JEE Math Practice Question

For x2 $$ \ne $$ n$$\pi $$ + 1, n $$ \in $$ N (the set of natural numbers), the integral $$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $$ is equal to : (where c is a constant of integration)

1. A.$${\log _e}\left| {{1 \over 2}{{\sec }^2}\left( {{x^2} - 1} \right)} \right| + c$$
2. B.$${1 \over 2}{\log _e}\left| {\sec \left( {{x^2} - 1} \right)} \right| + c$$
3. C.$${1 \over 2}{\log _e}\left| {{{\sec }^2}\left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$$
4. D.$${\log _e}\left| {\sec \left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$$