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Direktori : /lib/python3/dist-packages/sympy/integrals/tests/ |
Current File : //lib/python3/dist-packages/sympy/integrals/tests/test_rationaltools.py |
from sympy import (S, symbols, I, atan, log, Poly, sqrt, simplify, integrate, Rational, Dummy) 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', real=True, 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