Mistake in solving $-\int \frac{1}{x\sqrt{x^2-1}}$ I have this function $$f:(-\infty ,-1)\rightarrow \mathbb{R}, f(x)=\frac{1}{x\sqrt{x^{2}-1}}$$ and I need to find the primitives of $f(x)$.So because $x<-1$ I need to calculate $-\int \frac{1}{x\sqrt{x^2-1}}\,dx=-\arctan(\sqrt{x^2-1})+C$ but in my book the correct answer is $\arcsin\left(\frac{1}{x}\right)+C$
Where is my mistake?I solved the integral using $u=\sqrt{x^{2}-1}$

Hint: 
You might want to plot both $\arcsin(\frac{1}{x})$ and $-\arctan(\sqrt{x^2 - 1})$, from, say, $-3$ to $-1$. You could do this using Desmos (https://www.desmos.com/). You might even insert an extra minus sign here or there...
