Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

Question

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$

How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$)

If possible, could one also provide a proof please?

Answer 1

I figured out with the use of user1952009's comments the solution:

$$\frac{a}{b}=\frac{2^t}{(2^{1+t}-1)t!}$$

Answer 2

Using my newfound knowledge of Mellin Transforms, we can note that if we denote

$$F(s)=\int_0^\infty x^{s-1} \space f(x) \text{ d}x$$

Plugging in $f(x)=\text{csch} x$

One might note that $$\frac{a}{b}=F(s+1)$$

which is known as the result given.