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from sympy.concrete.summations import Sumfrom sympy.core.add import Addfrom sympy.core.function import (Derivative, Function, diff)from sympy.core.numbers import (I, Rational, pi)from sympy.core.relational import Nefrom sympy.core.symbol import (Symbol, symbols)from sympy.functions.elementary.exponential import (LambertW, exp, log)from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh, tanh)from sympy.functions.elementary.miscellaneous import sqrtfrom sympy.functions.elementary.piecewise import Piecewisefrom sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan)from sympy.functions.special.bessel import (besselj, besselk, bessely, jn)from sympy.functions.special.error_functions import erffrom sympy.integrals.integrals import Integralfrom sympy.simplify.ratsimp import ratsimpfrom sympy.simplify.simplify import simplifyfrom sympy.integrals.heurisch import components, heurisch, heurisch_wrapperfrom sympy.testing.pytest import XFAIL, skip, slow, ON_TRAVISfrom sympy.integrals.integrals import integratex, y, z, nu = symbols('x,y,z,nu')f = Function('f')
def test_components(): assert components(x*y, x) == {x} assert components(1/(x + y), x) == {x} assert components(sin(x), x) == {sin(x), x} assert components(sin(x)*sqrt(log(x)), x) == \ {log(x), sin(x), sqrt(log(x)), x} assert components(x*sin(exp(x)*y), x) == \ {sin(y*exp(x)), x, exp(x)} assert components(x**Rational(17, 54)/sqrt(sin(x)), x) == \ {sin(x), x**Rational(1, 54), sqrt(sin(x)), x}
assert components(f(x), x) == \ {x, f(x)} assert components(Derivative(f(x), x), x) == \ {x, f(x), Derivative(f(x), x)} assert components(f(x)*diff(f(x), x), x) == \ {x, f(x), Derivative(f(x), x), Derivative(f(x), x)}
def test_issue_10680(): assert isinstance(integrate(x**log(x**log(x**log(x))),x), Integral)
def test_issue_21166(): assert integrate(sin(x/sqrt(abs(x))), (x, -1, 1)) == 0
def test_heurisch_polynomials(): assert heurisch(1, x) == x assert heurisch(x, x) == x**2/2 assert heurisch(x**17, x) == x**18/18 # For coverage assert heurisch_wrapper(y, x) == y*x
def test_heurisch_fractions(): assert heurisch(1/x, x) == log(x) assert heurisch(1/(2 + x), x) == log(x + 2) assert heurisch(1/(x + sin(y)), x) == log(x + sin(y))
# Up to a constant, where C = pi*I*Rational(5, 12), Mathematica gives identical # result in the first case. The difference is because SymPy changes # signs of expressions without any care. # XXX ^ ^ ^ is this still correct? assert heurisch(5*x**5/( 2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12] assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
assert heurisch(1/x**2, x) == -1/x assert heurisch(-1/x**5, x) == 1/(4*x**4)
def test_heurisch_log(): assert heurisch(log(x), x) == x*log(x) - x assert heurisch(log(3*x), x) == -x + x*log(3) + x*log(x) assert heurisch(log(x**2), x) in [x*log(x**2) - 2*x, 2*x*log(x) - 2*x]
def test_heurisch_exp(): assert heurisch(exp(x), x) == exp(x) assert heurisch(exp(-x), x) == -exp(-x) assert heurisch(exp(17*x), x) == exp(17*x) / 17 assert heurisch(x*exp(x), x) == x*exp(x) - exp(x) assert heurisch(x*exp(x**2), x) == exp(x**2) / 2
assert heurisch(exp(-x**2), x) is None
assert heurisch(2**x, x) == 2**x/log(2) assert heurisch(x*2**x, x) == x*2**x/log(2) - 2**x*log(2)**(-2)
assert heurisch(Integral(x**z*y, (y, 1, 2), (z, 2, 3)).function, x) == (x*x**z*y)/(z+1) assert heurisch(Sum(x**z, (z, 1, 2)).function, z) == x**z/log(x)
def test_heurisch_trigonometric(): assert heurisch(sin(x), x) == -cos(x) assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
assert heurisch(cos(x), x) == sin(x) assert heurisch(tan(x), x) in [ log(1 + tan(x)**2)/2, log(tan(x) + I) + I*x, log(tan(x) - I) - I*x, ]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y) assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2] assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7 assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) - 2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \ + (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723 assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3 assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x) - 1) - atan(sqrt(2)*sin(x) + 1)
assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
def test_heurisch_hyperbolic(): assert heurisch(sinh(x), x) == cosh(x) assert heurisch(cosh(x), x) == sinh(x)
assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x) assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
assert heurisch( x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4 + x**2)/4
def test_heurisch_mixed(): assert heurisch(sin(x)*exp(x), x) == exp(x)*sin(x)/2 - exp(x)*cos(x)/2 assert heurisch(sin(x/sqrt(-x)), x) == 2*x*cos(x/sqrt(-x))/sqrt(-x) - 2*sin(x/sqrt(-x))
def test_heurisch_radicals(): assert heurisch(1/sqrt(x), x) == 2*sqrt(x) assert heurisch(1/sqrt(x)**3, x) == -2/sqrt(x) assert heurisch(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*sqrt(cos(x))**3/3 y = Symbol('y') assert heurisch(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \ 2*sqrt(x)*cos(y*sqrt(x))/y assert heurisch_wrapper(sin(y*sqrt(x)), x) == Piecewise( (-2*sqrt(x)*cos(sqrt(x)*y)/y + 2*sin(sqrt(x)*y)/y**2, Ne(y, 0)), (0, True)) y = Symbol('y', positive=True) assert heurisch_wrapper(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \ 2*sqrt(x)*cos(y*sqrt(x))/y
def test_heurisch_special(): assert heurisch(erf(x), x) == x*erf(x) + exp(-x**2)/sqrt(pi) assert heurisch(exp(-x**2)*erf(x), x) == sqrt(pi)*erf(x)**2 / 4
def test_heurisch_symbolic_coeffs(): assert heurisch(1/(x + y), x) == log(x + y) assert heurisch(1/(x + sqrt(2)), x) == log(x + sqrt(2)) assert simplify(diff(heurisch(log(x + y + z), y), y)) == log(x + y + z)
def test_heurisch_symbolic_coeffs_1130(): y = Symbol('y') assert heurisch_wrapper(1/(x**2 + y), x) == Piecewise( (log(x - sqrt(-y))/(2*sqrt(-y)) - log(x + sqrt(-y))/(2*sqrt(-y)), Ne(y, 0)), (-1/x, True)) y = Symbol('y', positive=True) assert heurisch_wrapper(1/(x**2 + y), x) == (atan(x/sqrt(y))/sqrt(y))
def test_heurisch_hacking(): assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \ x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14 assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \ x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \ sqrt(7)*asinh(sqrt(7)*x)/7 assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \ sqrt(7)*asin(sqrt(7)*x)/7
assert heurisch(exp(-7*x**2), x, hints=[]) == \ sqrt(7*pi)*erf(sqrt(7)*x)/14
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \ asin(x*Rational(2, 3))/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \ asinh(x*Rational(2, 3))/2
def test_heurisch_function(): assert heurisch(f(x), x) is None
@XFAILdef test_heurisch_function_derivative(): # TODO: it looks like this used to work just by coincindence and # thanks to sloppy implementation. Investigate why this used to # work at all and if support for this can be restored.
df = diff(f(x), x)
assert heurisch(f(x)*df, x) == f(x)**2/2 assert heurisch(f(x)**2*df, x) == f(x)**3/3 assert heurisch(df/f(x), x) == log(f(x))
def test_heurisch_wrapper(): f = 1/(y + x) assert heurisch_wrapper(f, x) == log(x + y) f = 1/(y - x) assert heurisch_wrapper(f, x) == -log(x - y) f = 1/((y - x)*(y + x)) assert heurisch_wrapper(f, x) == Piecewise( (-log(x - y)/(2*y) + log(x + y)/(2*y), Ne(y, 0)), (1/x, True)) # issue 6926 f = sqrt(x**2/((y - x)*(y + x))) assert heurisch_wrapper(f, x) == x*sqrt(-x**2/(x**2 - y**2)) \ - y**2*sqrt(-x**2/(x**2 - y**2))/x
def test_issue_3609(): assert heurisch(1/(x * (1 + log(x)**2)), x) == atan(log(x))
### These are examples from the Poor Man's Integrator### http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples/
def test_pmint_rat(): # TODO: heurisch() is off by a constant: -3/4. Possibly different permutation # would give the optimal result?
