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

#.........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)

#.........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=[])

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

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))

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

def test_issue_5414():
    assert ratint(1/(x**2 + 16), x) == atan(x/4)/4

    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.........

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)

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