INTEGRALS CONTAINING Tanh(ax)

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

where the constants B_n are the Bernoulli's numbers.

8.

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

9.

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

where the constants B_n are the Bernoulli's numbers.

10.

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

11.

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

[Tables]

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