def test_manualintegrate_derivative():
    assert manualintegrate(pi * Derivative(x**2 + 2*x + 3), x) == \
        pi * ((x**2 + 2*x + 3))
    assert manualintegrate(Derivative(x**2 + 2*x + 3, y), x) == \
        Integral(Derivative(x**2 + 2*x + 3, y))
    assert manualintegrate(Derivative(sin(x), x, x, x, y), x) == \
        Derivative(sin(x), x, x, y)

def test_manualintegrate_exponentials():
    assert manualintegrate(exp(2*x), x) == exp(2*x) / 2
    assert manualintegrate(2**x, x) == (2 ** x) / log(2)

    assert manualintegrate(1 / x, x) == log(x)
    assert manualintegrate(1 / (2*x + 3), x) == log(2*x + 3) / 2
    assert manualintegrate(log(x)**2 / x, x) == log(x)**3 / 3

def test_manualintegrate_orthogonal_poly():
    n = symbols('n')
    a, b = 7, S(5)/3
    polys = [jacobi(n, a, b, x), gegenbauer(n, a, x), chebyshevt(n, x),
        chebyshevu(n, x), legendre(n, x), hermite(n, x), laguerre(n, x),
        assoc_laguerre(n, a, x)]
    for p in polys:
        integral = manualintegrate(p, x)
        for deg in [-2, -1, 0, 1, 3, 5, 8]:
            # some accept negative "degree", some do not
            try:
                p_subbed = p.subs(n, deg)
            except ValueError:
                continue
            assert (integral.subs(n, deg).diff(x) - p_subbed).expand() == 0

        # can also integrate simple expressions with these polynomials
        q = x*p.subs(x, 2*x + 1)
        integral = manualintegrate(q, x)
        for deg in [2, 4, 7]:
            assert (integral.subs(n, deg).diff(x) - q.subs(n, deg)).expand() == 0

        # cannot integrate with respect to any other parameter
        t = symbols('t')
        for i in range(len(p.args) - 1):
            new_args = list(p.args)
            new_args[i] = t
            assert isinstance(manualintegrate(p.func(*new_args), t), Integral)

def test_issue_2850():
    assert manualintegrate(asin(x)*log(x), x) == -x*asin(x) - sqrt(-x**2 + 1) \
            + (x*asin(x) + sqrt(-x**2 + 1))*log(x) - Integral(sqrt(-x**2 + 1)/x, x)
    assert manualintegrate(acos(x)*log(x), x) == -x*acos(x) + sqrt(-x**2 + 1) + \
        (x*acos(x) - sqrt(-x**2 + 1))*log(x) + Integral(sqrt(-x**2 + 1)/x, x)
    assert manualintegrate(atan(x)*log(x), x) == -x*atan(x) + (x*atan(x) - \
            log(x**2 + 1)/2)*log(x) + log(x**2 + 1)/2 + Integral(log(x**2 + 1)/x, x)/2

def test_cyclic_parts():
    f = cos(x)*exp(x/4)
    F = 16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17
    assert manualintegrate(f, x) == F and F.diff(x) == f
    f = x*cos(x)*exp(x/4)
    F = (x*(16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17) -
        128*exp(x/4)*sin(x)/289 + 240*exp(x/4)*cos(x)/289)
    assert manualintegrate(f, x) == F and F.diff(x) == f

def test_manualintegrate_trig_substitution():
    assert manualintegrate(sqrt(16*x**2 - 9)/x, x) == \
        sqrt(16*x**2 - 9) - 3*acos(3/(4*x))
    assert manualintegrate(1/(x**4 * sqrt(25-x**2)), x) == \
        -sqrt(-x**2/25 + 1)/(125*x) - (-x**2/25 + 1)**(3*S.Half)/(15*x**3)
    assert manualintegrate(x**7/(49*x**2 + 1)**(3 * S.Half), x) == \
        ((49*x**2 + 1)**(5*S.Half)/28824005 -
         (49*x**2 + 1)**(3*S.Half)/5764801 +
         3*sqrt(49*x**2 + 1)/5764801 + 1/(5764801*sqrt(49*x**2 + 1)))

def test_issue_9858():
    assert manualintegrate(exp(x)*cos(exp(x)), x) == sin(exp(x))
    assert manualintegrate(exp(2*x)*cos(exp(x)), x) == \
        exp(x)*sin(exp(x)) + cos(exp(x))
    res = manualintegrate(exp(10*x)*sin(exp(x)), x)
    assert not res.has(Integral)
    assert res.diff(x) == exp(10*x)*sin(exp(x))
    # an example with many similar integrations by parts
    assert manualintegrate(sum([x*exp(k*x) for k in range(1, 8)]), x) == (
        x*exp(7*x)/7 + x*exp(6*x)/6 + x*exp(5*x)/5 + x*exp(4*x)/4 +
        x*exp(3*x)/3 + x*exp(2*x)/2 + x*exp(x) - exp(7*x)/49 -exp(6*x)/36 -
        exp(5*x)/25 - exp(4*x)/16 - exp(3*x)/9 - exp(2*x)/4 - exp(x))

