Help with the integral $\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$

Question

We have the integral : $$\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$$ We have: $$\frac{1}{\Gamma(y)}=\frac{i}{2\pi}\int_{C}(-t)^{-y}e^{-t}dt$$ Where the path $C$ encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. Thus: $$\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy=\frac{i}{2\pi}\int_{0}^{\infty}\int_{C}\left(-\frac{t}{x}\right)^{-y}e^{-t}\cos(y)dtdy$$ But : $$\int_{0}^{\infty}\left(-\frac{t}{x}\right)^{-y}\cos(y)dy=\frac{\log(-t/x)}{1+\log^{2}(-t/x)}$$ Thus, our integral becomes: $$\frac{i}{2\pi}\int_{C}\frac{\log(-t/x)}{1+\log^{2}(-t/x)}e^{-t}dt$$ And i am stuck here !!