Spaces:
Sleeping
Sleeping
| from sympy.core.numbers import (I, Rational) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import (Dummy, symbols) | |
| from sympy.functions.elementary.exponential import log | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import atan | |
| from sympy.integrals.integrals import integrate | |
| from sympy.polys.polytools import Poly | |
| from sympy.simplify.simplify import simplify | |
| from sympy.integrals.rationaltools import ratint, ratint_logpart, log_to_atan | |
| from sympy.abc import a, b, x, t | |
| half = S.Half | |
| def test_ratint(): | |
| assert ratint(S.Zero, x) == 0 | |
| assert ratint(S(7), x) == 7*x | |
| assert ratint(x, x) == x**2/2 | |
| assert ratint(2*x, x) == x**2 | |
| assert ratint(-2*x, x) == -x**2 | |
| assert ratint(8*x**7 + 2*x + 1, x) == x**8 + x**2 + x | |
| f = S.One | |
| g = x + 1 | |
| assert ratint(f / g, x) == log(x + 1) | |
| assert ratint((f, g), x) == log(x + 1) | |
| f = x**3 - x | |
| g = x - 1 | |
| assert ratint(f/g, x) == x**3/3 + x**2/2 | |
| f = x | |
| g = (x - a)*(x + a) | |
| assert ratint(f/g, x) == log(x**2 - a**2)/2 | |
| f = S.One | |
| g = x**2 + 1 | |
| assert ratint(f/g, x, real=None) == atan(x) | |
| assert ratint(f/g, x, real=True) == atan(x) | |
| assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2 | |
| f = S(36) | |
| g = x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2 | |
| assert ratint(f/g, x) == \ | |
| -4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1) | |
| f = x**4 - 3*x**2 + 6 | |
| g = x**6 - 5*x**4 + 5*x**2 + 4 | |
| assert ratint(f/g, x) == \ | |
| atan(x) + atan(x**3) + atan(x/2 - Rational(3, 2)*x**3 + S.Half*x**5) | |
| f = x**7 - 24*x**4 - 4*x**2 + 8*x - 8 | |
| g = x**8 + 6*x**6 + 12*x**4 + 8*x**2 | |
| assert ratint(f/g, x) == \ | |
| (4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x) | |
| assert ratint((x**3*f)/(x*g), x) == \ | |
| -(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \ | |
| 5*sqrt(2)*atan(x*sqrt(2)/2) + S.Half*x**2 - 3*log(2 + x**2) | |
| f = x**5 - x**4 + 4*x**3 + x**2 - x + 5 | |
| g = x**4 - 2*x**3 + 5*x**2 - 4*x + 4 | |
| assert ratint(f/g, x) == \ | |
| x + S.Half*x**2 + S.Half*log(2 - x + x**2) + (9 - 4*x)/(7*x**2 - 7*x + 14) + \ | |
| 13*sqrt(7)*atan(Rational(-1, 7)*sqrt(7) + 2*x*sqrt(7)/7)/49 | |
| assert ratint(1/(x**2 + x + 1), x) == \ | |
| 2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3 | |
| assert ratint(1/(x**3 + 1), x) == \ | |
| -log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3) | |
| /3 + 2*x*sqrt(3)/3)/3 | |
| assert ratint(1/(x**2 + x + 1), x, real=False) == \ | |
| -I*3**half*log(half + x - half*I*3**half)/3 + \ | |
| I*3**half*log(half + x + half*I*3**half)/3 | |
| assert ratint(1/(x**3 + 1), x, real=False) == log(1 + x)/3 + \ | |
| (Rational(-1, 6) + I*3**half/6)*log(-half + x + I*3**half/2) + \ | |
| (Rational(-1, 6) - I*3**half/6)*log(-half + x - I*3**half/2) | |
| # issue 4991 | |
| assert ratint(1/(x*(a + b*x)**3), x) == \ | |
| (3*a + 2*b*x)/(2*a**4 + 4*a**3*b*x + 2*a**2*b**2*x**2) + ( | |
| log(x) - log(a/b + x))/a**3 | |
| assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2 | |
| assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2 | |
| assert ratint((x/4 - 4/(1 - x)).diff(x), x) == x/4 + 4/(x - 1) | |
| ans = atan(x) | |
| assert ratint(1/(x**2 + 1), x, symbol=x) == ans | |
| assert ratint(1/(x**2 + 1), x, symbol='x') == ans | |
| assert ratint(1/(x**2 + 1), x, symbol=a) == ans | |
| # this asserts that as_dummy must return a unique symbol | |
| # even if the symbol is already a Dummy | |
| d = Dummy() | |
| assert ratint(1/(d**2 + 1), d, symbol=d) == atan(d) | |
| def test_ratint_logpart(): | |
| assert ratint_logpart(x, x**2 - 9, x, t) == \ | |
| [(Poly(x**2 - 9, x), Poly(-2*t + 1, t))] | |
| assert ratint_logpart(x**2, x**3 - 5, x, t) == \ | |
| [(Poly(x**3 - 5, x), Poly(-3*t + 1, t))] | |
| def test_issue_5414(): | |
| assert ratint(1/(x**2 + 16), x) == atan(x/4)/4 | |
| def test_issue_5249(): | |
| assert ratint( | |
| 1/(x**2 + a**2), x) == (-I*log(-I*a + x)/2 + I*log(I*a + x)/2)/a | |
| def test_issue_5817(): | |
| a, b, c = symbols('a,b,c', positive=True) | |
| assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \ | |
| sqrt(a)*atan(sqrt( | |
| b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b)) | |
| def test_issue_5981(): | |
| u = symbols('u') | |
| assert integrate(1/(u**2 + 1)) == atan(u) | |
| def test_issue_10488(): | |
| a,b,c,x = symbols('a b c x', positive=True) | |
| assert integrate(x/(a*x+b),x) == x/a - b*log(a*x + b)/a**2 | |
| def test_issues_8246_12050_13501_14080(): | |
| a = symbols('a', nonzero=True) | |
| assert integrate(a/(x**2 + a**2), x) == atan(x/a) | |
| assert integrate(1/(x**2 + a**2), x) == atan(x/a)/a | |
| assert integrate(1/(1 + a**2*x**2), x) == atan(a*x)/a | |
| def test_issue_6308(): | |
| k, a0 = symbols('k a0', real=True) | |
| assert integrate((x**2 + 1 - k**2)/(x**2 + 1 + a0**2), x) == \ | |
| x - (a0**2 + k**2)*atan(x/sqrt(a0**2 + 1))/sqrt(a0**2 + 1) | |
| def test_issue_5907(): | |
| a = symbols('a', nonzero=True) | |
| assert integrate(1/(x**2 + a**2)**2, x) == \ | |
| x/(2*a**4 + 2*a**2*x**2) + atan(x/a)/(2*a**3) | |
| def test_log_to_atan(): | |
| f, g = (Poly(x + S.Half, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX')) | |
| fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3) | |
| assert log_to_atan(f, g) == fg_ans | |
| assert log_to_atan(g, f) == -fg_ans | |
| def test_issue_25896(): | |
| # for both tests, C = 0 in log_to_real | |
| # but this only has a log result | |
| e = (2*x + 1)/(x**2 + x + 1) + 1/x | |
| assert ratint(e, x) == log(x**3 + x**2 + x) | |
| # while this has more | |
| assert ratint((4*x + 7)/(x**2 + 4*x + 6) + 2/x, x) == ( | |
| 2*log(x) + 2*log(x**2 + 4*x + 6) - sqrt(2)*atan( | |
| sqrt(2)*x/2 + sqrt(2))/2) | |