For complicated expression, the simplification does not seem to work out.
The beta_1 in the numerator and the denominator should have precisely
canceled out, however they do not.
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import sympy
from sympy import *
f = sympy.symbols('f', positive=True) #True Anomaly
t = sympy.symbols('t', positive=True) #Time
n = sympy.symbols('n', positive=True) #Period
f_p = sympy.symbols('f_p', positive=True) #True Anomaly prime due to tidal distortion
n_p = sympy.symbols('n_p', positive=True) #Mean motion prime
G = sympy.symbols('G', positive=True) #Gravitation constant
Ms = sympy.symbols('Ms', positive=True) #Mass of the sun
Mp = sympy.symbols('Mp', positive=True) #mass of the planet
R = sympy.symbols('R', positive=True) #Radius of the planet
r = sympy.symbols('r', positive=True) #distance of the planet
a = sympy.symbols('a', positive=True) #semi-major axis of the planet
e = sympy.symbols('e', positive=True) #eccentricity of the orbit0
r_p = sympy.symbols('r_p', positive=True) #distance of the planet
#New set of variables for defining integrand
rho = sympy.symbols('rho', positive=True) #Density of the planet
h2 = sympy.symbols('h2', positive=True) #h2 is the vertical displace number
g = sympy.symbols('g', positive=True) #g is the local acceleration due to gravity
beta_10 = Symbol('beta_1', positive=True)
Numerator = 3*sympy.pi*G**2*Ms**2*R**6*h2*n*rho
Denominator=5*beta_10*g*r**3*r_p**5
FinalValue = Mul(Rational(3, 5), pi, Pow(Symbol('G', positive=True), Integer(2)), Pow(Symbol('Ms', positive=True), Integer(2)), Pow(Symbol('R', positive=True), Integer(6)), Pow(Symbol('beta_1', Positive=True), Integer(-1)), Pow(Symbol('g', positive=True), Integer(-1)), Symbol('h2', positive=True), Symbol('n', positive=True), Pow(Symbol('r', positive=True), Integer(-3)), Pow(Symbol('r_p', positive=True), Integer(-5)), Symbol('rho', positive=True), Add(Mul(Integer(-1), Integer(4), Pow(Symbol('a', positive=True), Integer(2)), Pow(Symbol('beta', positive=True), Integer(2)), Pow(sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True))))), Mul(Integer(-1), Integer(4), Pow(Symbol('a', positive=True), Integer(2)), Pow(Symbol('beta', positive=True), Integer(2)), sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True))))), Mul(Integer(4), Pow(Symbol('a', positive=True), Integer(2)), Pow(Symbol('beta', positive=True), Integer(2)), sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(4), Pow(Symbol('a', positive=True), Integer(2)), Pow(Symbol('beta', positive=True), Integer(2)), sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True))))), Mul(Rational(3, 4), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), Add(sin(Add(Mul(Integer(2), Symbol('Delta', positive=True)), Mul(Integer(2), Symbol('f', positive=True)), Mul(Integer(-1), Integer(3), Symbol('f_p', positive=True)))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('Delta', positive=True)), Mul(Integer(2), Symbol('f', positive=True)), Mul(Integer(-1), Symbol('f_p', positive=True))))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('Delta', positive=True)), Mul(Integer(-1), Integer(2), Symbol('f', positive=True)), Mul(Integer(-1), Integer(3), Symbol('f_p', positive=True)), Mul(Integer(4), Symbol('n', positive=True), Symbol('t', positive=True))))), sin(Add(Mul(Integer(2), Symbol('Delta', positive=True)), Mul(Integer(-1), Integer(2), Symbol('f', positive=True)), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Integer(4), Symbol('n', positive=True), Symbol('t', positive=True)))))), Mul(Integer(-1), Integer(9), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), Pow(sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(-1), Integer(3), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), Pow(cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(5), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(-1), Integer(3), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), Pow(cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(5), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(-1), Integer(9), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), Pow(cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(5), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(5), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True)), Pow(cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(-1), Integer(5), Symbol('a', positive=True), Symbol('e', positive=True), Symbol('r_p', positive=True), sin(Symbol('f_p', positive=True))), Mul(Integer(4), Symbol('beta', positive=True), Pow(Symbol('r_p', positive=True), Integer(2)), Pow(sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True))))), Mul(Integer(4), Symbol('beta', positive=True), Pow(Symbol('r_p', positive=True), Integer(2)), sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True))))), Mul(Integer(-1), Integer(4), Symbol('beta', positive=True), Pow(Symbol('r_p', positive=True), Integer(2)), sin(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2))), Mul(Integer(-1), Integer(4), Symbol('beta', positive=True), Pow(Symbol('r_p', positive=True), Integer(2)), sin(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))), Pow(cos(Add(Symbol('f', positive=True), Mul(Integer(-1), Symbol('n', positive=True), Symbol('t', positive=True)))), Integer(2)), cos(Add(Symbol('Delta', positive=True), Mul(Integer(-1), Symbol('f_p', positive=True)), Mul(Symbol('n', positive=True), Symbol('t', positive=True)))))))
pprint(FinalValue)
FinalValue = powsimp(FinalValue*Denominator/Numerator)
pprint(FinalValue)
