Table of Integrals, Series, and Products, 8th Edition

2.47 Combinations of hyperbolic functions and powers

2.471

1.¹² 

$\begin{array}{c}{\displaystyle \int x^r{\text{sinh}}^p}x{\text{cosh}}^{q}x\text{d}x\hfill \\ =\hfill & \frac{1}{{\left(p+q\right)}^2}[\left(p+q\right)x^r{\text{sinh}}^{p+1}x{\text{cosh}}^{q-1}x\hfill \\ -r{x}^{r-1}{\text{sinh}}^{p}x{\text{cosh}}^{q}x+r\left(r+1\right){\displaystyle \int x^{r-2}{\text{sinh}}^{p}x{\text{cosh}}^{q}x\text{d}x}\hfill \\ +rp{\displaystyle \int x^{r-1}{\text{sinh}}^{p-1}x{\text{cosh}}^{q-1}}x\text{d}x+\left(q-1\right)\left(p+q\right){\displaystyle \int x^r{\text{sinh}}^{p}x{\text{cosh}}^{q-2}x\text{d}x}]\hfill \\ =\hfill & \frac{1}{{\left(p+q\right)}^2}[\left(p+q\right)x^r{\text{sinh}}^{p-1}x{\text{cosh}}^{q+1}x\hfill \\ -r{x}^{r-1}{\text{sinh}}^{p}x{\text{cosh}}^{q}x+r\left(r-1\right){\displaystyle \int x^{r-2}{\text{sinh}}^{p}x{\text{cosh}}^{q}x\text{d}x}\hfill \\ -rp{\displaystyle \int x^{r-1}{\text{sinh}}^{p-1}x{\text{cosh}}^{q-1}}x\text{d}x-\left(p-1\right)\left(p+q\right){\displaystyle \int x^r{\text{sinh}}^{p-2}x{\text{cosh}}^{q}x\text{d}x}]\hfill \end{array}$

GU (353)(1)

2. 

${\displaystyle \int x^n{\text{ sinh}}^{2m}x\text{ d}x={\left(-1\right)}^m\left({}_m^{2m}\right)\frac{x^{n+1}}{2^{2m}\left(n+1\right)}+\frac{1}{2^{2m–1}}{\displaystyle \sum _{k=0}^{m-1}{\left(-1\right)}^k\left({}_k^{2m}\right){\displaystyle \int x^n\text{ cosh}\left(2m-2k\right)x\text{ d}x}}}$

3. 

${\displaystyle \int x^n{\text{sinh}}^{2m+1}x\text{d}x}$