Answer
$$ - \coth \frac{{{x^2}}}{2} + C$$
Work Step by Step
$$\eqalign{
& \int {x{\text{csc}}{{\text{h}}^2}\frac{{{x^2}}}{2}dx} \cr
& {\text{Let }}u = \frac{{{x^2}}}{2},{\text{ }}du = xdx \cr
& {\text{Substituting}} \cr
& \int {x{\text{csc}}{{\text{h}}^2}\frac{{{x^2}}}{2}dx} = \int {x{\text{csc}}{{\text{h}}^2}u\left( {\frac{1}{x}} \right)du} \cr
& = \int {{\text{csc}}{{\text{h}}^2}udu} \cr
& {\text{Integrating }} \cr
& = - \coth u + C \cr
& {\text{Write in terms of }}x \cr
& = - \coth \frac{{{x^2}}}{2} + C \cr} $$
