MTtranslateService/Lib/site-packages/sympy/integrals/tests/test_failing_integrals.py

# A collection of failing integrals from the issues.

from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (csch, sech, sinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, atan, cos, sec, sin, tan)
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import (Integral, integrate)


from sympy.testing.pytest import XFAIL, SKIP, slow, skip, ON_TRAVIS

from sympy.abc import x, k, c, y, b, h, a, m, z, n, t


@SKIP("Too slow for @slow")
@XFAIL
def test_issue_3880():
    # integrate_hyperexponential(Poly(t*2*(1 - t0**2)*t0*(x**3 + x**2), t), Poly((1 + t0**2)**2*2*(x**2 + x + 1), t), [Poly(1, x), Poly(1 + t0**2, t0), Poly(t, t)], [x, t0, t], [exp, tan])
    assert not integrate(exp(x)*cos(2*x)*sin(2*x) * (x**3 + x**2)/(2*(x**2 + x + 1)), x).has(Integral)


@XFAIL
def test_issue_4212():
    assert not integrate(sign(x), x).has(Integral)


@XFAIL
def test_issue_4491():
    # Can be solved via variable transformation x = y - 1
    assert not integrate(x*sqrt(x**2 + 2*x + 4), x).has(Integral)


@XFAIL
def test_issue_4511():
    # This works, but gives a complicated answer.  The correct answer is x - cos(x).
    # If current answer is simplified, 1 - cos(x) + x is obtained.
    # The last one is what Maple gives.  It is also quite slow.
    assert integrate(cos(x)**2 / (1 - sin(x))) in [x - cos(x), 1 - cos(x) + x,
            -2/(tan((S.Half)*x)**2 + 1) + x]


@XFAIL
def test_integrate_DiracDelta_fails():
    # issue 6427
    assert integrate(integrate(integrate(
        DiracDelta(x - y - z), (z, 0, oo)), (y, 0, 1)), (x, 0, 1)) == S.Half


@XFAIL
@slow
def test_issue_4525():
    # Warning: takes a long time
    assert not integrate((x**m * (1 - x)**n * (a + b*x + c*x**2))/(1 + x**2), (x, 0, 1)).has(Integral)



@XFAIL
@slow
def test_issue_4540():
    if ON_TRAVIS:
        skip("Too slow for travis.")
    # Note, this integral is probably nonelementary
    assert not integrate(
        (sin(1/x) - x*exp(x)) /
        ((-sin(1/x) + x*exp(x))*x + x*sin(1/x)), x).has(Integral)


@XFAIL
@slow
def test_issue_4891():
    # Requires the hypergeometric function.
    assert not integrate(cos(x)**y, x).has(Integral)


@XFAIL
@slow
def test_issue_1796a():
    assert not integrate(exp(2*b*x)*exp(-a*x**2), x).has(Integral)


@XFAIL
def test_issue_4895b():
    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, 0)).has(Integral)


@XFAIL
def test_issue_4895c():
    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, oo)).has(Integral)


@XFAIL
def test_issue_4895d():
    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, 0, oo)).has(Integral)


@XFAIL
@slow
def test_issue_4941():
    if ON_TRAVIS:
        skip("Too slow for travis.")
    assert not integrate(sqrt(1 + sinh(x/20)**2), (x, -25, 25)).has(Integral)


@XFAIL
def test_issue_4992():
    # Nonelementary integral.  Requires hypergeometric/Meijer-G handling.
    assert not integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k), (x, 0, oo)).has(Integral)


@XFAIL
def test_issue_16396a():
    i = integrate(1/(1+sqrt(tan(x))), (x, pi/3, pi/6))
    assert not i.has(Integral)


@XFAIL
def test_issue_16396b():
    i = integrate(x*sin(x)/(1+cos(x)**2), (x, 0, pi))
    assert not i.has(Integral)


@XFAIL
def test_issue_16161():
    i = integrate(x*sec(x)**2, x)
    assert not i.has(Integral)
    # assert i == x*tan(x) + log(cos(x))


@XFAIL
def test_issue_16046():
    assert integrate(exp(exp(I*x)), [x, 0, 2*pi]) == 2*pi


@XFAIL
def test_issue_15925a():
    assert not integrate(sqrt((1+sin(x))**2+(cos(x))**2), (x, -pi/2, pi/2)).has(Integral)


