Real Area Hyperbolic Secant of x over a in Logarithm Form

Theorem

$\arsech \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 - x^2} } x}$

$\ds \arsech \frac x a$

$=$

$\ds \map \ln {\frac {1 + \sqrt {1 - \paren {\frac x a}^2} } {\frac x a} }$

$\ds $

$=$

$\ds \map \ln {\frac {a \paren {1 + \sqrt {\dfrac {a^2 - x^2} {a^2} } } } x}$

$\ds $

$=$

$\ds \map \ln {\frac {a \paren {\dfrac {a + \sqrt {a^2 - x^2} } a} } x}$

$\ds $

$=$

$\ds \map \ln {\frac {a + \sqrt {a^2 - x^2} } x}$

$\blacksquare$