$let Z = i^log⁡(i+1) ∴ logZ = log(1+i).logi$ $But log(i+1) = log√2 + itan^{-1}⁡1 = log√2 + iπ/4 and logi = i.( π)/2$ $∴ logZ = (log√2 + iπ/4 ). i.( π)/2 = \frac{[1/2 log2+i π/4 π)}{2} = (-π)^2/8+i π/4 log2 = e^{-π^2/8+iθ} = e^{-π^2/8.iθ} where θ= π/4 log2 = e^{-π^2/8}[cosθ+isinθisinθ$ $∴ Real part of Z = e^{-π^2/8}cosθ= e^{-π^2/8}cos(π/4 log2)∴nary part of Z = e^{-π^2/8}sin(π/4 log2)$