How can I find the explicit formula for the inverse of sinh?

Write y = sinh^(-1)(x) ie x = sinhy. Then writing this in terms of exponentials and multiplying by 2 we will get 2x = exp(y) - exp(-y). Multiply by exp(y) and rearrange to obtain exp(2y) - 2xexp(y) -1 = 0. Then this is is simply a quadratic in exp(y), so using the quadratic equation or completing the square we get exp(y) = x + sqrt(x^2 +1). Notice we take the positive square root since we must have exp(y) > 0. Then simply take logs of both sides to get the equation for y in terms of x. This is now the inverse of sinh.