def test_trigintegrate_mixed(): assert trigintegrate(sin(x)*sec(x), x) == -log(sin(x)**2 - 1)/2 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(cos(x)**2 - 1)/2 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)
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
def _trigpats(): global _trigpat a, b, c = symbols('a b c', cls=Wild) d = Wild('d', commutative=False) # for the simplifications like sinh/cosh -> tanh: # DO NOT REORDER THE FIRST 14 since these are assumed to be in this # order in _match_div_rewrite. matchers_division = ( (a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)), (a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)), (a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)), (a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)), (a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)), (a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)), (a*(cos(b) + 1)**c*(cos(b) - 1)**c, a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1), (a*(sin(b) + 1)**c*(sin(b) - 1)**c, a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1), (a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One), (a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One), (a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One), (a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One), (a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One), (a*coth(b)**c*tanh(b)**c, a, S.One, S.One), (c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)), tanh(a + b)*c, S.One, S.One), ) matchers_add = ( (c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d), (c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d), (c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d), (c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d), (c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d), (c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d), ) # for cos(x)**2 + sin(x)**2 -> 1 matchers_identity = ( (a*sin(b)**2, a - a*cos(b)**2), (a*tan(b)**2, a*(1/cos(b))**2 - a), (a*cot(b)**2, a*(1/sin(b))**2 - a), (a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))), (a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))), (a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))), (a*sinh(b)**2, a*cosh(b)**2 - a), (a*tanh(b)**2, a - a*(1/cosh(b))**2), (a*coth(b)**2, a + a*(1/sinh(b))**2), (a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))), (a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))), (a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))), ) # Reduce any lingering artifacts, such as sin(x)**2 changing # to 1-cos(x)**2 when sin(x)**2 was "simpler" artifacts = ( (a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos), (a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos), (a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin), (a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh), (a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh), (a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh), # same as above but with noncommutative prefactor (a*d - a*d*cos(b)**2 + c, a*d*sin(b)**2 + c, cos), (a*d - a*d*(1/cos(b))**2 + c, -a*d*tan(b)**2 + c, cos), (a*d - a*d*(1/sin(b))**2 + c, -a*d*cot(b)**2 + c, sin), (a*d - a*d*cosh(b)**2 + c, -a*d*sinh(b)**2 + c, cosh), (a*d - a*d*(1/cosh(b))**2 + c, a*d*tanh(b)**2 + c, cosh), (a*d + a*d*(1/sinh(b))**2 + c, a*d*coth(b)**2 + c, sinh), ) _trigpat = (a, b, c, d, matchers_division, matchers_add, matchers_identity, artifacts) return _trigpat
def test_harmonic_rational(): ne = S(6) no = S(5) pe = S(8) po = S(9) qe = S(10) qo = S(13) Heee = harmonic(ne + pe/qe) Aeee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8))) + pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8))) + 13944145/S(4720968)) Heeo = harmonic(ne + pe/qo) Aeeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13) + 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13) - 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13) + 2422020029/S(702257080)) Heoe = harmonic(ne + po/qe) Aeoe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4) + 11818877030/S(4286604231) - pi*sqrt(sqrt(5)/8 + 5/S(8))/(-sqrt(5)/2 + 1/S(2)) ) Heoo = harmonic(ne + po/qo) Aeoo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13) + 