Python trigonometry.trigintegrate函数代码示例

def test_trigintegrate_mixed():
    assert trigintegrate(sin(x)*sec(x), x) == -log(cos(x))
    assert trigintegrate(sin(x)*csc(x), x) == x
    assert trigintegrate(sin(x)*cot(x), x) == sin(x)

    assert trigintegrate(cos(x)*sec(x), x) == x
    assert trigintegrate(cos(x)*csc(x), x) == log(sin(x))
    assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
    assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
        - log(cos(x) + 1)/2 + cos(x)
    assert trigintegrate(cot(x)*cos(x)**2, x) == log(sin(x)) - sin(x)**2/2

def test_trigintegrate_even():
    assert trigintegrate(sin(x)**2, x) == x/2 - cos(x)*sin(x)/2
    assert trigintegrate(cos(x)**2, x) == x/2 + cos(x)*sin(x)/2

    assert trigintegrate(sin(3*x)**2, x) == x/2 - cos(3*x)*sin(3*x)/6
    assert trigintegrate(cos(3*x)**2, x) == x/2 + cos(3*x)*sin(3*x)/6
    assert trigintegrate(sin(x)**2 * cos(x)**2, x) == \
        x/8 - sin(2*x)*cos(2*x)/16

    assert trigintegrate(sin(x)**4 * cos(x)**2, x) == \
        x/16 - sin(x) *cos(x)/16 - sin(x)**3*cos(x)/24 + \
        sin(x)**5*cos(x)/6

    assert trigintegrate(sin(x)**2 * cos(x)**4, x) == \
        x/16 + cos(x) *sin(x)/16 + cos(x)**3*sin(x)/24 - \
        cos(x)**5*sin(x)/6

    assert trigintegrate(sin(x)**(-4), x) == -2*cos(x)/(3*sin(x)) \
        - cos(x)/(3*sin(x)**3)

    assert trigintegrate(cos(x)**(-6), x) == sin(x)/(5*cos(x)**5) \
        + 4*sin(x)/(15*cos(x)**3) + 8*sin(x)/(15*cos(x))

def test_trigintegrate_even():
    assert trigintegrate(sin(x)**2, x)  == x/2 - cos(x)*sin(x)/2
    assert trigintegrate(cos(x)**2, x)  == x/2 + cos(x)*sin(x)/2

    assert trigintegrate(sin(3*x)**2, x)== x/2 - cos(3*x)*sin(3*x)/6
    assert trigintegrate(cos(3*x)**2, x)== x/2 + cos(3*x)*sin(3*x)/6

    assert trigintegrate(sin(x)**2 * cos(x)**2, x) == x/8 - cos(x)**3*sin(x)/4 \
                                                          + cos(x)   *sin(x)/8

    assert trigintegrate(sin(x)**4 * cos(x)**2, x) == x/16- sin(x)   *cos(x)/16 \
                                                          - sin(x)**3*cos(x)/24 \
                                                          + sin(x)**5*cos(x)/6

    assert trigintegrate(sin(x)**2 * cos(x)**4, x) == x/16+ cos(x)   *sin(x)/16 \
                                                          + cos(x)**3*sin(x)/24 \
                                                          - cos(x)**5*sin(x)/6

def test_trigintegrate_symbolic():
    n = Symbol('n', integer=True)
    assert trigintegrate(cos(x)**n, x) is None
    assert trigintegrate(sin(x)**n, x) is None
    assert trigintegrate(cot(x)**n, x) is None

def test_trigintegrate_odd():
    assert trigintegrate(Rational(1), x) == x
    assert trigintegrate(x, x) is None
    assert trigintegrate(x**2, x) is None

    assert trigintegrate(sin(x), x) == -cos(x)
    assert trigintegrate(cos(x), x) == sin(x)

    assert trigintegrate(sin(3*x), x) == -cos(3*x)/3
    assert trigintegrate(cos(3*x), x) == sin(3*x)/3

