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

Theorem

 * $\ds \int \frac {\cosh^{-1} \dfrac x a \rd x} {x^2} = \begin {cases}

\dfrac {-\cosh^{-1} \dfrac x a} x - \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x} & : \cosh^{-1} \dfrac x a > 0 \\ \dfrac {-\cosh^{-1} \dfrac x a} x + \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x} & : \cosh^{-1} \dfrac x a < 0 \\ \end {cases}$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\sinh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\tanh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\coth^{-1} \frac x a} {x^2}$