Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 12

$$-\operatorname{csch}x+C$$

Given $$\int \frac{\cosh x}{\sinh^2 x} d x $$ Let $$ u= \sin h x \ \ \ \to du = \cosh x dx$$ Then \begin{aligned} \int \frac{\cosh x}{\sinh ^{2} x} d x &=\int \frac{1}{u^{2}} d u \\ &=\int u^{-2} d u \\ &=-u^{-1}+C \\ &=-\frac{1}{\sinh x}+C \\ &=-\operatorname{csch}x+C \end{aligned}