Answer
$$\frac{-1}{4} \operatorname{csch}^{-1} ( -1/4)+ \frac{-1}{4} \operatorname{csch}^{-1} ( -3/4)$$
Work Step by Step
We integrate as follows:
\begin{align*}
\int_{-3}^{-1} \frac{d x}{x \sqrt{x^{2}+16}}&=\int_{-3}^{-1} \frac{d x}{4x \sqrt{(x/4)^{2}+1}}\\
&= \frac{1}{4}\int_{-3}^{-1} \frac{(1/4)d x}{(x/4) \sqrt{(x/4)^{2}+1}}\\
&= \frac{-1}{4} \operatorname{csch}^{-1} ( x/4)\bigg|_{-3}^{-1}\\
&= \frac{-1}{4} \operatorname{csch}^{-1} ( -1/4)+ \frac{-1}{4} \operatorname{csch}^{-1} ( -3/4)
\end{align*}
