[Python] Unable to simplify an expression with Sympy

I need to evaluate many integrals, derivatives, etc. symbolically and I have no luck with Sympy. When it is able to perform an integral, the result is gigantic while it should be reducible, but I am totally unable to reach the simplest result. Here's an example:

import simpy as sp

# Define the variables
r, theta, H0, ri, ro, rho, T_0, Delta_T, S_xx, S_yy = sp.symbols('r theta H0 ri ro rho T_0 Delta_T S_xx S_yy')

# Define H0
H0_expr = -(S_xx - S_yy) / (4 * rho) * Delta_T / sp.log(ro/ri)

# Define the components of the vector field J_r and J_theta
J_r = 2 * H0 * (1/r - 1/(ri**2 + ro**2)*(r + ri**2*ro**2/r**3)) * sp.cos(2*theta)
J_theta = H0 * (1/(ri**2 + ro**2)*(2*r - 2*ri**2*ro**2/r**3)) * sp.sin(2*theta)

J_x = J_r * sp.cos(theta) - J_theta * sp.sin(theta)
dJ_x_dx = sp.diff(J_x, r) * sp.cos(theta) - 1/r * sp.sin(theta) * sp.diff(J_x, theta) * sp.sin(theta)

T = T_0 + Delta_T * sp.log(r/ro) / sp.log(ro/ri)

# Set up the integral for Q
Q = sp.integrate(r * T * (S_xx - S_yy) * dJ_x_dx, (theta, 0, sp.pi/2), (r, ri, ro))

# Substitute H0 in Q
Q_sub = Q.subs(H0, H0_expr)
Q_sub = sp.simplify(Q_sub)
Q_sub = sp.collect(Q_sub, [ri, ro, sp.log(ro/ri)])
Q_sub = sp.factor(Q_sub)
print(f"This is Q: {Q}")


Running the code yields:

This is Q: -Delta_T*(S_xx - S_yy)**2*(-16*Delta_T*ri**2*log(ri/ro)**2 + 15*pi*Delta_T*ri**2*log(ri/ro)**2 + 80*Delta_T*ri**2*log(ri/ro) + 30*pi*Delta_T*ri**2*log(ri/ro) - 64*Delta_T*ri**2 - 15*pi*Delta_T*ri**2 - 16*Delta_T*ro**2*log(ri/ro)**2 + 15*pi*Delta_T*ro**2*log(ri/ro)**2 + 48*Delta_T*ro**2*log(ri/ro) + 15*pi*Delta_T*ro**2 + 64*Delta_T*ro**2 - 32*T_0*ri**2*log(ri)*log(ro/ri) + 30*pi*T_0*ri**2*log(ri)*log(ro/ri) - 30*pi*T_0*ri**2*log(ro)*log(ro/ri) + 32*T_0*ri**2*log(ro)*log(ro/ri) + 32*T_0*ri**2*log(ro/ri) + 30*pi*T_0*ri**2*log(ro/ri) - 32*T_0*ro**2*log(ri)*log(ro/ri) + 30*pi*T_0*ro**2*log(ri)*log(ro/ri) - 30*pi*T_0*ro**2*log(ro)*log(ro/ri) + 32*T_0*ro**2*log(ro)*log(ro/ri) - 30*pi*T_0*ro**2*log(ro/ri) - 32*T_0*ro**2*log(ro/ri))/(480*rho*(ri**2 + ro**2)*log(ro/ri)**2).


Which is enormous. This should be equal to

$$\frac{\pi\,\Delta T\left(S_{xx}-S_{yy}\right)^2\left[\Delta T\left(r_{o}^{2}-r_{i}^{2}\right)-2\log\left(\frac{r_{o}}{r_{i}}\right)\left(\Delta T\,r_{i}^{2}+T_{0}\left(r_{o}^{2}-r_{i}^{2}\right)\right)\right]}{16\,\rho\left(r_{i}^{2}+r_{o}^{2}\right)\log^{2}\left(\frac{r_{o}}{r_{i}}\right)}$$


or

pi*Delta_T*(S_xx - S_yy)**2*(Delta_T*(-ri**2 + ro**2) - 2*(Delta_T*ri**2 + T_0*(-ri**2 + ro**2))*log(ro/ri))/(16*rho*(ri**2 + ro**2)*log(ro/ri)**2)


How can I reach the above "simple" expression using Sympy?

Continue reading...