INTEGRALS CONTAINING Tanh(ax)

1.
$\displaystyle\int\tanh ax dx=\displaystyle \frac{1}{a}\ln\cosh ax$

2.
$\displaystyle\int\tanh^2 axdx=x-\displaystyle \frac{\tanh ax}{a}$

3.
$\displaystyle\int\tanh^3 axdx=\displaystyle \frac{1}{a}\ln\cosh ax-\displaystyle \frac{\tanh^2 ax}{2a}$

4.
$\displaystyle\int\tanh^n ax\cosh^{-2} ax dx=\displaystyle \frac{\tanh^{n+1}ax}{(n+1)a}$

5.
$\displaystyle\int\displaystyle \frac{1}{\cosh^{2}ax\tanh ax}dx=\displaystyle \frac{1}{a}\ln\tanh ax$

6.
$\displaystyle\int\displaystyle \frac{dx}{\tanh ax}=\displaystyle \frac{1}{a}\ln\sinh ax$

7.
$\displaystyle\int x\tanh ax dx=\displaystyle \frac{1}{a^2}\left\{\displaystyle ...
...frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdot\cdot\cdot\right\}$

where the constants Bn are the Bernoulli's numbers.

8.
$\displaystyle\int x\tanh^2 axdx=\displaystyle \frac{x^2}{2}-\displaystyle \frac{x\tanh ax}{a}+\displaystyle \frac{1}{a^2}\ln\cosh ax$

9.
$\displaystyle\int\displaystyle \frac{\tanh ax}{x}dx= ax-\displaystyle \frac{(ax...
...displaystyle \frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdot$

where the constants Bn are the Bernoulli's numbers.

10.
$\displaystyle\int\displaystyle \frac{dx}{p+q\tanh ax}=\displaystyle \frac{px}{p^2-q^2}-\displaystyle \frac{q}{a(p^2-q^2)}\ln(q\sinh ax+p\cosh ax)$

11.
$\displaystyle\int\tanh^n axdx=\displaystyle \frac{-\tanh^{n-1}ax}{a(n-1)}+\int\tanh^{n-2}axdx$

[Tables]

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