Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 10

$$-\frac{1}{2} \operatorname{csch}\left(x^{2}\right)+C$$

Given $$\int x \operatorname{csch}\left(x^{2}\right) \operatorname{coth}\left(x^{2}\right) d x $$ Let $$ u= x^2 \ \ \ \to du = 2x dx$$ Then \begin{aligned} \int x \operatorname{csch}\left(x^{2}\right) \operatorname{coth}\left(x^{2}\right) d x &=-\frac{1}{2} \int d u \\ &=-\frac{1}{2} u+C \\ &=-\frac{1}{2} \operatorname{csch}\left(x^{2}\right)+C\end{aligned}