def test_NonElementaryIntegral(): assert isinstance(risch_integrate(exp(x**2), x), NonElementaryIntegral) assert isinstance(risch_integrate(x**x * log(x), x), NonElementaryIntegral) # Make sure methods of Integral still give back a NonElementaryIntegral assert isinstance( NonElementaryIntegral(x**x * t0, x).subs({t0: log(x)}), NonElementaryIntegral)
def test_risch_integrate(): assert risch_integrate(t0 * exp(x), x) == t0 * exp(x) assert risch_integrate( sin(x), x, rewrite_complex=True) == -exp(I * x) / 2 - exp(-I * x) / 2 # From my GSoC writeup assert risch_integrate((1 + 2*x**2 + x**4 + 2*x**3*exp(2*x**2)) / (x**4*exp(x**2) + 2*x**2*exp(x**2) + exp(x**2)), x) == \ NonElementaryIntegral(exp(-x**2), x) + exp(x**2)/(1 + x**2) assert risch_integrate(0, x) == 0 # These are tested here in addition to in test_DifferentialExtension above # (symlogs) to test that backsubs works correctly. The integrals should be # written in terms of the original logarithms in the integrands. # XXX: Unfortunately, making backsubs work on this one is a little # trickier, because x**x is converted to exp(x*log(x)), and so log(x**x) # is converted to x*log(x). (x**2*log(x)).subs({x*log(x): log(x**x)}) is # smart enough, the issue is that these splits happen at different places # in the algorithm. Maybe a heuristic is in order assert risch_integrate(log(x**x), x) == x**2 * log(x) / 2 - x**2 / 4 assert risch_integrate(log(x**y), x) == x * log(x**y) - x * y assert risch_integrate(log(sqrt(x)), x) == x * log(sqrt(x)) - x / 2
def test_risch_integrate(): assert risch_integrate(t0*exp(x), x) == t0*exp(x) assert risch_integrate(sin(x), x, rewrite_complex=True) == -exp(I*x)/2 - exp(-I*x)/2 # From my GSoC writeup assert risch_integrate((1 + 2*x**2 + x**4 + 2*x**3*exp(2*x**2)) / (x**4*exp(x**2) + 2*x**2*exp(x**2) + exp(x**2)), x) == \ NonElementaryIntegral(exp(-x**2), x) + exp(x**2)/(1 + x**2) assert risch_integrate(0, x) == 0 # These are tested here in addition to in test_DifferentialExtension above # (symlogs) to test that backsubs works correctly. The integrals should be # written in terms of the original logarithms in the integrands. # XXX: Unfortunately, making backsubs work on this one is a little # trickier, because x**x is converted to exp(x*log(x)), and so log(x**x) # is converted to x*log(x). (x**2*log(x)).subs({x*log(x): log(x**x)}) is # smart enough, the issue is that these splits happen at different places # in the algorithm. Maybe a heuristic is in order assert risch_integrate(log(x**x), x) == x**2*log(x)/2 - x**2/4 assert risch_integrate(log(x**y), x) == x*log(x**y) - x*y assert risch_integrate(log(sqrt(x)), x) == x*log(sqrt(x)) - x/2
def test_xtothex(): a = risch_integrate(x**x, x) assert a == NonElementaryIntegral(x**x, x) assert isinstance(a, NonElementaryIntegral)
def test_risch_integrate_float(): assert risch_integrate((-60 * exp(x) - 19.2 * exp(4 * x)) * exp(4 * x), x) == -2.4 * exp(8 * x) - 12.0 * exp(5 * x)
def _eval_integral(self, f, x, meijerg=None, risch=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 Diofant), 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 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-function methods so that this can be deleted. """ from diofant.integrals.deltafunctions import deltaintegrate from diofant.integrals.heurisch import heurisch, heurisch_wrapper from diofant.integrals.rationaltools import ratint from diofant.integrals.risch import risch_integrate if risch: try: return risch_integrate(f, x, conds=conds) except NotImplementedError: return # 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 diofant expression instead of a Polynomial. # # see Polynomial for details. if isinstance(f, Poly) and not meijerg: 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 and not meijerg: return poly.integrate().as_expr() if risch is not False: try: result, i = risch_integrate(f, x, separate_integral=True, conds=conds) except NotImplementedError: pass else: if i: # There was a nonelementary integral. Try integrating it. return result + i.doit(risch=False) else: return result # 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 # Note that in general, this is a bad idea, because Integral(g1) + # Integral(g2) might not be computable, even if Integral(g1 + g2) is. # For example, Integral(x**x + x**x*log(x)). But many heuristics only # work term-wise. So we compute this step last, after trying # risch_integrate. We also try risch_integrate again in this loop, # because maybe the integral is a sum of an elementary part and a # nonelementary part (like erf(x) + exp(x)). risch_integrate() is # quite fast, so this is acceptable. parts = [] args = Add.make_args(f) for g in args: coeff, g = g.as_independent(x) # g(x) = const if g is S.One and not meijerg: 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: parts.append(coeff * (h + self.func(order_term, *self.limits))) 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 # 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 = log(g.base) elif conds != 'piecewise': h = g.base**(g.exp + 1) / (g.exp + 1) else: h1 = 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 # algorithm. h = None else: h = None if meijerg is not False and h is None: # rewrite using G functions try: h = meijerint_indefinite(g, x) except NotImplementedError: from diofant.integrals.meijerint import _debug _debug('NotImplementedError from meijerint_definite') res = None 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 # at 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: # Note: risch will be identical on the expanded # expression, but maybe it will be able to pick out parts, # like x*(exp(x) + erf(x)). return self._eval_integral(f, x, meijerg=meijerg, risch=risch, conds=conds) if h is not None: parts.append(coeff * h) else: return return Add(*parts)
def test_NonElementaryIntegral(): assert isinstance(risch_integrate(exp(x**2), x), NonElementaryIntegral) assert isinstance(risch_integrate(x**x*log(x), x), NonElementaryIntegral) # Make sure methods of Integral still give back a NonElementaryIntegral assert isinstance(NonElementaryIntegral(x**x*t0, x).subs({t0: log(x)}), NonElementaryIntegral)
def test_risch_integrate_float(): assert risch_integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) == -2.4*exp(8*x) - 12.0*exp(5*x)