Inverse hyperbolic cosine is notated as arccosh(z) or cosh^(-1)(z). Using the latter notation, it is possible to confuse the inverse hyperbolic cosine with a hyperbolic cosine raised to the power of negative one, written as cosh(z)^(-1). For this reason, some people prefer the former definition.
The hyperbolic cosine is defined as cosh(z) = (e^z + e^(-z)) / 2, and cosh(0) = 1. The inverse hyperbolic cosine is defined as arccosh(z) = ln(z + sqrt(z^2 -1)), where z is greater than or equal to one.
The derivative of the inverse hyperbolic cosine is 1 / (z^2 - 1).
Integration with the inverse hyperbolic cosine in the answer uses the formula: S(1 / sqrt(u^2 - a^2) du = arccosh(u / a) + c = ln(u + sqrt(u^2 - a^2) + c. (S here stands for the integral sign.) As you can see from the equivalence, it is possible to write the answer to this integration with or without involving the inverse hyperbolic cosine.