2018 SRJC P1 Q5 (ii)

The curve \(C\) is defined by the parametric equations

\(x=2t-2\ln \left( t+1 \right)+2\), \(y=-2t-2\ln \left( t+1 \right)+1\) where \(t>-1\).

Another curve \(L\) is defined by the equation \(x=2{{\left( y+1 \right)}^{2}}-6\). The graphs of \(C\) and \(L\)intersect at the point \(A\left( 2,1 \right)\) as shown in the diagram below.

Find the area of the shaded region bounded by \(C\), \(L\) and the line \(y=-1-\ln 4\), express your answer in the form \(\frac{62}{3}+4A-2{{A}^{2}}-\frac{16{{A}^{3}}}{3}\), where \(A\) is an exact real constant.[6]