Theorem: Primitive of x over Power of Hyperbolic Sine of a x

:$\ds \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh^n a x} 
      | r = \int x \csch^n a x \rd x
      | c = {{Defof|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} a x} \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} a x}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}