# Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?

Recall the definitions of the sine and cosine integrals:
$$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$
Both functions are oscillating, with a countably infinite number of minima and maxima.

Note that
$$\lim_{x\to\infty}\operatorname{Si}(x)=\frac\pi2,\quad\lim_{x\to\infty}\operatorname{Ci}(x)=0.$$
Consider the following function:
$$f(x)=\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}.$$

It appears that the function $f(x)$ and all its derivatives are monotonic for $x>0$. Specifically, the function itself and all its derivatives of an even order are strictly decreasing, and all its derivative of an odd order are strictly increasing.

Is it actually true? If so, then how can we prove it?