173 lines
5.2 KiB
Python
173 lines
5.2 KiB
Python
from sympy.core.numbers import (I, Rational)
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from sympy.core.singleton import S
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from sympy.core.symbol import (Dummy, symbols)
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from sympy.functions.elementary.exponential import log
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import atan
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from sympy.integrals.integrals import integrate
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from sympy.polys.polytools import Poly
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from sympy.simplify.simplify import simplify
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from sympy.integrals.rationaltools import ratint, ratint_logpart, log_to_atan
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from sympy.abc import a, b, x, t
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half = S.Half
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def test_ratint():
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assert ratint(S.Zero, x) == 0
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assert ratint(S(7), x) == 7*x
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assert ratint(x, x) == x**2/2
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assert ratint(2*x, x) == x**2
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assert ratint(-2*x, x) == -x**2
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assert ratint(8*x**7 + 2*x + 1, x) == x**8 + x**2 + x
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f = S.One
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g = x + 1
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assert ratint(f / g, x) == log(x + 1)
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assert ratint((f, g), x) == log(x + 1)
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f = x**3 - x
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g = x - 1
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assert ratint(f/g, x) == x**3/3 + x**2/2
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f = x
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g = (x - a)*(x + a)
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assert ratint(f/g, x) == log(x**2 - a**2)/2
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f = S.One
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g = x**2 + 1
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assert ratint(f/g, x, real=None) == atan(x)
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assert ratint(f/g, x, real=True) == atan(x)
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assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2
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f = S(36)
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g = x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2
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assert ratint(f/g, x) == \
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-4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1)
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f = x**4 - 3*x**2 + 6
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g = x**6 - 5*x**4 + 5*x**2 + 4
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assert ratint(f/g, x) == \
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atan(x) + atan(x**3) + atan(x/2 - Rational(3, 2)*x**3 + S.Half*x**5)
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f = x**7 - 24*x**4 - 4*x**2 + 8*x - 8
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g = x**8 + 6*x**6 + 12*x**4 + 8*x**2
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assert ratint(f/g, x) == \
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(4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)
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assert ratint((x**3*f)/(x*g), x) == \
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-(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
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5*sqrt(2)*atan(x*sqrt(2)/2) + S.Half*x**2 - 3*log(2 + x**2)
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f = x**5 - x**4 + 4*x**3 + x**2 - x + 5
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g = x**4 - 2*x**3 + 5*x**2 - 4*x + 4
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assert ratint(f/g, x) == \
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x + S.Half*x**2 + S.Half*log(2 - x + x**2) + (9 - 4*x)/(7*x**2 - 7*x + 14) + \
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13*sqrt(7)*atan(Rational(-1, 7)*sqrt(7) + 2*x*sqrt(7)/7)/49
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assert ratint(1/(x**2 + x + 1), x) == \
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2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3
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assert ratint(1/(x**3 + 1), x) == \
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-log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3)
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/3 + 2*x*sqrt(3)/3)/3
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assert ratint(1/(x**2 + x + 1), x, real=False) == \
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-I*3**half*log(half + x - half*I*3**half)/3 + \
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I*3**half*log(half + x + half*I*3**half)/3
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assert ratint(1/(x**3 + 1), x, real=False) == log(1 + x)/3 + \
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(Rational(-1, 6) + I*3**half/6)*log(-half + x + I*3**half/2) + \
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(Rational(-1, 6) - I*3**half/6)*log(-half + x - I*3**half/2)
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# issue 4991
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assert ratint(1/(x*(a + b*x)**3), x) == \
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(3*a + 2*b*x)/(2*a**4 + 4*a**3*b*x + 2*a**2*b**2*x**2) + (
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log(x) - log(a/b + x))/a**3
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assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2
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assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2
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assert ratint((x/4 - 4/(1 - x)).diff(x), x) == x/4 + 4/(x - 1)
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ans = atan(x)
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assert ratint(1/(x**2 + 1), x, symbol=x) == ans
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assert ratint(1/(x**2 + 1), x, symbol='x') == ans
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assert ratint(1/(x**2 + 1), x, symbol=a) == ans
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# this asserts that as_dummy must return a unique symbol
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# even if the symbol is already a Dummy
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d = Dummy()
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assert ratint(1/(d**2 + 1), d, symbol=d) == atan(d)
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def test_ratint_logpart():
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assert ratint_logpart(x, x**2 - 9, x, t) == \
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[(Poly(x**2 - 9, x), Poly(-2*t + 1, t))]
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assert ratint_logpart(x**2, x**3 - 5, x, t) == \
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[(Poly(x**3 - 5, x), Poly(-3*t + 1, t))]
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def test_issue_5414():
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assert ratint(1/(x**2 + 16), x) == atan(x/4)/4
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def test_issue_5249():
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assert ratint(
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1/(x**2 + a**2), x) == (-I*log(-I*a + x)/2 + I*log(I*a + x)/2)/a
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def test_issue_5817():
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a, b, c = symbols('a,b,c', positive=True)
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assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
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sqrt(a)*atan(sqrt(
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b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))
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def test_issue_5981():
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u = symbols('u')
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assert integrate(1/(u**2 + 1)) == atan(u)
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def test_issue_10488():
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a,b,c,x = symbols('a b c x', positive=True)
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assert integrate(x/(a*x+b),x) == x/a - b*log(a*x + b)/a**2
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def test_issues_8246_12050_13501_14080():
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a = symbols('a', nonzero=True)
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assert integrate(a/(x**2 + a**2), x) == atan(x/a)
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assert integrate(1/(x**2 + a**2), x) == atan(x/a)/a
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assert integrate(1/(1 + a**2*x**2), x) == atan(a*x)/a
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def test_issue_6308():
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k, a0 = symbols('k a0', real=True)
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assert integrate((x**2 + 1 - k**2)/(x**2 + 1 + a0**2), x) == \
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x - (a0**2 + k**2)*atan(x/sqrt(a0**2 + 1))/sqrt(a0**2 + 1)
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def test_issue_5907():
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a = symbols('a', nonzero=True)
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assert integrate(1/(x**2 + a**2)**2, x) == \
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x/(2*a**4 + 2*a**2*x**2) + atan(x/a)/(2*a**3)
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def test_log_to_atan():
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f, g = (Poly(x + S.Half, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX'))
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fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
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assert log_to_atan(f, g) == fg_ans
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assert log_to_atan(g, f) == -fg_ans
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