When I choose arctanh or arccoth?

Question

I want to know When I choose $\operatorname{arctanh}$ or $\operatorname{arccoth}$ ?

E.x $$\int_0^3 \frac{1}{49-4x^2} dx$$ It will be $(1/14) \operatorname{arctanh} (2x/7)$ or $(1/14) \operatorname{arccoth} (2x/7)$.

Answer 1

Note that for real $x$, we always have $\coth x > \lvert \sinh x\rvert$, so the hyperbolic tangent attains only values with absolute value less than $1$,

$$\lvert \tanh x\rvert < 1,\quad x \in\mathbb{R},$$

and the hyperbolic cotangent attains only values with absolute value greater than $1$,

$$\lvert \coth x\rvert > 1, \quad x \in \mathbb{R}.$$

$\DeclareMathOperator{\atanh}{Ar tanh}\DeclareMathOperator{\acoth}{Ar coth}$

Therefore, whether you use $\atanh$ or $\acoth$ depends on whether the argument has absolute value less than or greater than $1$. In your case,

$$\int_0^3 \frac{dx}{49-4x^2},$$

the argument will be $\dfrac{2x}{7}$, which for $0\leqslant x \leqslant 3$ has absolute value less than $1$, hence you use $\atanh$,

$$\int_0^3\frac{dx}{49-4x^2} = \frac{1}{14}\atanh \left(\frac{2x}{7}\right)\Bigl\lvert_0^3 = \frac{\atanh \frac67}{14}.$$