manualintegrate.py 文件源码

def find_substitutions(integrand, symbol, u_var):
    results = []

    def test_subterm(u, u_diff):
        substituted = integrand / u_diff
        if symbol not in substituted.free_symbols:
            # replaced everything already
            return False

        substituted = substituted.subs(u, u_var).cancel()
        if symbol not in substituted.free_symbols:
            return substituted.as_independent(u_var, as_Add=False)

        return False

    def possible_subterms(term):
        if isinstance(term, (TrigonometricFunction,
                             sympy.asin, sympy.acos, sympy.atan,
                             sympy.exp, sympy.log, sympy.Heaviside)):
            return [term.args[0]]
        elif isinstance(term, sympy.Mul):
            r = []
            for u in term.args:
                r.append(u)
                r.extend(possible_subterms(u))
            return r
        elif isinstance(term, sympy.Pow):
            if term.args[1].is_constant(symbol):
                return [term.args[0]]
            elif term.args[0].is_constant(symbol):
                return [term.args[1]]
        elif isinstance(term, sympy.Add):
            r = []
            for arg in term.args:
                r.append(arg)
                r.extend(possible_subterms(arg))
            return r
        return []

    for u in possible_subterms(integrand):
        if u == symbol:
            continue
        u_diff = manual_diff(u, symbol)
        new_integrand = test_subterm(u, u_diff)
        if new_integrand is not False:
            constant, new_integrand = new_integrand
            if new_integrand == integrand.subs(symbol, u_var):
                continue
            substitution = (u, constant, new_integrand)
            if substitution not in results:
                results.append(substitution)

    return results