Will by parts be helpful?

If the integral $\int_0^{\frac{\pi}{4}}y^2\ln\left(\frac{\cos y}{\ln 2}\right)dy=\frac{3\pi}{b}\zeta(z)+ \frac{\pi^2}{b}G-\frac{\ln(\ln4)}{c}\pi^z-\frac{1}{d}\left(\psi^3\left(\frac{z}{4}\right)-\psi^3\left(\frac{1}{4}\right)\right)$ where $a,b,c,d$ and $z$ are positive integers with $z$ being prime, then find the value of $a+b+c+d+z$ .

Notation: $G$ is Catalan's Constant , $\psi^m(.)$ is Polygamma function and $\zeta(.)$ is Riemann zeta function .

Inspired by How many integrals