I was playing around with some integrals I made up myself and was trying to find a closed-form for
$$ \int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty $$
I'm aware that if we let $t \to \infty$ then a solution can be obtained in terms of a modified Bessel function, but I couldn't seem to get a result for the finite-limit version. By plugging the integral into WolframAlpha it replies that the integral above doesn't have an elementary solution. However, I would like to find any possible closed form, even if it's in terms of special functions (which will very likely be necessary if such a closed-form exists).
I tried doing integration by parts, which resulted in the identity $$ \int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx = - e^{-\left( t + \frac{1}{t}\right)} +\int_{\frac{1}{t}}^{\infty} e^{-\left( u + \frac{1}{u}\right)} \ du $$ which makes the problem equivalent to finding a solution to $\int_{\tau}^{\infty} e^{-\left( x + \frac{1}{x}\right)} \ dx$, but I again didn't see how this could help much.
Does anyone know any way to get a closed-form for said integral? Or alternatively, is there a way to prove that $F(t) = \int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx$ is "independent" of other special functions, and as such, would need to be defined as its own "new kind" of special function?