How to evaluate $\int_0^{\pi/6} \frac{x}{\sin x}$?

Question

A well-know representation of the Catalan constant is given by $$C=\frac12\int_0^{\pi/2}\frac{x}{\sin x}\,dx.$$ So, I asked myself if there are other values of $x$ that make the integral $$f(x)=\int_0^x \frac{t}{\sin t}\,dt$$ take well-known closed form values. Experimenting with Mathematica (see this), I found that $$f\left(\frac{\pi}{6}\right)=\frac{4}{3}C-\frac{\pi}{6} \cosh^{-1}(2).$$ However, I failed to prove this result. So, any help would be very appriciated.

Answer 1

Utilize $\int_0^{\pi/12} \ln(\tan x)\,dx =-\frac{2}3C$ to obtan \begin{align} f\left(\frac{\pi}{6}\right)=&\int_0^{\pi/6} \frac{t}{\sin t}\,dt = \int_0^{\pi/6}t\ d(\ln(\tan\frac t2) )\\ \overset{ibp}=&\ \frac{\pi}6 \ln(\tan\frac{\pi}{12})-\int_0^{\pi/6} \ln(\tan\frac t2) dt =-\frac{\pi}{6} \cosh^{-1}2+ \frac{4}{3}C\\ \end{align}