Integration mania!

Calculus Level 5

If the value of the integral \[I=\displaystyle\int_{0}^{1} \dfrac{\log\left({3t^2+4t+3}\right)-\log{(3(t^2+1))}}{t}dt\]

Is of the form

\[\frac{\arcsin {\left(\frac{p}{q}\right)}\left(k\pi-\arcsin {\left(\frac{p}{q}\right)}\right)}{r}\],

where \(p\) and \(q\) are co-prime integers, then find \(p+q+r+k\).

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