eval_integral(x)

        # let's cut it short if `f` does not depend on `x`
        if not f.has(x):
            return f*x

        # try to convert to poly(x) and then integrate if successful (fast)
        poly = f.as_poly(x)

        if poly is not None:
            return poly.integrate(x).as_basic()

        # since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ...
        # we are going to handle Add terms separately,
        # if `f` is not Add -- we only have one term
        parts = []
        for g in make_list(f, Add):
            coeff, g = g.as_independent(x)

            # g(x) = const
            if g is S.One:
                parts.append(coeff * x)
                continue

            #               c
            # g(x) = (a*x+b)
            if g.is_Pow and not g.exp.has(x):
                a = Wild('a', exclude=[x])
                b = Wild('b', exclude=[x])

                M = g.base.match(a*x + b)

                if M is not None:
                    if g.exp == -1:
                        h = C.log(g.base)
                    else:
                        h = g.base**(g.exp+1) / (g.exp+1)

                    parts.append(coeff * h / M[a])
                    continue

            #        poly(x)
            # g(x) = -------
            #        poly(x)
            if g.is_rational_function(x):
                parts.append(coeff * ratint(g, x))
                continue

            # g(x) = Mul(trig)
#.........这里部分代码省略.........
开发者ID:smichr,项目名称:sympy-live,代码行数:101,代码来源:integrals.py


示例9: test_ratint

def test_ratint():
    assert ratint(S(0), x) == 0
    assert ratint(S(7), x) == 7*x

    assert ratint(x, x) == x**2/2
    assert ratint(2*x, x) == x**2

    assert ratint(8*x**7+2*x+1, x) == x**8+x**2+x

    f = S(1)
    g = x + 1

    assert ratint(f / g, x) == log(x + 1)
    assert ratint((f,g), x) == log(x + 1)

    f = x**3 - x
    g = x - 1

    assert ratint(f/g, x) == x**3/3 + x**2/2

    f = x
    g = (x - a)*(x + a)

    assert ratint(f/g, x) == log(x**2 - a**2)/2

    f = S(1)
    g = x**2 + 1

    assert ratint(f/g, x, real=None) == atan(x)
    assert ratint(f/g, x, real=True) == atan(x)

    assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2

    f = S(36)
    g = x**5-2*x**4-2*x**3+4*x**2+x-2

    assert ratint(f/g, x) == \
        -4*log(1 + x) + 4*log(-2 + x) - (6 + 12*x)/(1 - x**2)

    f = x**4-3*x**2+6
    g = x**6-5*x**4+5*x**2+4

    assert ratint(f/g, x) == \
        atan(x) + atan(x**3) + atan(x/2 - 3*x**S(3)/2 + S(1)/2*x**5)

    f = x**7-24*x**4-4*x**2+8*x-8
    g = x**8+6*x**6+12*x**4+8*x**2

    assert ratint(f/g, x) == \
        (4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)

    assert ratint((x**3*f)/(x*g), x) == \
        -(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
        5*2**(S(1)/2)*atan(x*2**(S(1)/2)/2) + S(1)/2*x**2 - 3*log(2 + x**2)

    f = x**5-x**4+4*x**3+x**2-x+5
    g = x**4-2*x**3+5*x**2-4*x+4

    assert ratint(f/g, x) == \
        x + S(1)/2*x**2 + S(1)/2*log(2-x+x**2) + (9-4*x)/(14-7*x+7*x**2) + \
        13*7**(S(1)/2)*atan(-S(1)/7*7**(S(1)/2) + 2*x*7**(S(1)/2)/7)/49

    assert ratint(1/(x**2+x+1), x) == \
        2*3**(S(1)/2)*atan(3**(S(1)/2)/3 + 2*x*3**(S(1)/2)/3)/3

    assert ratint(1/(x**3+1), x) == \
        -log(1 - x + x**2)/6 + log(1 + x)/3 + 3**(S(1)/2)*atan(-3**(S(1)/2)/3 + 2*x*3**(S(1)/2)/3)/3

    assert ratint(1/(x**2+x+1), x, real=False) == \
        -I*3**half*log(half + x - half*I*3**half)/3 + \
        I*3**half*log(half + x + half*I*3**half)/3

    assert ratint(1/(x**3+1), x, real=False) == log(1 + x)/3 - \
        (S(1)/6 - I*3**half/6)*log(-half + x + I*3**half/2) - \
        (S(1)/6 + I*3**half/6)*log(-half + x - I*3**half/2)
开发者ID:Aang,项目名称:sympy,代码行数:75,代码来源:test_rationaltools.py


示例10: test_issue_2150

def test_issue_2150():
    assert ratint(1/(x**2 + a**2), x) == \
        sqrt(-1/a**2)*log(x + a**2*sqrt(-1/a**2))/2 - sqrt(-1/a**2)*log(x -
        a**2*sqrt(-1/a**2))/2
开发者ID:101man,项目名称:sympy,代码行数:4,代码来源:test_rationaltools.py



注:本文中的sympy.integrals.rationaltools.ratint函数示例由纯净天空整理自Github/MSDocs等源码及文档管理平台,相关代码片段筛选自各路编程大神贡献的开源项目,源码版权归原作者所有,传播和使用请参考对应项目的License;未经允许,请勿转载。


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