Integral 2

Calculus Level 3

\[\large \int_{0}^{\pi/2}\frac{d\theta}{{(\cos^3{\theta}+\sin^3{\theta})}^{2/3}} = \frac{a\pi^2/b}{\Gamma^3{(c/d)}} \]

The equation above holds true for positive integers \(a\), \(b\), \(c\), and \(d\), where \(\gcd(a,b)=\gcd(c,d)= 1\) and \(c\) is minimized. Find \(1000a+100b+10c+d\).

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

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