Symbolic Toolbox: integration

>> int('cos(x^2)^2')
Warning: Explicit integral could not be found.  

ans =

int(cos(x^2)^2, x)

>> int('cos(2*x^2)')

ans =

(pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/2

But there is a formula cos(2y)=2cos(y)^2)-1 and in first case MATLAB can't solve..

May be MuPAD forced to solve?

[Walter Roberson]

It works in Maple.

[Walter Roberson]

You could _try_

int(simple('cos(x^2)^2'),'x')

[Andrei Bobrov]

Hi Walter! Small typo:

int(simplify('cos(x^2)^2'),'x')

[Walter Roberson]

No, simple() is correct rather than simplify() . simple() goes through a variety of conversions to find the "best" one, telling you along the way what it is doing.

[Andrei Bobrov]

in "my" MATLAB with Maple Toolbox

http://imageshack.us/photo/my-images/838/symint.png/

[Andrei Bobrov]

in "my" MATLAB with MuPAD (I use MATLAB 7.10)

http://imageshack.us/photo/my-images/542/symint2.png/

[Walter Roberson]

int(simple(sym('cos(x^2)^2')),'x')

[Andrei Bobrov]

I'm agree , Walter!

(Maple Toolbox)

http://imageshack.us/photo/my-images/843/symint3.png/

[Andrei Bobrov]

Last comment :)

http://imageshack.us/photo/my-images/402/symint4.png/

(for Maple Toolbox)

Answer 1

[bym]

Apparently you have to do the substitution manually

int(simple(cos(x^2)^2))
Warning: Explicit integral could not be found.

Using the sincos version returned by

simple(cos(x^2)^2)

yields this

int(cos(2*x^2)/2 + 1/2)

ans =

x/2 + (pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/4

I could not find where the sincos 'simplification' is available outside the simple command as it is in Maple...maybe this is the reason Maple returns the result automagically ;)

[edit for completeness]

int(feval(symengine, 'combine', cos(x^2)^2, 'sincos'))

ans =

x/2 + (pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/4

[Walter Roberson]

http://www.mathworks.com/help/toolbox/mupad/stdlib/combine.html

combine(cos(x^2)^2,sincos)

Note that this is at the MuPAD level, not the MATLAB level, so you may need to use feval() or similar to access it

feval(symengine, 'combine', cos(x^2)^2, 'sincos')

[bym]

works well, my local documentation does not turn up an obvious help topic when searching for 'combine'

feval(symengine, 'combine', cos(x^2)^2, 'sincos')

ans =

cos(2*x^2)/2 + 1/2

thanks Walter

[Igor]

Yes, yes...

good construction with "combine"/"sincos"

But from the technical calculation point of view (as semi-analitical and fast integration is needed)

it's not good to think about integration procedure details...

And if more complex integral occures??

To ask own intuition?

[Walter Roberson]

Humans are better at seeing patterns or understanding context than machines are. It is *often* the case that a human will have to apply knowledge of theoretical equivalences in order to make progress in integration.

MuPAD is not as strong as some of the other products around, but _all_ of them are missing a lot of patterns.