$$y^n-x^m-\int^{n}_{m}x^\phi\ln{x}\text{ d}\phi\doteq\sum^{n}_{i=1}\sum^i_{\rho=0}\left((-1)^\rho\cdot\frac{\prod\limits_{j_1=\rho+1}^{i}(j_1)}{(i-\rho)!y^\rho}y^ix^\rho\right)\frac{\prod\limits_{j_2=i+1}^{n}(j_2)}{(n-i)!x^i}x^n$$
I'm pretty sure it's correct, but I have no way of verifying/proving.