What is this integral

[JulieK]

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

 

[pasmith]

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]

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

 

[HakimPhilo]

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

 

[TheDemx27]

Just starting with Mathematica, I type in:

Integrate[((ArcSinh[a * x])/ a * x)^b, x]

and I get out:

\[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x

Is there some reason I am getting a different output?