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368 lines
12 KiB
368 lines
12 KiB
from sympy.concrete.summations import Sum
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from sympy.core.add import Add
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from sympy.core.function import (Derivative, Function, diff)
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from sympy.core.numbers import (I, Rational, pi)
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from sympy.core.relational import Ne
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.functions.elementary.exponential import (LambertW, exp, log)
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from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh, tanh)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.piecewise import Piecewise
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from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan)
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from sympy.functions.special.bessel import (besselj, besselk, bessely, jn)
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from sympy.functions.special.error_functions import erf
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from sympy.integrals.integrals import Integral
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from sympy.simplify.ratsimp import ratsimp
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from sympy.simplify.simplify import simplify
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from sympy.integrals.heurisch import components, heurisch, heurisch_wrapper
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from sympy.testing.pytest import XFAIL, skip, slow, ON_CI
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from sympy.integrals.integrals import integrate
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x, y, z, nu = symbols('x,y,z,nu')
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f = Function('f')
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def test_components():
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assert components(x*y, x) == {x}
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assert components(1/(x + y), x) == {x}
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assert components(sin(x), x) == {sin(x), x}
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assert components(sin(x)*sqrt(log(x)), x) == \
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{log(x), sin(x), sqrt(log(x)), x}
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assert components(x*sin(exp(x)*y), x) == \
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{sin(y*exp(x)), x, exp(x)}
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assert components(x**Rational(17, 54)/sqrt(sin(x)), x) == \
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{sin(x), x**Rational(1, 54), sqrt(sin(x)), x}
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assert components(f(x), x) == \
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{x, f(x)}
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assert components(Derivative(f(x), x), x) == \
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{x, f(x), Derivative(f(x), x)}
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assert components(f(x)*diff(f(x), x), x) == \
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{x, f(x), Derivative(f(x), x), Derivative(f(x), x)}
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def test_issue_10680():
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assert isinstance(integrate(x**log(x**log(x**log(x))),x), Integral)
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def test_issue_21166():
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assert integrate(sin(x/sqrt(abs(x))), (x, -1, 1)) == 0
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def test_heurisch_polynomials():
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assert heurisch(1, x) == x
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assert heurisch(x, x) == x**2/2
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assert heurisch(x**17, x) == x**18/18
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# For coverage
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assert heurisch_wrapper(y, x) == y*x
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def test_heurisch_fractions():
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assert heurisch(1/x, x) == log(x)
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assert heurisch(1/(2 + x), x) == log(x + 2)
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assert heurisch(1/(x + sin(y)), x) == log(x + sin(y))
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# Up to a constant, where C = pi*I*Rational(5, 12), Mathematica gives identical
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# result in the first case. The difference is because SymPy changes
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# signs of expressions without any care.
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# XXX ^ ^ ^ is this still correct?
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assert heurisch(5*x**5/(
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2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12]
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assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
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assert heurisch(1/x**2, x) == -1/x
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assert heurisch(-1/x**5, x) == 1/(4*x**4)
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def test_heurisch_log():
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assert heurisch(log(x), x) == x*log(x) - x
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assert heurisch(log(3*x), x) == -x + x*log(3) + x*log(x)
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assert heurisch(log(x**2), x) in [x*log(x**2) - 2*x, 2*x*log(x) - 2*x]
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def test_heurisch_exp():
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assert heurisch(exp(x), x) == exp(x)
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assert heurisch(exp(-x), x) == -exp(-x)
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assert heurisch(exp(17*x), x) == exp(17*x) / 17
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assert heurisch(x*exp(x), x) == x*exp(x) - exp(x)
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assert heurisch(x*exp(x**2), x) == exp(x**2) / 2
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assert heurisch(exp(-x**2), x) is None
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assert heurisch(2**x, x) == 2**x/log(2)
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assert heurisch(x*2**x, x) == x*2**x/log(2) - 2**x*log(2)**(-2)
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assert heurisch(Integral(x**z*y, (y, 1, 2), (z, 2, 3)).function, x) == (x*x**z*y)/(z+1)
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assert heurisch(Sum(x**z, (z, 1, 2)).function, z) == x**z/log(x)
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# https://github.com/sympy/sympy/issues/23707
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anti = -exp(z)/(sqrt(x - y)*exp(z*sqrt(x - y)) - exp(z*sqrt(x - y)))
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assert heurisch(exp(z)*exp(-z*sqrt(x - y)), z) == anti
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def test_heurisch_trigonometric():
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assert heurisch(sin(x), x) == -cos(x)
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assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
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assert heurisch(cos(x), x) == sin(x)
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assert heurisch(tan(x), x) in [
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log(1 + tan(x)**2)/2,
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log(tan(x) + I) + I*x,
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log(tan(x) - I) - I*x,
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]
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assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
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assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
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# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
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assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
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assert heurisch(cos(x)/sin(x), x) == log(sin(x))
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assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
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assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
