The curve \(C\) has equation \(y=-\ln \left(1-x^{2}\right)\) for \(-\frac{1}{2} \leqslant x \leqslant \frac{1}{2}\). Show that
\(1+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}=\left(\frac{1+x^{2}}{1-x^{2}}\right)^{2}\)

Show further that \(\frac{1+x^{2}}{1-x^{2}}\) may be expressed in the form \(\frac{P}{1+x}+\frac{Q}{1-x}+R\), where \(P, Q\) and \(R\) are constants to be determined.

Find the exact arc length of \(C\).