Consider the function $f: D \to \mathbb{C}$ defined by $f(z) = \log_e(z^5+1)$. Obviously, to check that it is analytic we can let $z = x+iy$ and the verify where the Cauchy-Riemann equations are satisfied. This involves the expansion of $(x+iy)^5$ however, and I don't think anyone wants to do that.
Is there a faster method?