integration - Integrating $int_0^{frac{pi}{2}} frac{x}{cos(x)+sin(x)} mathrm dx$ using complex analysis

I am struggling with the integration of

$$\text{I} = \int_0^{\frac{\pi}{2}} \frac{x}{\cos(x)+\sin(x)} \mathrm dx$$ using complex analysis. I used the substitution $z=e^{ix} \rightarrow \mathrm dx = \frac{1}{iz}\mathrm dz$ and hence $\cos(x)+\sin(x)= ((z+\frac{1}{z})-i (z-\frac{1}{z}))$ because we have $\cos(x)=\frac{1}{2}(e^{ix}+ e^{-ix})$, and $\sin(x)=\frac{1}{2i}(e^{ix} -e^{-ix})$, and $x= -i \ln(z)$. However I am not successful to obtain the correct result when using real analysis, which gives $$\text{I}= \displaystyle\int_{0}^{\frac{\pi}{2}}\dfrac{x}{\sin\,x + \cos\,x}\,dx = 0.978959918$$

Any idea how to calculate this integral using complex analysis and Cauchy integral theorem?