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Calculus Level 5

\[ \large \displaystyle \int_{0}^{\infty}e^{-x^{8}}\cos(x^{8})x^{3}\ln(x)\, dx \] If the integral above can be expressed as \[ \dfrac{-\pi^{\frac{1}{A}}}{B}\left[ \sqrt{\sqrt{A}+C}\left(\gamma^{C}+\dfrac{D}{A}\ln(A)\right)+\dfrac{\pi^{C}}{E}\sqrt{\sqrt{A}-C}\right] \]

with positive integers \( A,B,C,D,E \) and all fractions are irreducible and \(A\) being square free, then evaluate \(A+B+C+D+E\).

\[\] Notation: \(\gamma \approx 0.5772 \) denotes the Euler-Mascheroni constant.

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