integration - Integrating definite integral $int_a^infty frac{1}{log (t)} , dt$ and then $int_a^infty frac{e^{-st}}{log (t)} , dt$

The differentiation of $\operatorname{Li}(t)$ gives

$$\frac{1}{\log t}$$

In mathematica using D[LogIntegral[t], t] it confirms this as one would expect.

But I'm having difficulty integrating:

$$\int_a^\infty \frac 1 {\log (t)} \, dt$$

I'm not sure by hand how to do this and it certainly won't compute in mathematica; yet it differentiates $\operatorname{Li}(t).$ I can see that with limit a set to zero there would be a problem with it going to infinity as it approaches the $y$-axis, something like $a=2$ would not present that problem.

A similar question was posed here:
Convergence or Divergence using Limits
but only established that it is divergent and did not explain that it came from $\operatorname{Li}(t)$ and what would be needed to get back.

Further I cannot integrate

$$\int_a^\infty \frac{e^{-st}}{\log (t)} \, dt$$
again with $a=0$ this would be a problem I suppose, but $a=2$ should be okay.

Answer

$$\int_a^\infty \frac{dt}{\log t} \tag{$*$}$$ does not converge. $$\log t < t \to \frac{1}{\log t}>\frac{1}{t}$$ and as $$\int_a^\infty \, \frac{dt}{t}$$ diverges, so does the $(*)$

The second integral converges for any $s>0;\;a>1$, but I can't find a closed form and I think It doesn't exist

Hope this helps