# What is this integral

What is this integral
$\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx$
where a and b are constants.

pasmith
Homework Helper
The substitution $ax = \sinh t$ yields $$\int \left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^b\,dx = \int \left(\frac{t}{\sinh t}\right)^b \frac{\cosh t}{a}\,dt \\ = \left[ \frac{1}{a(1-b)}\frac{t^b}{(\sinh t)^{b-1}}\right] + \frac{b}{a(b - 1)} \int \left(\frac{t}{\sinh t}\right)^{b-1}\,dt \\$$ on integration by parts. Unfortunately that seems to be as far as one can get.

Philip Wood
Gold Member
The wonderful Wolfram online integrator can't do it, so there's not much hope...

I confirm, Mathematica replies: "no result found in terms of standard mathematical functions" which is true in most cases.

TheDemx27
Gold Member
Just starting with Mathematica, I type in:
Code:
Integrate[((ArcSinh[a * x])/ a * x)^b, x]
and I get out:
Code:
\[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x
Is there some reason I am getting a different output?