{{begin-eqn}} {{eqn 	| l = \int \frac {x \rd x} {\sinh^n a x}  	| r = \int x \csch^n a x \rd x 	| c = Definition of Hyperbolic Cosecant }} {{eqn 	| r = \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta 	| c = Substitution of $a x \to \theta$ }} {{eqn 	| ll= \leadsto         | l = \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta         | r = \frac 1 {a^2} \int \csch^2 \theta \times \theta \csch^{n - 2} \theta \rd \theta         | c =  $\rd u = \csch^2 \theta$ and $v = \theta \csch^{n - 2} \theta$ }} {{eqn 	| r = \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta - \int - \coth \theta \paren {- \paren {n - 2} \theta \coth \theta + 1} \csch^{n - 2} \theta \rd \theta} 	| c = Integration by Parts }} {{eqn 	| r = \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta  \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \coth^2 \theta \csch^{n - 2} \theta \rd \theta} 	| c = simplifying }} {{eqn 	| r = \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta  \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \paren {1 + \csch^2 \theta} \csch^{n - 2} \theta \rd \theta} 	| c = Difference of Squares of Hyperbolic Cotangent and Cosecant }} {{eqn 	| r = \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta  \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^n \theta \rd \theta} 	| c = simplifying }} {{eqn 	| ll= \leadsto         | l = \frac {1 + \paren {n - 2} } {a^2} \int \theta \csch^n \theta \rd \theta         | r = \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta  \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta} 	| c = adding end term to both sides }} {{eqn 	| ll= \leadsto         | l = \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta         | r = \frac 1 {a^2 \paren {n - 1} } \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta  \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta} 	| c = rearranging for the intended primitive }} {{eqn 	| r = \frac 1 {a^2 \paren {n - 1} } \paren {- \theta \coth \theta \csch^{n - 2} \theta + \paren {- \frac {\csch^{n - 2} \theta } { \paren {n - 2} } } - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta} 	| c = Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x }} {{eqn 	| r = - \frac {\theta \cosh \theta} {a^2 \paren {n - 1} \sinh^{n - 1} \theta } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} \theta } - \frac {n - 2} {a^2 \paren {n - 1} } \int \frac {\theta} {\sinh^{n - 2} \theta} \rd \theta 	| c = simplifying }} {{eqn 	| ll= \leadsto         | l = \int \frac {x \rd x} {\sinh^n a x}          | r = - \frac {a x \cosh a x} {a^2 \paren {n - 1} \sinh^{n - 1} a x } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} ax } - \frac {n - 2} {n - 1} \int \frac x {\sinh^{n - 2} ax} \rd x 	| c = Substituting back $\theta \to ax$ }} {{eqn   | r = \frac {- x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} ax } - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} ax} + C 	| c = simplifying and adding integration constant }} {{end-eqn}} {{qed}}