Counterintuitive integration

Calculus Level 5

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

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.
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