Theorem: Moment Generating Function of Logistic Distribution

Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \begin {cases} \map \exp {\mu t}  \map \Beta {\paren {1 - s t}, \paren {1 + s t} } & \size t < \dfrac 1 s  \\ \text {does not exist} & \size t \ge \dfrac 1 s \end {cases}$
where $\Beta$ denotes the beta function

Proof:
From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$
So: 
:$\ds \map {M_X} t = \frac 1 s \int_{-\infty}^\infty \dfrac {\map \exp {t x} \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}
      | c = Integration by Substitution
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = -\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-2} \paren {-\frac 1 s \map \exp {-\dfrac {\paren {x - \mu} } s} }
      | c = Power Rule for Derivatives, Chain Rule for Derivatives and Derivative of Exponential Function: Corollary 1
}}
{{eqn | ll= \leadsto
      | l = \dfrac 1 u - 1
      | r = \paren {\map \exp {-\dfrac {\paren {x - \mu} } s} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1}
      | r = \map \exp {-\dfrac \mu s} \paren {\map \exp {-\dfrac {\paren {x - \mu} } s} }
      | c = multiplying both sides by $\map \exp {-\dfrac \mu s}$
}}
{{eqn | ll= \leadsto
      | l = \map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1}
      | r = \paren {\map \exp {-\dfrac x s} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \paren {\map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1} }^{-s t}
      | r = \paren {\map \exp {-\dfrac x s} }^{-s t}
      | c = raising both sides to the $-s t$ power
}}
{{eqn | ll= \leadsto
      | l = \map \exp {\mu t} \paren {\dfrac {1 - u} u}^{-s t}
      | r = \map \exp {x t}
      | c = 
}}
{{end-eqn}}
and also:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to -\infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}
      | r = 0
      | c = 
}}
{{eqn | l = \lim_{x \mathop \to \infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}
      | r = 1
      | c = 
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn	| l = \map {M_X} t
        | r = \map \exp {\mu t} \int_{\to 0}^{\to 1}  \paren {\dfrac {1 - u} u}^{-s t} \rd u
	| c = 
}}
{{eqn	| r = \map \exp {\mu t} \int_{\to 0}^{\to 1}  \paren {1 - u }^{-s t} u^{s t} \rd u
	| c = 
}}
{{eqn	| r = \map \exp {\mu t} \map \Beta {\paren {1 - s t}, \paren {1 + s t} }
	| c = {{Defof|Beta Function}} and Commutativity of Parameters of Beta Function
}}
{{end-eqn}}
Note that by definition of the beta function, $\Beta: \C \times \C \to \C$ is defined only for $\map \Re x, \map \Re y > 0$:
Therefore:
:$1 - s t > 0 \leadsto t < \dfrac 1 s$
and:
:$1 + s t > 0 \leadsto t > -\dfrac 1 s$
Therefore, the moment generating function is defined only when $\size t < \dfrac 1 s$.
{{qed}}