Answer
$$\frac{1}{4}\left[\operatorname{csch}^{-1}\left(-\frac{3}{4}\right)-\operatorname{csch}^{-1}\left(-\frac{1}{4}\right)\right]$$
Work Step by Step
\begin{aligned} \int_{-3}^{-1} \frac{1}{x \sqrt{x^{2}+16}} d x &=\int_{-3}^{-1} \frac{1}{x \sqrt{16\left(\frac{x^{2}}{16}+1\right)}} d x \\ &=\int_{-3}^{-1} \frac{1}{4 x \sqrt{\left(\left(\frac{x}{4}\right)^{2}+1\right.})} d x \\ &=\int_{-3}^{-1} \frac{\frac{1}{4}}{\left.4 \cdot \frac{x}{4} \sqrt{\left(\left(\frac{x}{4}\right)^{2}+1\right.}\right)} d x \\ &=\frac{1}{4}\left[-\operatorname{csch}^{-1}\left(\frac{x}{4}\right)\right]_{-3}^{-1} \\ &=\frac{1}{4}\left[\operatorname{csch}^{-1}\left(-\frac{3}{4}\right)-\operatorname{csch}^{-1}\left(-\frac{1}{4}\right)\right] \end{aligned}
