A3. What is the value of the expression $\frac{\sin 2 \alpha}{1+\tan \alpha}+\left(\cos \alpha+\sin ^{2} \alpha\right)^{2}$ for $\alpha=\frac{\pi}{4}$?
(A) $\frac{\sqrt{2}+5}{4}$
(B) $\frac{\sqrt{2}+6}{4}$
(C) $\frac{5+2 \sqrt{2}}{4}$
(D) $\frac{5-2 \sqrt{2}}{4}$
(E) $5+2 \sqrt{2}$