Does $$\int_0^a e^{-bt-c/t} dt \tag{1}$$ have a closed-form expression?
Note: $a, b,$ and $c$ are all positive constants.
The following is my trials.
(i) Let $x = -bt-\frac{c}{t}$. But I can't express $t$ in terms of $x$.
(ii) Let $y = \frac{1}{t}$. Then $t = \frac{1}{y}$, $\frac{dy}{dt} = - \frac{1}{t^2} = -y^2 \Rightarrow dt = - \frac{dy}{y^2}$, and $t = 0 \Rightarrow y=\frac{1}{0}$. But zero cannot appear in the denominator.
Then I don't know how to continue.