Primitive of Inverse Hyperbolic Secant of x over a over x/Lemma

Lemma for Primitive of Inverse Hyperbolic Secant of x over a over x

 * $\map {\ln^2} {\dfrac x {2 a} } = \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \map {\ln^2} 2$

Proof
$$\ln \left(\frac{a}{x}\right) \ln \left(\frac{4 a}{x}\right)-\ln ^2\left(\frac{x}{2 a}\right)= \ln \left(\frac{a}{x}\right) \left[\ln \left(\frac{a}{x}\right)+2 \log (2)\right]-\left[\ln \left(\frac{a}{x}\right)+\ln (2)\right]^2 =-\ln ^2(2)$$

Then:

Hence the result.