Make Mupad use trig identities to simplify integrations

Hello, here is my problem:

Integrals such as:

int(sinh(A*x)*sin(B*x),x)

or

int(sinh(A*x)*sin(B*x),x)

give very clear, easily coded solutions in MuPad. However

int(sinh(A*x)*sin(B*x)*cos(C*x),x)

generates a horrid solution involving terms such as sin(A x i), when it is clear as an integral of real functions on real variables the result should be real. (yes I have used assume(x, Type::Real)..)

By hand, there is a simple way to break down the third integral to a sum of the form of the first integral using sin(A)cos(B)=(sin(A+B)+sin(A-B))/2 i.e:

int(sinh(A*x)*(sin((B+C)*x)+sin((B-C)*x))/2,x)

but that still gives me a similarly horrible answer. In fact I have to seperate the integral entirely and by hand to

int(sinh(A*x)*sin((B+C)*x)/2,x)+int(sinh(A*x)*sin((B-C)*x)/2,x)

So my questions are:

1: Is it possible to get MuPad to do the trig conversion above automatically? I've only found functions to do the opposite so far.

2: Can integration be persuaded to keep the expanded/simplified form of these integrals so that it gives more elegant results?

Obviously the examples above are quite simple, but my problem involves many such terms in a bit of a 'worms nest', which all need expanding to fourier series coefficients, hence the form of the above equations and me wanting Mupad to save me a lot of time and possible error!

Answer 1

[Walter Roberson]

I tried this out in Maple, and it has similar difficulties. The work-around I found in Maple was

t := int(EXPRESSION,x);
combine(convert(t,trig),trig);

Unfortunately I have not found the MuPad equivalent of Maple's combine(), and MuPad's equivalent of convert() are not as easy to access as in Maple.

For the record, the result I got was

(1/2)*(-2*A*B*C*cosh(A*x)*sin(B*x+C*x) + B^2*C*sinh(A*x)*cos(B*x+C*x) + C^2*B*sinh(A*x)*cos(B*x+C*x)+B^2*A*cosh(A*x)*sin(B*x+C*x) + 2*A*B*C*cosh(A*x)*sin(B*x-C*x) + C^2*A*cosh(A*x)*sin(B*x+C*x) - A^2*B*sinh(A*x)*cos(B*x+C*x) + C^3*sinh(A*x)*cos(B*x-C*x) - B^2*C*sinh(A*x)*cos(B*x-C*x) - B^3*sinh(A*x)*cos(B*x-C*x) - A^2*C*sinh(A*x)*cos(B*x+C*x) + A^2*C*sinh(A*x)*cos(B*x-C*x) - A^2*B*sinh(A*x)*cos(B*x-C*x) + A^3*cosh(A*x)*sin(B*x-C*x) + C^2*B*sinh(A*x)*cos(B*x-C*x) + C^2*A*cosh(A*x)*sin(B*x-C*x) - C^3*sinh(A*x)*cos(B*x+C*x) + B^2*A*cosh(A*x)*sin(B*x-C*x) - B^3*sinh(A*x)*cos(B*x+C*x) + A^3*cosh(A*x)*sin(B*x+C*x)) / (B^4+C^4+2*A^2*B^2+2*A^2*C^2-2*B^2*C^2+A^4)

[Alexander]

This answer works well - which is ironic because I originally moved over to MuPad because it seemed to handle simpler trig products much better than Maple.

thanks

Answer 2

[Andrei Bobrov]

variant

syms x A B C
int(subs(sinh(A*x)*sin(B*x)*cos(C*x),sinh(A*x),1/2*(exp(A*x)-exp(-A*x))),x)

or so

eval(['f1 = @(x,A,B,C)' vectorize(char(int(subs(sinh(A*x)*sin(B*x)*cos(C*x),sinh(A*x),1/2*(exp(A*x)-exp(-A*x))),x)))]);
>> f1(3,2,5,1)
ans =
       -36.21

or so 2: idea of Walter Roberson, use Maple Toolbox for MATLAB:

Ex = int(sinh(A*x)*sin(B*x)*cos(C*x),x);
count =  maple(['simplify(combine(convert(' char(Ex) ',trig),trig),size)'])
count =
          2          2
  1/2 (-(A  + (C + B) ) (-C + B) sinh(x A) cos(x (-C + B))

                           2          2                      2           2
           + A cosh(x A) (A  + (C + B) ) sin(x (-C + B)) + (A  + (-C + B) )

                                                                              /
          (-sinh(x A) (C + B) cos(x (C + B)) + cosh(x A) sin(x (C + B)) A))  /
                                                                            /

             2           2    2          2
          ((A  + (-C + B) ) (A  + (C + B) ))

[Alexander]

Your first suggestion works well and seems to be the better answer for my purposes as it is straightforward to add the additional complexity I need and to integrate into Matlab - hopefully without typing any of the expressions out myself! It also avoids using Maple, which helps me when I work at home. It would be nice to stick to sinh/cosh notation but at this stage I am perfectly with exp if it gets the job done..

For my expression I had to apply 4 substitutions; for sinh and the equivalent in cosh each with two different operands.

It turns out I can only 'accept' one answer but this answer is just as good if not better!

thanks