Riemann's zeta function in a definite integral

Question

I am trying to evaluate the value of the definite integral

$ I= \int_0^1((1-\delta)\log[p]-\delta\log[1-p])^4dp$

Using Binomial expansion I get

$I=(1-\delta)^4\int_0^1(\log[p])^4dp-4(1-\delta)^3\delta\int_0^1(\log[p])^3(\log[1-p])dp+\\6(1-\delta)^2\delta^2\int_0^1(\log[p])^2(\log[1-p])^2dp-4(1-\delta)\delta^3\int_0^1(\log[p])(\log[1-p])^3dp+\delta^4\int_0^1(\log[1-p])^4dp$

Let $\psi(j,k)=\int_0^1(\log[p])^j(1-\log[1-p])^kdp$

Then

$I=(1-\delta)^4\psi(4,0)-4(1-\delta)^3\delta\psi(3,1)+6(1-\delta)^2\delta^2\psi(2,2)- 4(1-\delta)\delta^3\psi(1,3)+\delta^4\psi(0,4)$

Next, how do I evaluate this expression? This is done in a paper but the authors have not given the details. The final expression involves Riemann's zeta function too. I need some help here.

Edit : In particular, how do I evaluate $\psi(j,k) $?