If the primitive of $$\cos (\log x)$$ is $$f(x)\{\cos (g(x))+\sin (h(x))\}$$, then which among the following is true?

$$\int \frac{\sec x}{\sqrt{\sin (2 x+\theta)+\sin \theta}} d x$$ is equal to

Given, $$\frac{3 x-2}{(x+1)^2(x+3)}=\frac{A}{x+1} +\frac{B}{(x+1)^2}+\frac{C}{x+3}$$, then $$4 A+2 B+4 C$$ is equal to

$$\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} d \alpha$$ is equal to