For $x^2\neq n\pi+1,n \in N$ (the set of natural numbers), the integral

$$\int x \sqrt{{2\sin{(x^2-1)}-\sin2{(x^2-1)}}\over{2\sin{(x^2-1)}+\sin2{(x^2-1)}}}dx$$

is equal to :

(where $c$ is a constant of integration)

(1) $\log_e|{\sec{({x^2-1\over2})}}|+c$

(2) $\log_e|{{1\over2}\sec^2{(x^2-1)}}|+c$

(3) ${1\over2}\log_e|{\sec^2{({x^2-1\over2})}}|+c$

(4) ${1\over2}\log_e|{\sec{(x^2-1)}}|+c$

$$\int x \sqrt{{2\sin{(x^2-1)}-\sin2{(x^2-1)}}\over{2\sin{(x^2-1)}+\sin2{(x^2-1)}}}dx$$

is equal to :

(where $c$ is a constant of integration)

(1) $\log_e|{\sec{({x^2-1\over2})}}|+c$

(2) $\log_e|{{1\over2}\sec^2{(x^2-1)}}|+c$

(3) ${1\over2}\log_e|{\sec^2{({x^2-1\over2})}}|+c$

(4) ${1\over2}\log_e|{\sec{(x^2-1)}}|+c$