Minor simplification problem in integral
11-01-2018, 08:55 PM (This post was last modified: 11-02-2018 06:21 AM by ijabbott.)
Post: #1
 ijabbott Senior Member Posts: 785 Joined: Jul 2015
Minor simplification problem in integral

I think it should have cancelled the outer 'a' with the inner denominators.

MathJax version for posterity:

$\int\frac{a}{{(a^2+x^2)}^2}dx \to a* \left(\frac{x}{2*a^2*(a^2+x^2)}+\frac{\mathsf{ATAN}(\frac{x}{a})}{2*a^3}\right)$

— Ian Abbott
11-02-2018, 07:18 AM
Post: #2
 parisse Senior Member Posts: 1,053 Joined: Dec 2013
RE: Minor simplification problem in integral
There is no simplification on the output, and a is factored out during integration.
11-02-2018, 10:26 PM (This post was last modified: 11-03-2018 01:09 AM by ijabbott.)
Post: #3
 ijabbott Senior Member Posts: 785 Joined: Jul 2015
RE: Minor simplification problem in integral
Yes, I see the 'a' is factored out as an independent variable early on. Is there some magic method to simplify the result further? The 'simplify' function arguably makes it worse. (It puts everything under a common denominator, which is good, but still does not cancel an 'a' common to the numerator and denominator.)

EDIT: Actually, I notice there isn't in fact a common 'a' in the numerator and denominator after using 'simplify', although the result is kind of messy so it took me a while to spot my mistake:

$\frac{a^2*\mathsf{ATAN}\left(\frac{x}{a}\right)+a*x+ x^2*\mathsf{ATAN}\left(\frac{x}{a}\right)}{2*a^4+2*a^2*x^2}$

I suppose I can't begrudge the CAS considering that simpler by the rules it is following (putting things under a common denominator where possible), although to my aging eyes and addled brain it looks more complicated!

— Ian Abbott
11-02-2018, 10:39 PM
Post: #4
 Carsen Member Posts: 189 Joined: Jan 2017
RE: Minor simplification problem in integral
Hi ijabbott,

Try using the Partial Fractions command. I believe it accomplishes what you are aiming for.
11-03-2018, 12:37 AM
Post: #5
 ijabbott Senior Member Posts: 785 Joined: Jul 2015
RE: Minor simplification problem in integral
(11-02-2018 10:39 PM)Carsen Wrote:  Hi ijabbott,

Try using the Partial Fractions command. I believe it accomplishes what you are aiming for.

Nearly. It produces a complex fraction:

$\frac{\mathsf{ATAN}\left(\frac{x}{a}\right)}{2*a^2}+\frac{\frac{x}{2*a}}{a^2+x^2​}$

Using 'expand' on that produces:

$\frac{\frac{1}{2}*x}{a*(a^2+x^2)}+\frac{\frac{1}{2}*\mathsf{ATAN}\left(\frac{x}{​a}\right)}{a^2}$

Which is pretty close to what I was expecting, although the 2s in $$\frac{1}{2}$$ could be moved to the bottom.

— Ian Abbott
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