By using the properties of definite integrals,evaluate the integral $ \begin{align*}\int\limits_0^\frac{\pi}{4} log (1+\tan x)\;dx \end{align*}$

Question

$\begin{array}{1 1} \frac{\pi}{8} \log 2 \\ \frac{\pi}{4} \log 16 \\ \frac{\pi}{8} \log 8 \\ \frac{\pi}{4} \log 8 \end{array} $