Combine $\sin 3x+\sin x=2\sin 2x\cos x$. So the integrand $=|\sin 2x|\cdot |2\cos x+1|=2|\sin x||\cos x|\cdot |2\cos x+1|$, and on $[0,\pi ]$ where $\sin x\ge 0$:

$I=12{\int }_0^{\pi }\sin x|\cos x(2\cos x+1)|dx=12{\int }_0^{\pi }\sin x|2{\cos }^{2}x+\cos x|dx$.

Substitute $t=\cos x$, $dt=-\sin xdx$. Limits go from $1$ to $-1$:

$I=12{\int }_{-1}^1|2t^2+t|dt$.

$2t^2+t=t(2t+1)$ changes sign at $t=0$ and $t=-1/2$. Split the integral over $[-1,-1/2]$, $[-1/2,0]$, $[0,1]$ and compute each elementary integral; the total equals $17$.