Does analytical solution to following integral exist?
$$\int_0^\infty { \frac{\exp\left[ {-\left(\frac{1}{t}+t+t^3\right)}\right]}{\sqrt{t}} }\mathrm{d}t$$
It converges numerically. I tested in Mathematica. So it looks like there is some kind of coordinate transformation that makes the pole at t=0 integrable. I cannot see which kind of transformation does it. I am looking for the analytical result. Can anyone help me?
By setting $t=x^2$ the given integral becomes $$ \int_{-\infty}^{+\infty}\exp\left[-\left(\frac{1}{x^2}+x^2+x^6\right)\right]\,dx \tag{1}$$ and by Glasser's master theorem (it is enough to set $x-\frac{1}{x}=u$) $$ \int_{-\infty}^{+\infty}\exp\left[-\left(\frac{1}{x^2}+x^2\right)\right]\,dx = \frac{\sqrt{\pi}}{2e^2}\tag{2}$$ so my suggestion is to perform the same substitution ($x-\frac{1}{x}=u$) in $(1)$ then switch to a series expansion. It looks that $(1)$ does not have a nice closed form.