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2*sin(x) + 2*x*cos(x))
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assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
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+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
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assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723
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assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3
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assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x)
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- 1) - atan(sqrt(2)*sin(x) + 1)
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assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
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def test_heurisch_hyperbolic():
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assert heurisch(sinh(x), x) == cosh(x)
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assert heurisch(cosh(x), x) == sinh(x)
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assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
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assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
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assert heurisch(
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x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4 + x**2)/4
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def test_heurisch_mixed():
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assert heurisch(sin(x)*exp(x), x) == exp(x)*sin(x)/2 - exp(x)*cos(x)/2
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assert heurisch(sin(x/sqrt(-x)), x) == 2*x*cos(x/sqrt(-x))/sqrt(-x) - 2*sin(x/sqrt(-x))
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def test_heurisch_radicals():
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assert heurisch(1/sqrt(x), x) == 2*sqrt(x)
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assert heurisch(1/sqrt(x)**3, x) == -2/sqrt(x)
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assert heurisch(sqrt(x)**3, x) == 2*sqrt(x)**5/5
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assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*sqrt(cos(x))**3/3
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y = Symbol('y')
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assert heurisch(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
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2*sqrt(x)*cos(y*sqrt(x))/y
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assert heurisch_wrapper(sin(y*sqrt(x)), x) == Piecewise(
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(-2*sqrt(x)*cos(sqrt(x)*y)/y + 2*sin(sqrt(x)*y)/y**2, Ne(y, 0)),
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(0, True))
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y = Symbol('y', positive=True)
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assert heurisch_wrapper(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
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2*sqrt(x)*cos(y*sqrt(x))/y
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def test_heurisch_special():
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assert heurisch(erf(x), x) == x*erf(x) + exp(-x**2)/sqrt(pi)
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assert heurisch(exp(-x**2)*erf(x), x) == sqrt(pi)*erf(x)**2 / 4
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def test_heurisch_symbolic_coeffs():
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assert heurisch(1/(x + y), x) == log(x + y)
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assert heurisch(1/(x + sqrt(2)), x) == log(x + sqrt(2))
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assert simplify(diff(heurisch(log(x + y + z), y), y)) == log(x + y + z)
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def test_heurisch_symbolic_coeffs_1130():
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y = Symbol('y')
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assert heurisch_wrapper(1/(x**2 + y), x) == Piecewise(
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(log(x - sqrt(-y))/(2*sqrt(-y)) - log(x + sqrt(-y))/(2*sqrt(-y)),
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Ne(y, 0)), (-1/x, True))
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y = Symbol('y', positive=True)
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assert heurisch_wrapper(1/(x**2 + y), x) == (atan(x/sqrt(y))/sqrt(y))
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def test_heurisch_hacking():
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assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
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x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
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assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
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x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
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assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
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sqrt(7)*asinh(sqrt(7)*x)/7
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assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
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sqrt(7)*asin(sqrt(7)*x)/7
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assert heurisch(exp(-7*x**2), x, hints=[]) == \
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sqrt(7*pi)*erf(sqrt(7)*x)/14
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assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
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asin(x*Rational(2, 3))/2
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assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
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asinh(x*Rational(2, 3))/2
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assert heurisch(1/sqrt(3*x**2-4), x, hints=[]) == \
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sqrt(3)*log(3*x + sqrt(3)*sqrt(3*x**2 - 4))/3
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def test_heurisch_function():
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assert heurisch(f(x), x) is None
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@XFAIL
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def test_heurisch_function_derivative():
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# TODO: it looks like this used to work just by coincindence and
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# thanks to sloppy implementation. Investigate why this used to
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# work at all and if support for this can be restored.
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df = diff(f(x), x)
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assert heurisch(f(x)*df, x) == f(x)**2/2
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assert heurisch(f(x)**2*df, x) == f(x)**3/3
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assert heurisch(df/f(x), x) == log(f(x))
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def test_heurisch_wrapper():
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f = 1/(y + x)
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assert heurisch_wrapper(f, x) == log(x + y)
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f = 1/(y - x)
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assert heurisch_wrapper(f, x) == -log(x - y)
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f = 1/((y - x)*(y + x))
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assert heurisch_wrapper(f, x) == Piecewise(
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(-log(x - y)/(2*y) + log(x + y)/(2*y), Ne(y, 0)), (1/x, True))
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# issue 6926
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f = sqrt(x**2/((y - x)*(y + x)))
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assert heurisch_wrapper(f, x) == x*sqrt(-x**2/(x**2 - y**2)) \
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- y**2*sqrt(-x**2/(x**2 - y**2))/x
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def test_issue_3609():
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assert heurisch(1/(x * (1 + log(x)**2)), x) == atan(log(x))
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### These are examples from the Poor Man's Integrator
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### http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples/
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def test_pmint_rat():
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# TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
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# would give the optimal result?
