def test_integrate_nonlinear_no_specials():
a, d, = (
Poly(
x ** 2 * t ** 5
+ x * t ** 4
- nu ** 2 * t ** 3
- x * (x ** 2 + 1) * t ** 2
- (x ** 2 - nu ** 2) * t
- x ** 5 / 4,
t,
),
Poly(
x ** 2 * t ** 4
+ x ** 2 * (x ** 2 + 2) * t ** 2
+ x ** 2
+ x ** 4
+ x ** 6 / 4,
t,
),
)
# f(x) == phi_nu(x), the logarithmic derivative of J_v, the Bessel function,
# which has no specials (see Chapter 5, note 4 of Bronstein's book).
f = Function("phi_nu")
DE = DifferentialExtension(
extension={
"D": [Poly(1, x), Poly(-(t ** 2) - t / x - (1 - nu ** 2 / x ** 2), t)],
"Tfuncs": [f],
}
)
assert integrate_nonlinear_no_specials(a, d, DE) == (
-log(1 + f(x) ** 2 + x ** 2 / 2) / 2
- (4 + x ** 2) / (4 + 2 * x ** 2 + 4 * f(x) ** 2),
True,
)
assert integrate_nonlinear_no_specials(Poly(t, t), Poly(1, t), DE) == (0, False)
def test_integrate_nonlinear_no_specials():
a, d, = Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 -
nu**2)*t - x**5/4, t), Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t)
# f(x) == phi_nu(x), the logarithmic derivative of J_v, the Bessel function,
# which has no specials (see Chapter 5, note 4 of Bronstein's book).
f = Function('phi_nu')
DE = DifferentialExtension(extension={'D': [Poly(1, x),
Poly(-t**2 - t/x - (1 - nu**2/x**2), t)], 'Tfuncs': [f]})
assert integrate_nonlinear_no_specials(a, d, DE) == \
(-log(1 + f(x)**2 + x**2/2)/2 - (4 + x**2)/(4 + 2*x**2 + 4*f(x)**2), True)
assert integrate_nonlinear_no_specials(Poly(t, t), Poly(1, t), DE) == \
(0, False)
