Evaluate
\left\{\begin{matrix}\frac{8a^{2}\sin(x)\sin(2ax)+2a\cos(x\left(2a+1\right))+2a\cos(x\left(2a-1\right))-16a^{3}\cos(x)-2\sin(x)\sin(2ax)-\cos(x\left(2a+1\right))+\cos(x\left(2a-1\right))+4a\cos(x)}{8a\left(4a^{2}-1\right)}+С,&a\neq 0\text{ and }|a|\neq \frac{1}{2}\\-\left(\cos(\frac{x}{2})\right)^{4}+С,&|a|=\frac{1}{2}\\-\cos(x)+С,&a=0\end{matrix}\right.
Differentiate w.r.t. x
\left\{\begin{matrix}\frac{4a\sin(x)\cos(2ax)+2\cos(x)\sin(2ax)-\sin(x\left(2a+1\right))-\sin(x\left(2a-1\right))+4a\sin(x)}{8a},&a\neq 0\text{ and }|a|\neq \frac{1}{2}\\2\sin(\frac{x}{2})\left(\cos(\frac{x}{2})\right)^{3},&|a|=\frac{1}{2}\\\sin(x),&a=0\end{matrix}\right.
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