Generalized Sophomore's dream. Question about originality

Although the OP has stated that he/she has already derived the more general form of the so-called "Sophomore's Dream," I thought that it might benefit others to see the development herein. So, here we go ...

$$\begin{align} \int_{-s}^{a-s} \frac{a^t}{(t+s)^t}dt&=a\int_0^1 t^st^{-at}dt \tag1\\\\ &=\sum_{n=0}^{\infty}\frac{(-1)^na^{n+1}}{n!}\int_0^1t^{n+s}\log^nt\,dt \tag2\\\\ &=\sum_{n=0}^{\infty}\frac{(-1)^na^{n+1}}{n!}\frac{(-1)^n}{(n+1)^{n+1}}\int_0^{\infty} t^ne^{\frac{n+1+s}{n+1}x}dx \tag3\\\\ &=\sum_{n=0}^{\infty}\frac{a^{n+1}}{n!(n+1+s)^{n+1}}\int_0^{\infty}t^ne^{-t}dt \tag4\\\\ &=\sum_{n=0}^{\infty}\frac{a^{n+1}}{(n+1+s)^{n+1}}\tag5\\\\ &=\sum_{n=1}^{\infty}\frac{a^{n}}{(n+s)^{n}}\tag6 \end{align}$$

as was to be shown!

NOTES:

$(1)$

We enforced the substitution $t \to at-s$

$(2)$

We wrote $t^{-at}=e^{-at\log t}$ and used the power series representation for $e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$. We also used the uniform convergence of the power series to justify interchanging the integral and summation.

$(3)$

We enforced the substitution $t\to e^{-t/(n+1)}$.

$(4)$

We enforced the substitution $t\to \frac{n+1}{n+1+s}t$.

$(5)$

We used the integral representation of the Gamma Function $\Gamma (z)=\int_0^{\infty}t^{z-1}e^{-t}dt$, which for $z=n+1$ is $\Gamma (n+1)=n!$.

$(6)$

We shifted the index of summation using $n\to n-1$