How to solve this indefinite integral? $\int {t}{\sqrt{1+\cos(t)}} \,dt$

I tried solving the following integral:

$\int {(t+\sin(t))}{\sqrt{1+\cos(t)}} \,dt$

I split the problem in two parts.

The first part gave me this:

$\int {\sin(t)}{\sqrt{1+\cos(t)}} \,dt = -\frac{2}{3}{(1+\cos(t))}^{\frac{3}2}+c$

I didn't know how to solve the second part:

$\int {t}{\sqrt{1+\cos(t)}} \,dt$

I tried substitution but couldn't find a way to integrate

HINT

You may be interested in the following identity: \begin{align*} 1 + \cos(t) = 2\cos^{2}\left(\frac{t}{2}\right) \end{align*}

Hence the proposed integral reduces to \begin{align*} \int t\sqrt{1 + \cos(t)}\mathrm{d}t = \pm\sqrt{2}\int t\cos\left(\frac{t}{2}\right)\mathrm{d}t \end{align*}

which can be easily solved through IBP.

Can you take it from here?