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13) - 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2 - 2*log(sin(2*pi/13))*cos(3*pi/13) + 11669332571/S(3628714320)) Hoee = harmonic(no + pe/qe) Aoee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8))) + pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8))) + 779405/S(277704)) Hoeo = harmonic(no + pe/qo) Aoeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13) + 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13) - 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13) + 53857323/S(16331560)) Hooe = harmonic(no + po/qe) Aooe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8))) + 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4) + 486853480/S(186374097) - pi*sqrt(sqrt(5)/8 + 5/S(8))/(2*(-sqrt(5)/4 + 1/S(4)))) Hooo = harmonic(no + po/qo) Aooo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13) + 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13) - 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2 - 2*log(sin(2*pi/13))*cos(3*pi/13) + 383693479/S(125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e/a) == 1 assert h.n() == a.n()
def _trigpats(): global _trigpat a, b, c = symbols('a b c', cls=Wild) d = Wild('d', commutative=False) # for the simplifications like sinh/cosh -> tanh: # DO NOT REORDER THE FIRST 14 since these are assumed to be in this # order in _match_div_rewrite. matchers_division = ( (a * sin(b)**c / cos(b)**c, a * tan(b)**c, sin(b), cos(b)), (a * tan(b)**c * cos(b)**c, a * sin(b)**c, sin(b), cos(b)), (a * cot(b)**c * sin(b)**c, a * cos(b)**c, sin(b), cos(b)), (a * tan(b)**c / sin(b)**c, a / cos(b)**c, sin(b), cos(b)), (a * cot(b)**c / cos(b)**c, a / sin(b)**c, sin(b), cos(b)), (a * cot(b)**c * tan(b)**c, a, sin(b), cos(b)), (a * (cos(b) + 1)**c * (cos(b) - 1)**c, a * (-sin(b)**2)**c, cos(b) + 1, cos(b) - 1), (a * (sin(b) + 1)**c * (sin(b) - 1)**c, a * (-cos(b)**2)**c, sin(b) + 1, sin(b) - 1), (a * sinh(b)**c / cosh(b)**c, a * tanh(b)**c, S.One, S.One), (a * tanh(b)**c * cosh(b)**c, a * sinh(b)**c, S.One, S.One), (a * coth(b)**c * sinh(b)**c, a * cosh(b)**c, S.One, S.One), (a * tanh(b)**c / sinh(b)**c, a / cosh(b)**c, S.One, S.One), (a * coth(b)**c / cosh(b)**c, a / sinh(b)**c, S.One, S.One), (a * coth(b)**c * tanh(b)**c, a, S.One, S.One), (c * (tanh(a) + tanh(b)) / (1 + tanh(a) * tanh(b)), tanh(a + b) * c, S.One, S.One), ) matchers_add = ( (c * sin(a) * cos(b) + c * cos(a) * sin(b) + d, sin(a + b) * c + d), (c * cos(a) * cos(b) - c * sin(a) * sin(b) + d, cos(a + b) * c + d), (c * sin(a) * cos(b) - c * cos(a) * sin(b) + d, sin(a - b) * c + d), (c * cos(a) * cos(b) + c * sin(a) * sin(b) + d, cos(a - b) * c + d), (c * sinh(a) * cosh(b) + c * sinh(b) * cosh(a) + d, sinh(a + b) * c + d), (c * cosh(a) * cosh(b) + c * sinh(a) * sinh(b) + d, cosh(a + b) * c + d), ) # for cos(x)**2 + sin(x)**2 -> 1 matchers_identity = ( (a * sin(b)**2, a - a * cos(b)**2), (a * tan(b)**2, a * (1 / cos(b))**2 - a), (a * cot(b)**2, a * (1 / sin(b))**2 - a), (a * sin(b + c), a * (sin(b) * cos(c) + sin(c) * cos(b))), (a * cos(b + c), a * (cos(b) * cos(c) - sin(b) * sin(c))), (a * tan(b + c), a * ((tan(b) + tan(c)) / (1 - tan(b) * tan(c)))), (a * sinh(b)**2, a * cosh(b)**2 - a), (a * tanh(b)**2, a - a * (1 / cosh(b))**2), (a * coth(b)**2, a + a * (1 / sinh(b))**2), (a * sinh(b + c), a * (sinh(b) * cosh(c) + sinh(c) * cosh(b))), (a * cosh(b + c), a * (cosh(b) * cosh(c) + sinh(b) * sinh(c))), (a * tanh(b + c), a * ((tanh(b) + tanh(c)) / (1 + tanh(b) * tanh(c)))), ) # Reduce any lingering artifacts, such as sin(x)**2 changing # to 1-cos(x)**2 when sin(x)**2 was "simpler" artifacts = ( (a - a * cos(b)**2 + c, a * sin(b)**2 + c, cos), (a - a * (1 / cos(b))**2 + c, -a * tan(b)**2 + c, cos), (a - a * (1 / sin(b))**2 + c, -a * cot(b)**2 + c, sin), (a - a * cosh(b)**2 + c, -a * sinh(b)**2 + c, cosh), (a - a * (1 / cosh(b))**2 + c, a * tanh(b)**2 + c, cosh), (a + a * (1 / sinh(b))**2 + c, a * coth(b)**2 + c, sinh), # same as above but with noncommutative prefactor (a * d - a * d * cos(b)**2 + c, a * d * sin(b)**2 + c, cos), (a * d - a * d * (1 / cos(b))**2 + c, -a * d * tan(b)**2 + c, cos), (a * d - a * d * (1 / sin(b))**2 + c, -a * d * cot(b)**2 + c, sin), (a * d - a * d * cosh(b)**2 + c, -a * d * sinh(b)**2 + c, cosh), (a * d - a * d * (1 / cosh(b))**2 + c, a * d * tanh(b)**2 + c, cosh), (a * d + a * d * (1 / sinh(b))**2 + c, a * d * coth(b)**2 + c, sinh), ) _trigpat = (a, b, c, d, matchers_division, matchers_add, matchers_identity, artifacts) return _trigpat