def test_issue_6746():
    y = Symbol('y')
    n = Symbol('n')
    assert manualintegrate(y**x, x) == \
        Piecewise((x, Eq(log(y), 0)), (y**x/log(y), True))
    assert manualintegrate(y**(n*x), x) == \
        Piecewise(
            (x, Eq(n, 0)),
            (Piecewise(
                (n*x, Eq(log(y), 0)),
                (y**(n*x)/log(y), True))/n, True))
    assert manualintegrate(exp(n*x), x) == \
        Piecewise((x, Eq(n, 0)), (exp(n*x)/n, True))

    y = Symbol('y', positive=True)
    assert manualintegrate((y + 1)**x, x) == (y + 1)**x/log(y + 1)
    y = Symbol('y', zero=True)
    assert manualintegrate((y + 1)**x, x) == x
    y = Symbol('y')
    n = Symbol('n', nonzero=True)
    assert manualintegrate(y**(n*x), x) == \
        Piecewise((n*x, Eq(log(y), 0)), (y**(n*x)/log(y), True))/n
    y = Symbol('y', positive=True)
    assert manualintegrate((y + 1)**(n*x), x) == \
        (y + 1)**(n*x)/(n*log(y + 1))
    a = Symbol('a', negative=True)
    assert manualintegrate(1 / (a + b*x**2), x) == \
        Piecewise((atan(x/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0), \
        (-acoth(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 > -a/b)), \
        (-atanh(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 < -a/b)))

def test_issue_6746():
    y = Symbol('y')
    n = Symbol('n')
    assert manualintegrate(y**x, x) == \
        Piecewise((x, Eq(log(y), 0)), (y**x/log(y), True))
    assert manualintegrate(y**(n*x), x) == \
        Piecewise(
            (x, Eq(n, 0)),
            (Piecewise(
                (n*x, Eq(log(y), 0)),
                (y**(n*x)/log(y), True))/n, True))
    assert manualintegrate(exp(n*x), x) == \
        Piecewise((x, Eq(n, 0)), (exp(n*x)/n, True))

    y = Symbol('y', positive=True)
    assert manualintegrate((y + 1)**x, x) == (y + 1)**x/log(y + 1)
    y = Symbol('y', zero=True)
    assert manualintegrate((y + 1)**x, x) == x
    y = Symbol('y')
    n = Symbol('n', nonzero=True)
    assert manualintegrate(y**(n*x), x) == \
        Piecewise((n*x, Eq(log(y), 0)), (y**(n*x)/log(y), True))/n
    y = Symbol('y', positive=True)
    assert manualintegrate((y + 1)**(n*x), x) == \
        (y + 1)**(n*x)/(n*log(y + 1))
    a = Symbol('a', negative=True)
    assert manualintegrate(1 / (a + b*x**2), x) == \
        Integral(1/(a + b*x**2), x)

def test_issue_6746():
    assert manualintegrate(y**x, x) == \
        Piecewise((x, Eq(log(y), 0)), (y**x/log(y), True))
    assert manualintegrate(y**(n*x), x) == \
        Piecewise(
            (x, Eq(n, 0)),
            (Piecewise(
                (n*x, Eq(log(y), 0)),
                (y**(n*x)/log(y), True))/n, True))
    assert manualintegrate(exp(n*x), x) == \
        Piecewise((x, Eq(n, 0)), (exp(n*x)/n, True))

    with assuming(~Q.zero(log(y))):
        assert manualintegrate(y**x, x) == y**x/log(y)
    with assuming(Q.zero(log(y))):
        assert manualintegrate(y**x, x) == x
    with assuming(~Q.zero(n)):
        assert manualintegrate(y**(n*x), x) == \
            Piecewise((n*x, Eq(log(y), 0)), (y**(n*x)/log(y), True))/n
    with assuming(~Q.zero(n) & ~Q.zero(log(y))):
        assert manualintegrate(y**(n*x), x) == \
            y**(n*x)/(n*log(y))
    with assuming(Q.negative(a)):
        assert manualintegrate(1 / (a + b*x**2), x) == \
            Integral(1/(a + b*x**2), x)

def test_manualintegrate_trigonometry():
    assert manualintegrate(sin(x), x) == -cos(x)
    assert manualintegrate(tan(x), x) == -log(cos(x))

    assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
    assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))

    assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
    assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
    assert manualintegrate(csc(x) * cot(x), x) == -csc(x)

    assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2))/2
    assert manualintegrate(cos(x)*csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x)))

def test_manualintegrate_inversetrig():
    assert manualintegrate(exp(x) / (1 + exp(2*x)), x) == atan(exp(x))
    assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x/2) / 6
    assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
    assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
    assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2*x) / 2
    assert manualintegrate(1/(a + b*x**2), x) == \
        Piecewise((atan(x*sqrt(b/a))/(a*sqrt(b/a)), And(a > 0, b > 0)))
    assert manualintegrate(1/(4 + b*x**2), x) == \
        Piecewise((atan(sqrt(b)*x/2)/(2*sqrt(b)), b > 0))
    assert manualintegrate(1/(a + 4*x**2), x) == \
        Piecewise((atan(2*x*sqrt(1/a))/(2*a*sqrt(1/a)), a > 0))
    assert manualintegrate(1/(4 + 4*x**2), x) == atan(x) / 4

def test_manualintegrate_Heaviside():
    assert manualintegrate(Heaviside(x), x) == x*Heaviside(x)
    assert manualintegrate(x*Heaviside(2), x) == x**2/2
    assert manualintegrate(x*Heaviside(-2), x) == 0
    assert manualintegrate(x*Heaviside( x), x) == x**2*Heaviside( x)/2
    assert manualintegrate(x*Heaviside(-x), x) == x**2*Heaviside(-x)/2
    assert manualintegrate(Heaviside(2*x + 4), x) == (x+2)*Heaviside(2*x + 4)
    assert manualintegrate(x*Heaviside(x), x) == x**2*Heaviside(x)/2
    assert manualintegrate(Heaviside(x + 1)*Heaviside(1 - x)*x**2, x) == \
        ((x**3/3 + S(1)/3)*Heaviside(x + 1) - S(2)/3)*Heaviside(-x + 1)

    y = Symbol('y')
    assert manualintegrate(sin(7 + x)*Heaviside(3*x - 7), x) == \
            (- cos(x + 7) + cos(S(28)/3))*Heaviside(3*x - S(7))

    assert manualintegrate(sin(y + x)*Heaviside(3*x - y), x) == \
            (cos(4*y/3) - cos(x + y))*Heaviside(3*x - y)

def test_issue_6799():
    r, x, phi = map(Symbol, 'r x phi'.split())
    n = Symbol('n', integer=True, positive=True)

    integrand = (cos(n*(x-phi))*cos(n*x))
    limits = (x, -pi, pi)
    assert manualintegrate(integrand, x).has(Integral)
    assert r * integrate(integrand.expand(trig=True), limits) / pi == r * cos(n * phi)
    assert not integrate(integrand, limits).has(Dummy)

def test_manualintegrate_polynomials():
    assert manualintegrate(y, x) == x*y
    assert manualintegrate(exp(2), x) == x * exp(2)
    assert manualintegrate(x**2, x) == x**3 / 3
    assert manualintegrate(3 * x**2 + 4 * x**3, x) == x**3 + x**4

    assert manualintegrate((x + 2)**3, x) == (x + 2)**4 / 4
    assert manualintegrate((3*x + 4)**2, x) == (3*x + 4)**3 / 9

    assert manualintegrate((u + 2)**3, u) == (u + 2)**4 / 4
    assert manualintegrate((3*u + 4)**2, u) == (3*u + 4)**3 / 9

def test_issue_6799():
    r, x, phi = map(Symbol, 'r x phi'.split())
    n = Symbol('n', integer=True, positive=True)

    integrand = (cos(n*(x-phi))*cos(n*x))
    limits = (x, -pi, pi)
    assert manualintegrate(integrand, x) == \
        ((n*x/2 + sin(2*n*x)/4)*cos(n*phi) - sin(n*phi)*cos(n*x)**2/2)/n
    assert r * integrate(integrand, limits).trigsimp() / pi == r * cos(n * phi)
    assert not integrate(integrand, limits).has(Dummy)

def test_manualintegrate_trigpowers():
    assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
    assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
        x / 8 - sin(4*x) / 32
    assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
    assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
        cos(x)**5 / 5 - cos(x)**3 / 3

    assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3/3 - sec(x)
    assert manualintegrate(tan(x) * sec(x) **2, x) == sec(x)**2/2

    assert manualintegrate(cot(x)**5 * csc(x), x) == \
        -csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
    assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
        -cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3

