manualintegrate.py 文件源码

def _parts_rule(integrand, symbol):
    # LIATE rule:
    # log, inverse trig, algebraic, trigonometric, exponential
    def pull_out_algebraic(integrand):
        integrand = integrand.cancel().together()
        algebraic = [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)]
        if algebraic:
            u = sympy.Mul(*algebraic)
            dv = (integrand / u).cancel()
            if not u.is_polynomial() and isinstance(dv, sympy.exp):
                return
            return u, dv

    def pull_out_u(*functions):
        def pull_out_u_rl(integrand):
            if any([integrand.has(f) for f in functions]):
                args = [arg for arg in integrand.args
                        if any(isinstance(arg, cls) for cls in functions)]
                if args:
                    u = reduce(lambda a,b: a*b, args)
                    dv = integrand / u
                    return u, dv

        return pull_out_u_rl

    liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos),
                   pull_out_algebraic, pull_out_u(sympy.sin, sympy.cos),
                   pull_out_u(sympy.exp)]


    dummy = sympy.Dummy("temporary")
    # we can integrate log(x) and atan(x) by setting dv = 1
    if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)):
        integrand = dummy * integrand

    for index, rule in enumerate(liate_rules):
        result = rule(integrand)

        if result:
            u, dv = result

            # Don't pick u to be a constant if possible
            if symbol not in u.free_symbols and not u.has(dummy):
                return

            u = u.subs(dummy, 1)
            dv = dv.subs(dummy, 1)

            for rule in liate_rules[index + 1:]:
                r = rule(integrand)
                # make sure dv is amenable to integration
                if r and sympy.simplify(r[0].subs(dummy, 1)) == sympy.simplify(dv):
                    du = u.diff(symbol)
                    v_step = integral_steps(sympy.simplify(dv), symbol)
                    v = _manualintegrate(v_step)

                    return u, dv, v, du, v_step