The sine integral $\text{Si}(x)$ is defined as $$\text{Si}(x):=\int_0^x \frac{\sin(t)}{t}dt$$
and the cosine integral $\text{Ci}(x)$ is defined as $$\text{Ci}(x):=\gamma+\int_0^x \frac{\cos(t)-1}{t} dt+\ln(x)$$
$\gamma$ is the Euler-Macheroni-constant.
These functions are needed to calculate the following definite integrals :
$$\int_0^1 \ln(x)\cos(x) dx=-\text{Si}(1)$$
$$\int_0^1 \ln(x)\sin(x) dx=\text{Ci}(1)-\gamma$$
Is it known, whether the numbers $\text{Si}(1)$ , $\text{Ci}(1)$ and $\text{Ci}(1)-\gamma$ are rational , irrational algebraic or transcendental ?
It is not known whether $\gamma$ is rational, but this does not rule out the possibility that the status of $\text{Ci}(1)-\gamma$ is known.