def test_manualintegrate_parts():
    assert manualintegrate(exp(x) * sin(x), x) == \
        (exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
    assert manualintegrate(2*x*cos(x), x) == 2*x*sin(x) + 2*cos(x)
    assert manualintegrate(x * log(x), x) == x**2*log(x)/2 - x**2/4
    assert manualintegrate(log(x), x) == x * log(x) - x
    assert manualintegrate((3*x**2 + 5) * exp(x), x) == \
        -6*x*exp(x) + (3*x**2 + 5)*exp(x) + 6*exp(x)
    assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2

def test_manualintegrate_parts():
    assert manualintegrate(exp(x) * sin(x), x) == \
        (exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
    assert manualintegrate(2*x*cos(x), x) == 2*x*sin(x) + 2*cos(x)
    assert manualintegrate(x * log(x), x) == x**2*log(x)/2 - x**2/4
    assert manualintegrate(log(x), x) == x * log(x) - x
    assert manualintegrate((3*x**2 + 5) * exp(x), x) == \
        3*x**2*exp(x) - 6*x*exp(x) + 11*exp(x)
    assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2
    # Make sure _parts_rule does not go into an infinite loop here
    assert manualintegrate(log(1/x)/(x + 1), x).has(Integral)

    # Make sure _parts_rule doesn't pick u = constant but can pick dv =
    # constant if necessary, e.g. for integrate(atan(x))
    assert _parts_rule(cos(x), x) == None
    assert _parts_rule(exp(x), x) == None
    assert _parts_rule(x**2, x) == None
    result = _parts_rule(atan(x), x)
    assert result[0] == atan(x) and result[1] == 1

    def _eval_integral(self, f, x, meijerg=None, risch=None, manual=None,
                       conds='piecewise'):
        """
        Calculate the anti-derivative to the function f(x).

        The following algorithms are applied (roughly in this order):

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

           - most frequently used functions (e.g. polynomials, products of trig functions)

        2. Integration of rational functions:

           - A complete algorithm for integrating rational functions is
             implemented (the Lazard-Rioboo-Trager algorithm).  The algorithm
             also uses the partial fraction decomposition algorithm
             implemented in apart() as a preprocessor to make this process
             faster.  Note that the integral of a rational function is always
             elementary, but in general, it may include a RootSum.

        3. Full Risch algorithm:

           - The Risch algorithm is a complete decision
             procedure for integrating elementary functions, which means that
             given any elementary function, it will either compute an
             elementary antiderivative, or else prove that none exists.
             Currently, part of transcendental case is implemented, meaning
             elementary integrals containing exponentials, logarithms, and
             (soon!) trigonometric functions can be computed.  The algebraic
             case, e.g., functions containing roots, is much more difficult
             and is not implemented yet.

           - If the routine fails (because the integrand is not elementary, or
             because a case is not implemented yet), it continues on to the
             next algorithms below.  If the routine proves that the integrals
             is nonelementary, it still moves on to the algorithms below,
             because we might be able to find a closed-form solution in terms
             of special functions.  If risch=True, however, it will stop here.

        4. The Meijer G-Function algorithm:

           - This algorithm works by first rewriting the integrand in terms of
             very general Meijer G-Function (meijerg in SymPy), integrating
             it, and then rewriting the result back, if possible.  This
             algorithm is particularly powerful for definite integrals (which
             is actually part of a different method of Integral), since it can
             compute closed-form solutions of definite integrals even when no
             closed-form indefinite integral exists.  But it also is capable
             of computing many indefinite integrals as well.

           - Another advantage of this method is that it can use some results
             about the Meijer G-Function to give a result in terms of a
             Piecewise expression, which allows to express conditionally
             convergent integrals.

           - Setting meijerg=True will cause integrate() to use only this
             method.

        5. The "manual integration" algorithm:

           - This algorithm tries to mimic how a person would find an
             antiderivative by hand, for example by looking for a
             substitution or applying integration by parts. This algorithm
             does not handle as many integrands but can return results in a
             more familiar form.

           - Sometimes this algorithm can evaluate parts of an integral; in
             this case integrate() will try to evaluate the rest of the
             integrand using the other methods here.

           - Setting manual=True will cause integrate() to use only this
             method.

        6. The Heuristic Risch algorithm:

           - This is a heuristic version of the Risch algorithm, meaning that
             it is not deterministic.  This is tried as a last resort because
             it can be very slow.  It is still used because not enough of the
             full Risch algorithm is implemented, so that there are still some
             integrals that can only be computed using this method.  The goal
             is to implement enough of the Risch and Meijer G methods so that
             this can be deleted.

        """
        from sympy.integrals.risch import risch_integrate

        manual = True # force manual integration
        if risch:
            try:
                return risch_integrate(f, x, conds=conds)
            except NotImplementedError:
                return None

        if manual:
            try:
                result = manualintegrate(f, x)
                if result is not None and result.func != Integral:
                    return result
            except (ValueError, PolynomialError):
                pass
#.........这里部分代码省略.........