"I am Back" says Integration Part 4

$I=\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\ \log^ 2 (\sin(x)) \ \mathrm d x }$

If $16I$ can be represented in the form of $a\pi \zeta(b) - c\pi\log^d (f)$

Find $a+b+c+d+f$

Details and Assumptions

• $a,b,c,d,f$ are positive integers not neccasarily distinct.

• $\zeta$ denote the Riemann Zeta Function.

• $\log$ is the natural logarithm.

• $f$ is not a multiple of a perfect power of any integer greater than $1$ .