Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 30

$$\frac{1}{2}\left[-\operatorname{csch}^{-1}(81)+\operatorname{csch}^{-1}(1)\right] $$

\begin{aligned} \int_{1}^{9} \frac{1}{x \sqrt{x^{4}+1}} d x &=\int_{1}^{9} \frac{1}{2} \cdot \frac{2 x}{x^{2} \sqrt{\left(x^{2}\right)^{2}+1}} d x \\ &=\frac{1}{2}\left[-\operatorname{csch}^{-1}\left(x^{2}\right)\right]_{1}^{9} \\ &=\frac{1}{2}\left[-\operatorname{csch}^{-1}(81)+\operatorname{csch}^{-1}(1)\right] \end{aligned}