1. **State the problem:** Simplify the expression $$\cos\left(2\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)\bigg/\sqrt{\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)^2 +1}\right)$$. 2. **Identify the structure:** The expression inside the cosine is a fraction where the numerator is $$2\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)$$ and the denominator is $$\sqrt{\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)^2 +1}$$. 3. **Simplify the fraction:** Let $$A = x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}$$. Then the expression becomes $$\cos\left(\frac{2A}{\sqrt{A^2+1}}\right)$$. 4. **Rewrite the denominator:** Note that $$\sqrt{A^2+1} = \sqrt{1 + A^2}$$. 5. **No further algebraic simplification is possible without additional context or values for $p$ and $x$.** **Final simplified form:** $$\cos\left(\frac{2\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)}{\sqrt{\left(x+2p(x^2+1)\sin(2x)+p\frac{x\cos(2x)}{(x^2+1)^{3/2}}\right)^2 +1}}\right)$$