Chebyshev coefficients for the Clausen function \(Cl_ 2(x)\) (Q1917924)

The Clausen function \(Cl_2(x)= - \int^x_0 \ln|2\sin (t/2)|dt\) is expanded in the forms \[ Cl_2(x)= \begin{cases} x- x\ln |x|+ \textstyle{{1\over 2}} x^3 \displaystyle{ \sum^\infty_{n= 0}} a_n T_{2n}(2x/\pi)\quad&\text{for }|x|\leq \pi/2\text{ and}\\ (\pi- x) \sum^\infty_{n= 0} b_n T_{2n}(2x/\pi)\quad&\text{for }\pi/2 < x\leq 3\pi/2.\end{cases} \] The numerical values of the coefficients \(a_n\), \(b_n\) (twenty digits) are given for evaluating the series.