"I am Back" says Integration Part 4

Calculus Level 5

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

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 $$1$$.

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