Here is another answer by me, in which I will give an explicit Mathematica code for evaluating integrals over surreal numbers. It is based on the same approach as my prevuous answer, but uses a more general formula.
It should be noted that my approach does not satisfy the property of linearity against an infinite coefficient. In other words, moving an infinite factor from under the integral changes the value of the expression: $\alpha\int f\ne \int (\alpha f)$. The property of linearity against infinite factors has been included in other approaches to surreal integration problem, but I consider it unnatural.
In the code below, for simplicity, in input w corresponds to $\omega$ and W to $\omega_1$. The input accepts any surreal numbers composed of these values as integration limits and any surreal-valued function of integration variable x as the expression under integral. In output, derivation is denoted as ꝺ.
f[x_] = w^W;
a = 0;
b = w;
f[x_] = f[x] /. W -> W[w];
Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
Fin[t_] := (PowerExpand[f[t]] /. Log[w] -> 0 /. w -> 0) +
Limit[Evaluate[LaplaceTransform[ D[f[t], w], t, x]] +
Evaluate[LaplaceTransform[ D[f[t], w], t, -x]], x -> 0]/
2 /. \[Infinity] -> 0 /. -\[Infinity] -> 0 //
FullSimplify (*Finding the finite part so to integrate separately*)
int = \[Pi] W Integrate[D[f[t], w], {t, a, b}] +
Integrate[Fin[t], {t, a, b}] // FullSimplify;
int = If[a == b, \[Pi] D[f[t], w], int, int] /.
Derivative[1][W][w] -> ꝺ[W] /. W[w_] :> W // Normal;
Print[Inactivate[
Integrate[
f[x] /. W[w] -> W /. w -> \[Omega] /.
W -> Subscript[\[Omega], 1], {x,
a /. w -> \[Omega] /. W -> Subscript[\[Omega], 1],
b /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]}],
Integrate], "=",
int /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]]
The code correctly gives all the results from the other my answer, but also can find more complicated results, such as:
$\int _0^{\omega }e^{c x \omega }dx=\frac{c \omega ^3+\omega +\pi \left(e^{c \omega ^2} \left(c \omega ^2-1\right)+1\right) \omega _1}{c \omega ^2}$
$\int _0^{\omega }e^{c x^2 \omega }dx=\omega +\frac{1}{2} e^{c \omega ^3} \pi \omega _1-\frac{\pi ^{3/2} \text{erfi}\left(\sqrt{c} \omega ^{3/2}\right) \omega _1}{4 \sqrt{c} \omega ^{3/2}}$
$\int _0^{\omega }\log (c x \omega )dx=\omega (\log (c)+\log (\omega )-1)+\pi \omega _1$
$\int _0^{\omega }\omega ^xdx=\frac{\frac{\pi \left((\omega \log (\omega )-1) \omega ^{\omega }+1\right) \omega _1}{\omega }+1}{\log ^2(\omega )}$
$\int _0^{\omega }\omega ^{\omega }dx=\pi (\log (\omega )+1) \omega _1 \omega ^{\omega +1}+\omega$
$\int _0^{\omega }p \omega ^{q \omega }dx=p \omega \left(\pi q (\log (\omega )+1) \omega _1 \omega ^{q \omega }+1\right)$
$\int _0^{\omega }\omega ^{\omega _1}dx=\pi \omega _1 \left(\omega \partial \left(\omega _1\right) \log (\omega )+\omega _1\right) \omega ^{\omega _1}$
$\int _0^{\omega }\omega _1^{\omega }dx=\pi \omega \left(\frac{\omega \partial\left(\omega _1\right)}{\omega _1}+\log \left(\omega _1\right)\right) \omega _1^{\omega +1}+\omega$
$\int _0^{\omega }\omega _1^xdx=\frac{\omega _1^{\omega }-1}{\log \left(\omega _1\right)}+\frac{\partial\left(\omega _1\right) \left(\pi \left(\omega \log \left(\omega _1\right)-1\right) \omega _1^{\omega +1}+\pi \omega _1+\omega \right)}{\log ^2\left(\omega _1\right) \omega _1}$
$\int _0^{\omega _1}\omega ^xdx=\frac{\omega _1 \left(\pi \left(\log (\omega ) \omega _1-1\right) \omega ^{\omega _1}+\pi +1\right)}{\omega \log ^2(\omega )}$
$\int _0^{\omega }\log \left(\omega _1\right)dx=\omega \left(\log \left(\omega _1\right)+\pi \partial\left(\omega _1\right) \omega _1\right)$
$\int _0^{\omega }\log \left(\omega _1\right)dx=\omega \left(\log \left(\omega _1\right)+\frac{\pi \partial\left(\omega _1\right) \omega _1}{\omega _1(\omega )}\right)$
etc.