Who's up to the challenge? 5

Calculus Level 4

$\large \int_0^1 x^2 \cos x \sin x \, dx$

If the integral above is in the form of

$\dfrac {-A + B\sin C - D\cos C}E ,$

where $A,B,C,D$ and $E$ are positive integers, find the minimum value of $A+B+C+D+E$ .

Clarification: Angles are measured in radians.