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Alternatively, you can write your expression out using Sympy, a symbolics package that's included with Sage. (Sympy Documentation)

sage: from sympy import *
sage: T = Symbol('T')
sage: p = Symbol('p')
sage: z = Symbol('z')
sage: X = (5/(exp(T*p) - 1) + 1/(exp(T*p) - exp(2*T)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: X.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that Sympy correctly identifies that exp(2*T*p) == z^2. In some cases, Sympy can work better than the core of most of Sage's symbolics: Maxima. You should try both and see which one performs to your liking. You can easily switch from Sage's symbolic expressions to Sympy's by doing the following:

sage: import sympy
sage: var('T,p,z')
sage: X = (5/(e^(T*p) - 1) + 1/(e^(T*p) - e^(2*T)))*e^(T*p)/(80*e^(T*p) + e^(2*T*p))
sage: type(X)
<type 'sage.symbolic.expression.Expression'>
sage: Y = sy.sympify(X)
sage: Y
(1/(-exp(2*T) + exp(T*p)) - 5/(1 - exp(T*p)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: type(Y)
<class 'sympy.core.mul.Mul'>


You can now treat Y similarly as above.

sage: Y.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that we didn't have to convert T,p,z, nor exp to Sympy data types. Isn't that great?

Alternatively, you can write your expression out using Sympy, a symbolics package that's included with Sage. (Sympy Documentation)

sage: from sympy import *
sage: T = Symbol('T')
sage: p = Symbol('p')
sage: z = Symbol('z')
sage: X = (5/(exp(T*p) - 1) + 1/(exp(T*p) - exp(2*T)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: X.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that Sympy correctly identifies that exp(2*T*p) == z^2. In some cases, Sympy can work better than the core of most of Sage's symbolics: Maxima. You should try both and see which one performs to your liking. You can easily switch from Sage's symbolic expressions to Sympy's by doing the following:

sage: import sympy
sage: var('T,p,z')
sage: X = (5/(e^(T*p) - 1) + 1/(e^(T*p) - e^(2*T)))*e^(T*p)/(80*e^(T*p) + e^(2*T*p))
sage: type(X)
<type 'sage.symbolic.expression.Expression'>
sage: Y = sy.sympify(X)
sympy.sympify(X)
sage: Y
(1/(-exp(2*T) + exp(T*p)) - 5/(1 - exp(T*p)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: type(Y)
<class 'sympy.core.mul.Mul'>


You can now treat Y similarly as above.

sage: Y.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that we didn't have to convert T,p,z, nor exp to Sympy data types. Isn't that great?

Alternatively, you can write your expression out using Sympy, a symbolics package that's included with Sage. (Sympy Documentation)

sage: from sympy import *
sage: T = Symbol('T')
sage: p = Symbol('p')
sage: z = Symbol('z')
sage: X = (5/(exp(T*p) - 1) + 1/(exp(T*p) - exp(2*T)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: X.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that Sympy correctly identifies that exp(2*T*p) == z^2. In some cases, Sympy can work better than the core of most of Sage's symbolics: Maxima. You should try both and see which one performs to your liking. You can easily switch from Sage's symbolic expressions to Sympy's by doing the following:

sage: import sympy
sage: var('T,p,z')
sage: X = (5/(e^(T*p) - 1) + 1/(e^(T*p) - e^(2*T)))*e^(T*p)/(80*e^(T*p) + e^(2*T*p))
sage: type(X)
<type 'sage.symbolic.expression.Expression'>
sage: Y = sympy.sympify(X)
sage: Y
(1/(-exp(2*T) + exp(T*p)) - 5/(1 - exp(T*p)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: type(Y)
<class 'sympy.core.mul.Mul'>


You can now treat Y similarly as above.

sage: Y.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that we didn't have to convert T,p,z, nor exp to Sympy data types. Isn't that great?

EDIT: You can convert a Sympy expression back to a Sage symbolic expression by doing the following:

sage: Z = SR(Y.subs(exp(T*p,z)))
sage: Z
(5/(z - 1) + 1/(z - e^(2*T)))*z/(z^2 + 80*z)
sage: Z(z=-1)
-1/79/(e^(2*T) + 1) - 5/158


Alternatively, you can write your expression out using Sympy, a symbolics package that's included with Sage. (Sympy Documentation)

sage: from sympy import *
sage: T = Symbol('T')
sage: p = Symbol('p')
sage: z = Symbol('z')
sage: X = (5/(exp(T*p) - 1) + 1/(exp(T*p) - exp(2*T)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: X.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that Sympy correctly identifies that exp(2*T*p) == z^2. In some cases, Sympy can work better than the core of most of Sage's symbolics: Maxima. You should try both and see which one performs to your liking. You can easily switch from Sage's symbolic expressions to Sympy's by doing the following:

sage: import sympy
sage: var('T,p,z')
sage: X = (5/(e^(T*p) - 1) + 1/(e^(T*p) - e^(2*T)))*e^(T*p)/(80*e^(T*p) + e^(2*T*p))
sage: type(X)
<type 'sage.symbolic.expression.Expression'>
sage: Y = sympy.sympify(X)
sage: Y
(1/(-exp(2*T) + exp(T*p)) - 5/(1 - exp(T*p)))*exp(T*p)/(80*exp(T*p) + exp(2*T*p))
sage: type(Y)
<class 'sympy.core.mul.Mul'>


You can now treat Y similarly as above.

sage: Y.subs(exp(T*p),z)
z*(1/(z - exp(2*T)) - 5/(1 - z))/(80*z + z**2)


Note that we didn't have to convert T,p,z, nor exp to Sympy data types. Isn't that great?

EDIT: You can convert a Sympy expression back to a Sage symbolic expression by doing the following:

sage: Z = SR(Y.subs(exp(T*p,z)))
SR(Y.subs(exp(T*p),z))
sage: Z
(5/(z - 1) + 1/(z - e^(2*T)))*z/(z^2 + 80*z)
sage: Z(z=-1)
-1/79/(e^(2*T) + 1) - 5/158