JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

Let $n$ be the number of integral values of $x$, which satisfy the equation

$\tan ^{-1} x-\sin ^{-1} x=\cos ^{-1} x-\cot ^{-1} x$.

Let $\vec{c}=|\vec{a} \times \vec{b}| \vec{a}+(\vec{a} \cdot \vec{b}) \vec{b} \quad$ where $\vec{a}$ and $\vec{b}$ are unit vectors and angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$

Let the value of definite integral $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\ln (\sec x)}{(2013)^{\sin x}+1} d x$ equals $\frac{\pi}{2} \ln k$, then the value of $\left(\mathrm{n}+2|\overrightarrow{\mathrm{c}}|^2+\mathrm{k}\right)$ is equal to