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## 2.4 Hyperbolic Functions

### 2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x

2.411 2.412

1.

$\begin{array}{lll}{\int }^{\text{​}}{\text{sinh}}^{p}x{\text{cosh}}^{2n}x\text{d}x\hfill & =\hfill & \frac{{\text{sinh}}^{p+1}}{2n+p}\left[{\text{cosh}}^{2n-1}x\hfill \\ +\sum _{k=1}^{n-1}\frac{\left(2n-1\right)\left(2n-3\right)\dots \left(2n-2k+1\right)}{\left(2n+p-2\right)\left(2n+p-4\right)\dots \left(2n+p-2k\right)}{\text{cosh}}^{2n-2k-1}x\right]\hfill \\ +\frac{\left(2n-1\right)!!}{\left(2n+p\right)\left(2n+p-2\right)\dots \left(p-2\right)}{\int }^{\text{​}}{\text{sinh}}^{p}x\text{d}x\hfill \end{array}$

This formula is applicable for arbitrary real p except for the following negative even integers: ...

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