Whose integrals is it?

Calculus Level 5

\[\displaystyle \large\sum\limits_{k=1}^{2018} \int_0^{\infty} \sin^{2k} \bigg(\frac{1}{x}\bigg) \mathrm{d}x = \displaystyle a \sqrt{\pi} \cdot \dfrac{ \Gamma \big(\frac{b}{c} \big)}{\Gamma (d) } \]

The equation above holds true for positive integers \(a\), \(b\), \(c\) and \(d\) with \( \gcd(b,c) = 1 \). Find \( a+b+c+d \).

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Notation: \( \Gamma(\cdot) \) denotes the Gamma function.

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