def test_harmonic_rational(): ne = S(6) no = S(5) pe = S(8) po = S(9) qe = S(10) qo = S(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + pi * (1 / S(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / S(8))) + 13944145 / S(4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 2422020029 / S(702257080)) Heoe = harmonic(ne + po / qe) Aeoe = (-log(20) + 2 * (1 / S(4) + sqrt(5) / 4) * log(-1 / S(4) + sqrt(5) / 4) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 + 1 / S(4)) * log(1 / S(4) + sqrt(5) / 4) + 11818877030 / S(4286604231) - pi * sqrt(sqrt(5) / 8 + 5 / S(8)) / (-sqrt(5) / 2 + 1 / S(2))) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 11669332571 / S(3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + pi * (1 / S(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / S(8))) + 779405 / S(277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 53857323 / S(16331560)) Hooe = harmonic(no + po / qe) Aooe = (-log(20) + 2 * (1 / S(4) + sqrt(5) / 4) * log(-1 / S(4) + sqrt(5) / 4) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 + 1 / S(4)) * log(1 / S(4) + sqrt(5) / 4) + 486853480 / S(186374097) - pi * sqrt(sqrt(5) / 8 + 5 / S(8)) / (2 * (-sqrt(5) / 4 + 1 / S(4)))) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 383693479 / S(125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert h.n() == a.n()
def test_harmonic_rational(): ne = S(6) no = S(5) pe = S(8) po = S(9) qe = S(10) qo = S(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * sqrt(2 * sqrt(5) / 5 + 1) / 2 + Rational(13944145, 4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(4, 13)) + 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(32, 13)) + 2 * log(sin(pi * Rational(5, 13))) * cos(pi * Rational(80, 13)) - 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(5, 13)) - 2 * log(sin(pi * Rational(4, 13))) * cos(pi / 13) + pi * cot(pi * Rational(5, 13)) / 2 - 2 * log(sin(pi / 13)) * cos(pi * Rational(3, 13)) + Rational(2422020029, 702257080)) Heoe = harmonic(ne + po / qe) Aeoe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(11818877030, 4286604231) + pi * sqrt(2 * sqrt(5) + 5) / 2) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(54, 13)) + 2 * log(sin(pi * Rational(4, 13))) * cos(pi * Rational(6, 13)) + 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(108, 13)) - 2 * log(sin(pi * Rational(5, 13))) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(pi * Rational(5, 13)) + pi * cot(pi * Rational(4, 13)) / 2 - 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(3, 13)) + Rational(11669332571, 3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * sqrt(2 * sqrt(5) / 5 + 1) / 2 + Rational(779405, 277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(4, 13)) + 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(32, 13)) + 2 * log(sin(pi * Rational(5, 13))) * cos(pi * Rational(80, 13)) - 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(5, 13)) - 2 * log(sin(pi * Rational(4, 13))) * cos(pi / 13) + pi * cot(pi * Rational(5, 13)) / 2 - 2 * log(sin(pi / 13)) * cos(pi * Rational(3, 13)) + Rational(53857323, 16331560)) Hooe = harmonic(no + po / qe) Aooe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(486853480, 186374097) + pi * sqrt(2 * sqrt(5) + 5) / 2) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(54, 13)) + 2 * log(sin(pi * Rational(4, 13))) * cos(pi * Rational(6, 13)) + 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(108, 13)) - 2 * log(sin(pi * Rational(5, 13))) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(pi * Rational(5, 13)) + pi * cot(pi * Rational(4, 13)) / 2 - 2 * log(sin(pi * Rational(2, 13))) * cos(3 * pi / 13) + Rational(383693479, 125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert abs(h.n() - a.n()) < 1e-12
def Trig_Check(s): if sin(s.args[0])/s is S.One or cos(s.args[0])/s is S.One \ or csc(s.args[0])/s is S.One or sec(s.args[0])/s is S.One \ or tan(s.args[0])/s is S.One or cot(s.args[0])/s is S.One: return True