$\int_0^\infty \frac{dt}{(1+t)(2+t)...(n+t)}$

Question

Good evening,

I am trying to calculate the following integral : $\int_0^\infty \frac{dt}{(1+t)(2+t)...(n+t)}$ for $n > 1$.

Using the decomposition of the rational fraction, and then integrating, I have the following expression, which I can't simplify :

$ln(\frac{\Pi_{l=1}^{\lfloor(n-1)/2\rfloor}(2l+1)^{\frac{1}{(n-2l-1)!(2l)!}}}{\Pi_{l=1}^{\lfloor n/2\rfloor}(2l)^{\frac{1}{(n-2l)!(2l-1)!}}})$

How can I proceed ?

Thank you very much, Respectfully,

AF