def drop_const(expr, x): if expr.is_Add: return Add(*[ arg for arg in expr.args if arg.has(x) ]) else: return expr
f = (x**7 - 24*x**4 - 4*x**2 + 8*x - 8)/(x**8 + 6*x**6 + 12*x**4 + 8*x**2) g = (4 + 8*x**2 + 6*x + 3*x**3)/(x**5 + 4*x**3 + 4*x) + log(x)
assert drop_const(ratsimp(heurisch(f, x)), x) == g
def test_pmint_trig(): f = (x - tan(x)) / tan(x)**2 + tan(x) g = -x**2/2 - x/tan(x) + log(tan(x)**2 + 1)/2
assert heurisch(f, x) == g
@slow # 8 seconds on 3.4 GHzdef test_pmint_logexp(): if ON_TRAVIS: # See https://github.com/sympy/sympy/pull/12795 skip("Too slow for travis.")
f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x g = log(x + exp(x) + log(x)) + 1/(x + exp(x) + log(x))
assert ratsimp(heurisch(f, x)) == g
def test_pmint_erf(): f = exp(-x**2)*erf(x)/(erf(x)**3 - erf(x)**2 - erf(x) + 1) g = sqrt(pi)*log(erf(x) - 1)/8 - sqrt(pi)*log(erf(x) + 1)/8 - sqrt(pi)/(4*erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_pmint_LambertW(): f = LambertW(x) g = x*LambertW(x) - x + x/LambertW(x)
assert heurisch(f, x) == g
def test_pmint_besselj(): f = besselj(nu + 1, x)/besselj(nu, x) g = nu*log(x) - log(besselj(nu, x))
assert heurisch(f, x) == g
f = (nu*besselj(nu, x) - x*besselj(nu + 1, x))/x g = besselj(nu, x)
assert heurisch(f, x) == g
f = jn(nu + 1, x)/jn(nu, x) g = nu*log(x) - log(jn(nu, x))
assert heurisch(f, x) == g
@slowdef test_pmint_bessel_products(): # Note: Derivatives of Bessel functions have many forms. # Recurrence relations are needed for comparisons. if ON_TRAVIS: skip("Too slow for travis.")
f = x*besselj(nu, x)*bessely(nu, 2*x) g = -2*x*besselj(nu, x)*bessely(nu - 1, 2*x)/3 + x*besselj(nu - 1, x)*bessely(nu, 2*x)/3
assert heurisch(f, x) == g
f = x*besselj(nu, x)*besselk(nu, 2*x) g = -2*x*besselj(nu, x)*besselk(nu - 1, 2*x)/5 - x*besselj(nu - 1, x)*besselk(nu, 2*x)/5
assert heurisch(f, x) == g
@slow # 110 seconds on 3.4 GHzdef test_pmint_WrightOmega(): if ON_TRAVIS: skip("Too slow for travis.") def omega(x): return LambertW(exp(x))
f = (1 + omega(x) * (2 + cos(omega(x)) * (x + omega(x))))/(1 + omega(x))/(x + omega(x)) g = log(x + LambertW(exp(x))) + sin(LambertW(exp(x)))
assert heurisch(f, x) == g
def test_RR(): # Make sure the algorithm does the right thing if the ring is RR. See # issue 8685. assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \ 0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
# TODO: convert the rest of PMINT tests:# Airy functions# f = (x - AiryAi(x)*AiryAi(1, x)) / (x**2 - AiryAi(x)**2)# g = Rational(1,2)*ln(x + AiryAi(x)) + Rational(1,2)*ln(x - AiryAi(x))# f = x**2 * AiryAi(x)# g = -AiryAi(x) + AiryAi(1, x)*x# Whittaker functions# f = WhittakerW(mu + 1, nu, x) / (WhittakerW(mu, nu, x) * x)# g = x/2 - mu*ln(x) - ln(WhittakerW(mu, nu, x))
def test_issue_22527(): t, R = symbols(r't R') z = Function('z')(t) def f(x): return x/sqrt(R**2 - x**2) Uz = integrate(f(z), z) Ut = integrate(f(t), t) assert Ut == Uz.subs(z, t)
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