INTEGRALS CONTAINING $\displaystyle \frac{1}{\cosh ax}$

1.

$\displaystyle\int\displaystyle \frac{dx}{\cosh ax}=\displaystyle \frac{2}{a}\tan^{-1}e^{ax}$

2.

$\displaystyle\int\displaystyle \frac{dx}{\cosh^2 ax}=\displaystyle \frac{\tanh ax}{a}$

3.

$\displaystyle\int\displaystyle \frac{dx}{\cosh^3 ax}=\displaystyle \frac{\tanh ax}{2a\cosh ax}+\displaystyle \frac{1}{2a}\tan^{-1}\sinh ax$

4.

$\displaystyle\int\displaystyle \frac{\tanh ax}{\cosh^n ax}dx=-\displaystyle \frac{1}{na\cosh^n ax}$

5.

$\displaystyle\int\cosh ax dx=\displaystyle \frac{\sinh ax}{a}$

6.

$\displaystyle\int\displaystyle \frac{x dx}{\cosh ax}=\displaystyle \frac{1}{a^2... ...displaystyle \frac{(-1)^n E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdot\cdot\cdot \right\}$

where the constants E_n are the Euler's numbers.

7.

$\displaystyle\int\displaystyle \frac{x dx}{\cosh^2 ax}=\displaystyle \frac{x\tanh ax}{a}-\displaystyle \frac{1}{a^2}\ln\cosh ax$

8.

$\displaystyle\int\displaystyle \frac{dx}{x\cosh ax}=\ln x -\displaystyle \frac{... ...ax)^6}{4320}+ \cdot\cdot\cdot \displaystyle \frac{(-1)^n E_n(ax)^{2n}}{2n(2n)!}$

where the constants E_n are the Euler's numbers.

9.

$\displaystyle\int\displaystyle \frac{dx}{q+p/ \cosh ax}=\displaystyle \frac{x}{q}-\displaystyle \frac{p}{q}\int\displaystyle \frac{dx}{p+q\cosh ax}$

10.

$\displaystyle\int\displaystyle \frac{dx}{\cosh^n ax}=\displaystyle \frac{\tanh ... ...-2}ax)}+\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{dx}{\cosh^{n-2}ax}$