Fermion-to-boson thermal integral ratio
= Fermion-to-boson thermal integral ratio
For $s>0$,
$$
\int_0^\infty\frac{x^s\,dx}{e^x+1}
=\left(1-2^{-s}\right)
\int_0^\infty\frac{x^s\,dx}{e^x-1}.
$$
This follows from $(e^x+1)^{-1}=(e^x-1)^{-1}-2(e^{2x}-1)^{-1}$. In three spatial dimensions it gives the fermion-to-boson ratios $3/4$ for ultrarelativistic particle number and $7/8$ for energy density.