# Counterintuitive integration

**Calculus**Level 4

\[ \int_{\pi/2}^{\infty} \dfrac{x^4 \cos x + 5x \cos x - 4x^3 \sin x - 5 \sin x}{x^8 + 10x^5 + 25x^2} \, dx \]

If the above integral can be represented in the form

\[ \dfrac{-1}{a \left( \dfrac{\pi}{2} \right)^b + c \left( \dfrac{\pi}{2} \right)^d} \]

where \( b > d \), and \(a , b, c, d \) are positive integers, find \(a+b+c+d\).

**Details and assumptions:**

- You can try but neither Mathematica nor Wolfram Alpha will compute this for you.