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

Theorem

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

\dfrac {x^3} 3 \cosh^{-1} \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \dfrac x a > 0 \\ \dfrac {x^3} 3 \cosh^{-1} \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C & : \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 $x^2 \sinh^{-1} \dfrac x a$


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


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