TITLE: Integrating $\frac{x^k }{1+\cosh(x)}$
QUESTION [15 upvotes]: In the course of solving a certain problem, I've had to evaluate integrals of the form:
$$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$
for several values of k. I've noticed that that, for k a positive integer other than 1, the result is seemingly always a dyadic rational multiple of $\zeta(k)$, which is not particularly surprising given some of the identities for $\zeta$ (k=7 is the first noninteger value).
However, I've been unable to find a nice way to evaluate this integral. I'm reasonably sure there's a way to change this expression into $\int \frac{x^{k-1}}{e^x+1} \mathrm{d}x$, but all the things I tried didn't work. Integration by parts also got too messy quickly, and Mathematica couldn't solve it (though it could calculate for a particular value of k very easily).
So I'm looking for a simple way to evaluate the above integral.

REPLY [5 votes]: sos440 Added it to my collection $$\int\limits_0^\infty \frac{x^{s-1}}{e^x-1}dx=\zeta(s)\Gamma(s)$$
$$\int\limits_0^\infty \frac{x^{s-1}}{e^x+1}dx=\eta(s)\Gamma(s)$$
$$\int\limits_0^\infty \frac{x^{s}e^x}{(e^x-1)^2}dx=\zeta(s)\Gamma(s+1)$$
$$\int\limits_0^\infty \frac{x^{s}e^x}{(e^x+1)^2}dx=\eta(s)\Gamma(s+1)$$
$$\int\limits_0^1 \frac{x^s}{1-x}\log x dx = \sum\limits_{k=0}^\infty \frac{1}{(k+s)^2}$$
I'm guessing my memory is all right, but feel free to correct any mistakes!