Primitive of Inverse Hyperbolic Cosine of x over a over x squared/Mistake

Source Work

 * Chapter $14$: Indefinite Integrals
 * Integrals involving Inverse Hyperbolic Functions: $14.655$

This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).

Mistake

 * $\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = \dfrac {-\map {\cosh^{-1} } {x / a} } x \mp \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x}$ $\sqbrk {- \text { if } \map {\cosh^{-1} } {x / a} > 0, + \text { if } \map {\cosh^{-1} } {x / a} < 0}$

Correction
As demonstrated in Primitive of $\dfrac {\map {\cosh^{-1} } {x / a} } {x^2}$, this is incorrect.

It should be:


 * $\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = -\frac 1 x \cosh^{-1} \dfrac x a \pm \frac 1 a \arcsec \size {\frac x a}$ $\sqbrk {+ \text { if } \map {\cosh^{-1} } {x / a} > 0, - \text { if } \map {\cosh^{-1} } {x / a} < 0}$