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| from sympy.core import Ne, Rational, Symbol | |
| from sympy.functions import sin, cos, tan, csc, sec, cot, log, Piecewise | |
| from sympy.integrals.trigonometry import trigintegrate | |
| x = Symbol('x') | |
| def test_trigintegrate_odd(): | |
| assert trigintegrate(Rational(1), x) == x | |
| assert trigintegrate(x, x) is None | |
| assert trigintegrate(x**2, x) is None | |
| assert trigintegrate(sin(x), x) == -cos(x) | |
| assert trigintegrate(cos(x), x) == sin(x) | |
| assert trigintegrate(sin(3*x), x) == -cos(3*x)/3 | |
| assert trigintegrate(cos(3*x), x) == sin(3*x)/3 | |
| y = Symbol('y') | |
| assert trigintegrate(sin(y*x), x) == Piecewise( | |
| (-cos(y*x)/y, Ne(y, 0)), (0, True)) | |
| assert trigintegrate(cos(y*x), x) == Piecewise( | |
| (sin(y*x)/y, Ne(y, 0)), (x, True)) | |
| assert trigintegrate(sin(y*x)**2, x) == Piecewise( | |
| ((x*y/2 - sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (0, True)) | |
| assert trigintegrate(sin(y*x)*cos(y*x), x) == Piecewise( | |
| (sin(x*y)**2/(2*y), Ne(y, 0)), (0, True)) | |
| assert trigintegrate(cos(y*x)**2, x) == Piecewise( | |
| ((x*y/2 + sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (x, True)) | |
| y = Symbol('y', positive=True) | |
| # TODO: remove conds='none' below. For this to work we would have to rule | |
| # out (e.g. by trying solve) the condition y = 0, incompatible with | |
| # y.is_positive being True. | |
| assert trigintegrate(sin(y*x), x, conds='none') == -cos(y*x)/y | |
| assert trigintegrate(cos(y*x), x, conds='none') == sin(y*x)/y | |
| assert trigintegrate(sin(x)*cos(x), x) == sin(x)**2/2 | |
| assert trigintegrate(sin(x)*cos(x)**2, x) == -cos(x)**3/3 | |
| assert trigintegrate(sin(x)**2*cos(x), x) == sin(x)**3/3 | |
| # check if it selects right function to substitute, | |
| # so the result is kept simple | |
| assert trigintegrate(sin(x)**7 * cos(x), x) == sin(x)**8/8 | |
| assert trigintegrate(sin(x) * cos(x)**7, x) == -cos(x)**8/8 | |
| assert trigintegrate(sin(x)**7 * cos(x)**3, x) == \ | |
| -sin(x)**10/10 + sin(x)**8/8 | |
| assert trigintegrate(sin(x)**3 * cos(x)**7, x) == \ | |
| cos(x)**10/10 - cos(x)**8/8 | |
| # both n, m are odd and -ve, and not necessarily equal | |
| assert trigintegrate(sin(x)**-1*cos(x)**-1, x) == \ | |
| -log(sin(x)**2 - 1)/2 + log(sin(x)) | |
| def test_trigintegrate_even(): | |
| assert trigintegrate(sin(x)**2, x) == x/2 - cos(x)*sin(x)/2 | |
| assert trigintegrate(cos(x)**2, x) == x/2 + cos(x)*sin(x)/2 | |
| assert trigintegrate(sin(3*x)**2, x) == x/2 - cos(3*x)*sin(3*x)/6 | |
| assert trigintegrate(cos(3*x)**2, x) == x/2 + cos(3*x)*sin(3*x)/6 | |
| assert trigintegrate(sin(x)**2 * cos(x)**2, x) == \ | |
| x/8 - sin(2*x)*cos(2*x)/16 | |
| assert trigintegrate(sin(x)**4 * cos(x)**2, x) == \ | |
| x/16 - sin(x) *cos(x)/16 - sin(x)**3*cos(x)/24 + \ | |
| sin(x)**5*cos(x)/6 | |
| assert trigintegrate(sin(x)**2 * cos(x)**4, x) == \ | |
| x/16 + cos(x) *sin(x)/16 + cos(x)**3*sin(x)/24 - \ | |
| cos(x)**5*sin(x)/6 | |
| assert trigintegrate(sin(x)**(-4), x) == -2*cos(x)/(3*sin(x)) \ | |
| - cos(x)/(3*sin(x)**3) | |
| assert trigintegrate(cos(x)**(-6), x) == sin(x)/(5*cos(x)**5) \ | |
| + 4*sin(x)/(15*cos(x)**3) + 8*sin(x)/(15*cos(x)) | |
| def test_trigintegrate_mixed(): | |
| assert trigintegrate(sin(x)*sec(x), x) == -log(cos(x)) | |
| assert trigintegrate(sin(x)*csc(x), x) == x | |
| assert trigintegrate(sin(x)*cot(x), x) == sin(x) | |
| assert trigintegrate(cos(x)*sec(x), x) == x | |
| assert trigintegrate(cos(x)*csc(x), x) == log(sin(x)) | |
| assert trigintegrate(cos(x)*tan(x), x) == -cos(x) | |
| assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \ | |
| - log(cos(x) + 1)/2 + cos(x) | |
| assert trigintegrate(cot(x)*cos(x)**2, x) == log(sin(x)) - sin(x)**2/2 | |
| def test_trigintegrate_symbolic(): | |
| n = Symbol('n', integer=True) | |
| assert trigintegrate(cos(x)**n, x) is None | |
| assert trigintegrate(sin(x)**n, x) is None | |
| assert trigintegrate(cot(x)**n, x) is None | |