Nested Integrate and NIntegrate: Analytic and Numeric solutions?

Question

Here is a function $F(r)$ which contains double integrations

$$F(r)=\exp\left[ \int_{0}^r dw \,\exp\left(-\int_{0}^w ds \frac{s^2}{s^2+1} \left(1-\exp(- s)\right) \right) \right]$$

I am fine for either/both obtaining

(1) a analytic solutions $F(r)$ (say, in terms of just a function of $r$),

or

(2) a numerical function of $r$ (say, a x-y plot of x=$r$ v.s. y=$f(r)$, even just a selection of points running from $\{r, 0, 100, 1\}$)

I find the most challenge part is that the choices we can make for doing the Integrate and NIntegrate. How to wisely choose NIntegrate and Integrate to obtain the answer.

For example, what will be the plot of Plot[F(r),{r,0,100}] or ListPlot[F(r),{r,0,100,1}] looks like?

Answer 1

The inner integral can be performed symbolically by

si = Integrate[s^2 (1 - Exp[-s])/(s^2 + 1), {s, 0, w}, Assumptions -> w > 0]
(* w - ArcTan[w] + 1/2 (-2 + 2 E^-w - I E^I ExpIntegralEi[-I] + 
   I E^-I ExpIntegralEi[I] + I E^I ExpIntegralEi[-I - w] - I E^-I ExpIntegralEi[I - w]) *)

and the outer integral numerically by

s = NDSolveValue[{f'[w] == Exp[-si], f[0] == 0}, f, {w, 0, 10}];
F[r_] := Exp[s[r]]
Plot[F[r], {r, 0, 10}, PlotRange -> All, AxesLabel -> {r, "F"}, 
    ImageSize -> Large, LabelStyle -> Directive[Bold, Black, Medium]]

Addendum

Although certainly not necessary, it is possible to simplify si as follows.

FullSimplify[ComplexExpand[si, TargetFunctions -> {Re, Im}], w > 0];
% /. 1/2 I E^-I (E^(2 I) ExpIntegralEi[-I - w] - ExpIntegralEi[I - w]) 
    -> Im[Exp[-I] ExpIntegralEi[I - w]]
% /. CosIntegral[1] Sin[1] - 1/2 Cos[1] (Pi + 2 SinIntegral[1]) - 1 
    -> N[CosIntegral[1] Sin[1] - 1/2 Cos[1] (Pi + 2 SinIntegral[1]) - 1]
(* -2.07596 + E^-w + w - ArcTan[w] + Im[E^-I ExpIntegralEi[I - w]] *)

Answer 2

Thank you for your correction of the question and revising to replace r with s which changes the problem.

Exp[Integrate[Exp[-Integrate[s^2/(s^2+1)(1-Exp[-s]), {s,0,w}]], {w,0,r}]]

which gives

E^Integrate[ConditionalExpression[
   E^(-w + ArcTan[w] + (2 - 2/E^w + I*E^I*ExpIntegralEi[-I] - 
      (I*ExpIntegralEi[I])/E^I - I*E^I*ExpIntegralEi[-I - w] + 
      (I*ExpIntegralEi[I - w])/E^I)/2), Re[w] > 0 && Im[w]==0], {w,0,r}]

because MMA can't determine whether all your values are real. Adding assumptions such as bbgodfrey did can then simplify this to probably what you are looking for.