What is the characteristic function of a random variable with density
$$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$
My hopefully still shortenable result is so far
$$\phi_X(t) = E[e^{itX}] = \frac{b}{2} \int_{-\infty}^0 e^{itx} e^{bx} dx + \frac{a}{2} \int_0^{\infty} e^{itx} e^{-ax} dx$$
$$ = \frac{1}{2} \int_0^{\infty} (a e^{itx} e^{-ax} + b e^{-itx} e^{-bx}) dx $$
There is some analogy to the fact that a r.v. with above density and $a=b$, i.e. a Laplace distributed r.v., is distributed as the difference of two exponential $(a)$-distributed r.v.'s, which can be proved via the characteristic functions (see eg. c.f. of exp. dist. or on this forum).
It is therefore worth a try to suppose that a r.v. with the above density with $a \neq b$ is distributed as the difference $Z$ of an exponential $(a)$-distributed r.v. and an exponential $(b)$-distributed r.v.. Is it correct $Z$ has characteristic function $$\phi_Z(t) = \frac{a \,b}{ab + it(a-b) + t^2 } \quad \quad ?$$
(Remark. A r.v. distributed with the density $f_X$ is sometimes called asymmetric Laplace density, although this terminus was used mainly for a different formulation, see eg. Kozubowski & Podgorski (2000).)