Why is the output different for this integral in Mathematica?

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SUMMARY

The integral discussed is \(\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx\), where \(a\) and \(b\) are constants. The substitution \(ax = \sinh t\) leads to a complex expression that Mathematica fails to simplify into standard mathematical functions, returning "no result found." The user reports a different output when using the command Integrate[((ArcSinh[a * x])/ a * x)^b, x] in Mathematica, which results in \(\[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x\). This discrepancy arises from Mathematica's handling of symbolic integration and its inability to express the integral in simpler terms.

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What is this integral
\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx
where a and b are constants.
 
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The substitution ax = \sinh t yields <br /> \int \left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^b\,dx = \int \left(\frac{t}{\sinh t}\right)^b \frac{\cosh t}{a}\,dt \\ <br /> = \left[ \frac{1}{a(1-b)}\frac{t^b}{(\sinh t)^{b-1}}\right] <br /> + \frac{b}{a(b - 1)} \int \left(\frac{t}{\sinh t}\right)^{b-1}\,dt \\<br /> on integration by parts. Unfortunately that seems to be as far as one can get.
 
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.
 
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?
 

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