Intriguing integrals! - one

Calculus Level 3

\[ \large \int_0^3 \frac{x^3}{1+4x+6x^2+4x^3+x^4} \, dx \]

Consider the integral above, if this integral can be represented as \[\ln(a) - \frac{b}{c} \] for positive integers \(a,b,c\) with \(b,c\) coprime, find the value of \( a+b+c\).

Please post your diverse approaches.
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