El Niño integral

Calculus Level 4

\[\large \displaystyle{\int_{1}^{2} \dfrac{x}{2x^2+5x+7} \, dx = \frac{A}{B} \ln\left(\frac{C}{D}\right)+\frac{E\tan^{-1}\left(\frac{F}{\sqrt{G}}\right)}{H\sqrt{I}} - \frac{J\tan^{-1}\left(\frac{K}{\sqrt{L}} \right) }{M\sqrt{N}}}\]

The equation above holds true for positive integers \(A,B,C,D,E,F,G,H,I,J,K,L,M\) and \(N\) such that \[ \gcd(A,B) = \gcd(C,D) = \gcd(E,F) = \gcd(K,L) = \gcd(J,M ) = 1, \] with \(I \) and \(N\) square-free.

Find \(A+B+C+D+E+F+G+H+I+J+K+L+M+N\).

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