# 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.
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