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def drop_const(expr, x):
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if expr.is_Add:
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return Add(*[ arg for arg in expr.args if arg.has(x) ])
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else:
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return expr
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f = (x**7 - 24*x**4 - 4*x**2 + 8*x - 8)/(x**8 + 6*x**6 + 12*x**4 + 8*x**2)
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g = (4 + 8*x**2 + 6*x + 3*x**3)/(x**5 + 4*x**3 + 4*x) + log(x)
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assert drop_const(ratsimp(heurisch(f, x)), x) == g
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def test_pmint_trig():
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f = (x - tan(x)) / tan(x)**2 + tan(x)
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g = -x**2/2 - x/tan(x) + log(tan(x)**2 + 1)/2
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assert heurisch(f, x) == g
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@slow # 8 seconds on 3.4 GHz
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def test_pmint_logexp():
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if ON_CI:
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# See https://github.com/sympy/sympy/pull/12795
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skip("Too slow for CI.")
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f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x
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g = log(x + exp(x) + log(x)) + 1/(x + exp(x) + log(x))
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assert ratsimp(heurisch(f, x)) == g
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def test_pmint_erf():
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f = exp(-x**2)*erf(x)/(erf(x)**3 - erf(x)**2 - erf(x) + 1)
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g = sqrt(pi)*log(erf(x) - 1)/8 - sqrt(pi)*log(erf(x) + 1)/8 - sqrt(pi)/(4*erf(x) - 4)
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assert ratsimp(heurisch(f, x)) == g
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def test_pmint_LambertW():
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f = LambertW(x)
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g = x*LambertW(x) - x + x/LambertW(x)
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assert heurisch(f, x) == g
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def test_pmint_besselj():
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f = besselj(nu + 1, x)/besselj(nu, x)
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g = nu*log(x) - log(besselj(nu, x))
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assert heurisch(f, x) == g
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f = (nu*besselj(nu, x) - x*besselj(nu + 1, x))/x
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g = besselj(nu, x)
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assert heurisch(f, x) == g
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f = jn(nu + 1, x)/jn(nu, x)
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g = nu*log(x) - log(jn(nu, x))
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assert heurisch(f, x) == g
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@slow
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def test_pmint_bessel_products():
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# Note: Derivatives of Bessel functions have many forms.
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# Recurrence relations are needed for comparisons.
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if ON_CI:
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skip("Too slow for CI.")
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f = x*besselj(nu, x)*bessely(nu, 2*x)
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g = -2*x*besselj(nu, x)*bessely(nu - 1, 2*x)/3 + x*besselj(nu - 1, x)*bessely(nu, 2*x)/3
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assert heurisch(f, x) == g
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f = x*besselj(nu, x)*besselk(nu, 2*x)
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g = -2*x*besselj(nu, x)*besselk(nu - 1, 2*x)/5 - x*besselj(nu - 1, x)*besselk(nu, 2*x)/5
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assert heurisch(f, x) == g
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@slow # 110 seconds on 3.4 GHz
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def test_pmint_WrightOmega():
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if ON_CI:
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skip("Too slow for CI.")
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def omega(x):
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return LambertW(exp(x))
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f = (1 + omega(x) * (2 + cos(omega(x)) * (x + omega(x))))/(1 + omega(x))/(x + omega(x))
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g = log(x + LambertW(exp(x))) + sin(LambertW(exp(x)))
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assert heurisch(f, x) == g
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def test_RR():
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# Make sure the algorithm does the right thing if the ring is RR. See
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# issue 8685.
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assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \
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0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
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# TODO: convert the rest of PMINT tests:
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# Airy functions
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# f = (x - AiryAi(x)*AiryAi(1, x)) / (x**2 - AiryAi(x)**2)
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# g = Rational(1,2)*ln(x + AiryAi(x)) + Rational(1,2)*ln(x - AiryAi(x))
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# f = x**2 * AiryAi(x)
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# g = -AiryAi(x) + AiryAi(1, x)*x
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# Whittaker functions
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# f = WhittakerW(mu + 1, nu, x) / (WhittakerW(mu, nu, x) * x)
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# g = x/2 - mu*ln(x) - ln(WhittakerW(mu, nu, x))
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def test_issue_22527():
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t, R = symbols(r't R')
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z = Function('z')(t)
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def f(x):
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return x/sqrt(R**2 - x**2)
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Uz = integrate(f(z), z)
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Ut = integrate(f(t), t)
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assert Ut == Uz.subs(z, t)
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