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