    y = Symbol('y')
    assert trigintegrate(sin(y*x), x) == \
        Piecewise((0, Eq(y, 0)), (-cos(y*x)/y, True))
    assert trigintegrate(cos(y*x), x) == \
        Piecewise((x, Eq(y, 0)), (sin(y*x)/y, True))
    assert trigintegrate(sin(y*x)**2, x) == \
        Piecewise((0, Eq(y, 0)), ((x*y/2 - sin(x*y)*cos(x*y)/2)/y, True))
    assert trigintegrate(sin(y*x)*cos(y*x), x) == \
        Piecewise((0, Eq(y, 0)), (sin(x*y)**2/(2*y), True))
    assert trigintegrate(cos(y*x)**2, x) == \
        Piecewise((x, Eq(y, 0)), ((x*y/2 + sin(x*y)*cos(x*y)/2)/y, True))

    y = Symbol('y', positive=True)
    # TODO: remove conds='none' below. For this to work we would have to rule
    #       out (e.g. by trying solve) the condition y = 0, incompatible with
    #       y.is_positive being True.
    assert trigintegrate(sin(y*x), x, conds='none') == -cos(y*x)/y
    assert trigintegrate(cos(y*x), x, conds='none') == sin(y*x)/y

    assert trigintegrate(sin(x)*cos(x), x) == sin(x)**2/2
    assert trigintegrate(sin(x)*cos(x)**2, x) == -cos(x)**3/3
    assert trigintegrate(sin(x)**2*cos(x), x) == sin(x)**3/3

    # check if it selects right function to substitute,
    # so the result is kept simple
    assert trigintegrate(sin(x)**7 * cos(x), x) == sin(x)**8/8
    assert trigintegrate(sin(x) * cos(x)**7, x) == -cos(x)**8/8

    assert trigintegrate(sin(x)**7 * cos(x)**3, x) == \
        -sin(x)**10/10 + sin(x)**8/8
    assert trigintegrate(sin(x)**3 * cos(x)**7, x) == \
        cos(x)**10/10 - cos(x)**8/8

    # both n, m are odd and -ve, and not necessarily equal
    assert trigintegrate(sin(x)**-1*cos(x)**-1, x) == \
        -log(sin(x)**2 - 1)/2 + log(sin(x))

#.........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.........
                # 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=[])
                    else:
                        h = heurisch(g, x, hints=[])
                except PolynomialError:
                    # XXX: this exception means there is a bug in the
                    # implementation of heuristic Risch integration

def test_trigintegrate_odd():
    assert trigintegrate(Rational(1), x)    == x
    assert trigintegrate(x, x)      is None
    assert trigintegrate(x**2, x)   is None

    assert trigintegrate(sin(x), x) == -cos(x)
    assert trigintegrate(cos(x), x) ==  sin(x)

    assert trigintegrate(sin(3*x), x) == -cos(3*x)/3
    assert trigintegrate(cos(3*x), x) ==  sin(3*x)/3

    assert trigintegrate(sin(y*x), x) == -cos(y*x)/y
    assert trigintegrate(cos(y*x), x) ==  sin(y*x)/y

    assert trigintegrate(sin(x)*cos(x), x)      ==  sin(x)**2/2
    assert trigintegrate(sin(x)*cos(x)**2, x)   == -cos(x)**3/3
    assert trigintegrate(sin(x)**2*cos(x), x)   ==  sin(x)**3/3

    # check if it selects right function to substitute,
    # so the result is kept simple
    assert trigintegrate(sin(x)**7 * cos(x), x) ==  sin(x)**8/8
    assert trigintegrate(sin(x) * cos(x)**7, x) == -cos(x)**8/8

    assert trigintegrate(sin(x)**7 * cos(x)**3, x)  == -sin(x)**10/10 + sin(x)**8/8
    assert trigintegrate(sin(x)**3 * cos(x)**7, x)  ==  cos(x)**10/10 - cos(x)**8/8

def timeit_trigintegrate_sin3x():
    trigintegrate(sin(x)**3, x)

def timeit_trigintegrate_x2():
    trigintegrate(x**2, x)  # -> None

#.........这里部分代码省略.........
         (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)
            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

            # fall back to the more general algorithm
            h = heurisch(g, x, hints=[])

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

        return C.Add(*parts)