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Current File : //lib/python3/dist-packages/sympy/integrals/tests/test_failing_integrals.py

# A collection of failing integrals from the issues.

from sympy import (
    integrate, I, Integral, exp, oo, pi, sign, sqrt, sin, cos, Piecewise,
    tan, S, log, gamma, sinh, sec, acos, atan, sech, csch, DiracDelta, Rational,
    symbols
)

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))

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