def test_find_substitutions():
assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
[(cot(x), 1, -u**6 - 2*u**4 - u**2)]
assert find_substitutions(
(sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)), x,
u) == [(sec(x) + tan(x), 1, 1 / u)]
assert find_substitutions(x * exp(-x**2), x,
u) == [(-x**2, -S.Half, exp(u))]
def test_find_substitutions():
assert find_substitutions(
(cot(x) ** 2 + 1) ** 2 * csc(x) ** 2 * cot(x) ** 2, x, u
) == [(cot(x), 1, -(u ** 6) - 2 * u ** 4 - u ** 2)]
assert find_substitutions(
(sec(x) ** 2 + tan(x) * sec(x)) / (sec(x) + tan(x)), x, u
) == [(sec(x) + tan(x), 1, 1 / u)]
assert find_substitutions(x * exp(-(x ** 2)), x, u) == [
(-(x ** 2), Rational(-1, 2), exp(u))
]
def test_find_substitutions():
assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
[(cot(x), 1, -u**6 - 2*u**4 - u**2)]
assert find_substitutions((sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)),
x, u) == [(sec(x) + tan(x), 1, 1/u)]
assert find_substitutions(x * exp(-x**2), x, u) == [(-x**2, -S.Half, exp(u))]
