Why is the output different for this integral in Mathematica?

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Discussion Overview

The discussion revolves around the integral \(\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx\), where \(a\) and \(b\) are constants. Participants explore the challenges of evaluating this integral using Mathematica and compare outputs from the software.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes a substitution \(ax = \sinh t\) to transform the integral, leading to a new expression involving hyperbolic functions.
  • The same participant notes that integration by parts yields a complex expression but does not lead to a complete solution.
  • Another participant mentions that Mathematica fails to provide a result, indicating that the integral cannot be expressed in terms of standard mathematical functions.
  • A different participant shares their experience with Mathematica, noting that their output differs from the expected result, prompting a question about the reason for this discrepancy.

Areas of Agreement / Disagreement

Participants generally agree that the integral is complex and that Mathematica does not yield a standard result. However, there is a disagreement regarding the output generated by Mathematica, with one participant questioning why their result differs from others.

Contextual Notes

There are limitations regarding the assumptions made in the substitution and the dependence on the definitions of the functions involved. The discussion does not resolve the mathematical steps or the nature of the outputs from Mathematica.

Who May Find This Useful

This discussion may be useful for individuals interested in advanced integral calculus, the capabilities of computational software like Mathematica, and the challenges of evaluating complex integrals.

JulieK
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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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