Why does MATLAB (Symbolic Math Toolbox) not integrate this simple function.

I am trying to integrate the following function symbolically, but MATLAB won't resolve this.

syms x a
int((1-x^2/a^2)^(3/2),x,-a,a)

ans = 

It should be and Wolfram Alpha calculates it without problem. Is there a way to get this result in MATLAB too?

Answer 1

[John D'Errico]

Your problem is, you need to define a properly. So, if you do only this:

syms x a
I = int((1-x^2/a^2)^(3/2),x)

I = 

Now you see that MATLAB finds a solution, but it does not know anything about a. We can try this, but MATLAB is still confused.

simplify(subs(I,a) - subs(I,-a))

ans = 

The problem there is, if a takes on some general complex value, that result may not be a simple thing. The point being:

syms a
simplify(a*sqrt(-1/a^2))

ans = 

syms a real
simplify(a*sqrt(-1/a^2))

ans = 

For real a, that last one reduces to +/-i, depending on the sign of a.

syms a real positive
simplify(a*sqrt(-1/a^2))

ans = 

i

Only in the third case does a drop out completely.

So if we specify a more clearly.

syms x
syms a real positive
I = int((1-x^2/a^2)^(3/2),x)

I = 

simplify(subs(I,a) - subs(I,-a))

ans = 

Answer 2

[Walter Roberson]

integrate 0 to a and multiply the result by 2

[Sebastian Götz]

Thanks, for the workaround. However, shouldn't it work as well if I integrate directly from -a to a? Why can't MATLAB do it?

Answer 3

[Paul]

syms x a real
I = int(simplify((1-x^2/a^2)^(3/2)),x,-a,a)

I = 

[Sebastian Götz]

This works as I hoped. Thanks. It is still strange that it works with the exponent 1/2

syms x a
int((1-x^2/a^2)^(1/2),x,-a,a)

ans = 

and with the exponent 2

int((1-x^2/a^2)^(2),x,-a,a)

ans = 

without the simplify but not with 3/2.

int((1-x^2/a^2)^(3/2),x,-a,a)

ans = 

[Paul]

Sometimes int() works in mysterious ways, I suppose. I believe I've seen other case where int() does not return a solution w/o manipulating the integrand into a different form.