@XFAIL
@slow
def test_issue_15925b():
    if ON_TRAVIS:
        skip("Too slow for travis.")
    assert not integrate(sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2),
                         (x, 0, pi/6)).has(Integral)


@XFAIL
def test_issue_15925b_manual():
    assert not integrate(sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2),
                         (x, 0, pi/6), manual=True).has(Integral)


@XFAIL
@slow
def test_issue_15227():
    if ON_TRAVIS:
        skip("Too slow for travis.")
    i = integrate(log(1-x)*log((1+x)**2)/x, (x, 0, 1))
    assert not i.has(Integral)
    # assert i == -5*zeta(3)/4


@XFAIL
@slow
def test_issue_14716():
    i = integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1))
    assert not i.has(Integral)
    # Mathematica can not solve it either, but
    # integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1)).transform(x, y - 5).doit()
    # works
    # assert i == -log(Rational(11, 2))/pi - Si(pi*Rational(11, 2))/pi + Si(6*pi)/pi


@XFAIL
def test_issue_14709a():
    i = integrate(x*acos(1 - 2*x/h), (x, 0, h))
    assert not i.has(Integral)
    # assert i == 5*h**2*pi/16


@slow
@XFAIL
def test_issue_14398():
    assert not integrate(exp(x**2)*cos(x), x).has(Integral)


@XFAIL
def test_issue_14074():
    i = integrate(log(sin(x)), (x, 0, pi/2))
    assert not i.has(Integral)
    # assert i == -pi*log(2)/2


@XFAIL
@slow
def test_issue_14078b():
    i = integrate((atan(4*x)-atan(2*x))/x, (x, 0, oo))
    assert not i.has(Integral)
    # assert i == pi*log(2)/2


@XFAIL
def test_issue_13792():
    i =  integrate(log(1/x) / (1 - x), (x, 0, 1))
    assert not i.has(Integral)
    # assert i in [polylog(2, -exp_polar(I*pi)), pi**2/6]


@XFAIL
def test_issue_11845a():
    assert not integrate(exp(y - x**3), (x, 0, 1)).has(Integral)


@XFAIL
def test_issue_11845b():
    assert not integrate(exp(-y - x**3), (x, 0, 1)).has(Integral)


@XFAIL
def test_issue_11813():
    assert not integrate((a - x)**Rational(-1, 2)*x, (x, 0, a)).has(Integral)


@XFAIL
def test_issue_11742():
    i =  integrate(sqrt(-x**2 + 8*x + 48), (x, 4, 12))
    assert not i.has(Integral)
    # assert i == 16*pi


@XFAIL
def test_issue_11254a():
    assert not integrate(sech(x), (x, 0, 1)).has(Integral)


@XFAIL
def test_issue_11254b():
    assert not integrate(csch(x), (x, 0, 1)).has(Integral)


@XFAIL
def test_issue_10584():
    assert not integrate(sqrt(x**2 + 1/x**2), x).has(Integral)


@XFAIL
def test_issue_9723():
    assert not integrate(sqrt(x + sqrt(x))).has(Integral)


@XFAIL
def test_issue_9101():
    assert not integrate(log(x + sqrt(x**2 + y**2 + z**2)), z).has(Integral)


@XFAIL
def test_issue_7264():
    assert not integrate(exp(x)*sqrt(1 + exp(2*x))).has(Integral)


@XFAIL
def test_issue_7147():
    assert not integrate(x/sqrt(a*x**2 + b*x + c)**3, x).has(Integral)


@XFAIL
def test_issue_7109():
    assert not integrate(sqrt(a**2/(a**2 - x**2)), x).has(Integral)


@XFAIL
def test_integrate_Piecewise_rational_over_reals():
    f = Piecewise(
        (0,                                              t - 478.515625*pi <  0),
        (13.2075145209219*pi/(0.000871222*t + 0.995)**2, t - 478.515625*pi >= 0))

    assert abs((integrate(f, (t, 0, oo)) - 15235.9375*pi).evalf()) <= 1e-7


@XFAIL
def test_issue_4311_slow():
    # Not slow when bypassing heurish
    assert not integrate(x*abs(9-x**2), x).has(Integral)

@XFAIL
def test_issue_20370():
    a = symbols('a', positive=True)
    assert integrate((1 + a * cos(x))**-1, (x, 0, 2 * pi)) == (2 * pi / sqrt(1 - a**2))