TITLE: I need some help with the integral $\int_0^\infty \frac{x^3}{e^x+1} \, dx$
QUESTION [12 upvotes]: I've been stuck on an integral for days now and would love to get some help with it:
$$\int_0^\infty \frac{x^3}{e^x+1} \, dx$$
My teacher was also "kind" enough to give me the answer to another similiar integral: 
$$\int_0^\infty \frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}$$
So that I should (based on that answer) calculate the integral:
$$\int_0^\infty \frac{x^3}{e^x+1} \, dx$$
I can't see how we can simplify/substitute anything so that it'll match the "help integral" so that we can use it's value to calculate the actual integral.
First post on this page, so please be kind! :)

REPLY [2 votes]: If you're interested in an approach to get both integrals without knowing either, $$\int_0^\infty\frac{x^3e^{-x}dx}{1\pm e^{-x}}=\sum_{n\ge 1}(\mp 1)^{n-1}\int_0^\infty x^3e^{-nx}dx=6\sum_{n\ge 1}\frac{(\mp 1)^{n-1}}{n^4},$$giving either $6\zeta(4)$ or $6\eta(4)$.