I saw this question and I was able to prove that $f(x)= e^x \left(\frac{x+1}{x^2+1}\right)^x \frac{(x+1)}{e^{2\arctan(x)}} $ but a new question came to my mind : Does the sequence $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly to $f(x)$ on $\mathbb{R}^+$? I tried to use Cauchy criterion to prove this but it didn't lead to anything useful. I also tried to check the monotonicity of this sequence of functions to use Dini’s Theorem on every closed interval .
Dini’s Theorem:- Suppose that $f_n$ is a monotone sequence of continuous functions on $I := [a, b]$ that converges on $I$ to a continuous function $f$. Then the convergence of the sequence is uniform.
but graphing $f_5 , f_{10}$ proves that this claim is wrong.
