Neither square nor cube root. It's 10th!

Calculus Level 5

\[\large\displaystyle \int_{0}^{\infty} \dfrac{x^{\frac{1}{10}}}{x^2+1} \, dx = \dfrac{A\pi}{B}\sec \left(\dfrac{C\pi}{D}\right) \]

If the equation above holds true for positive integers \(A,B,C,D\) with \(\gcd(A,B)=\gcd(C,D)=1\) Compute \(A+B+C+D\).

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