integrate(sqrt(1+cos(x)^2),x)

On Thu, 31 Oct 2013 16:27:45 -0700
Raymond Toy <toy.raymond at
gmail.com<http://www.math.utexas.edu/mailman/listinfo/maxima>;
>
writes:

>>>>>* "Martin" == Martin Kraska <kraska at fh-brandenburg.de<http://www.math.utexas.edu/mailman/listinfo/maxima>>;
writes: *
    Martin> Hi,
    Martin> using Maxima 5.31.2 I tried this:

    Martin> integrate(sqrt(1+cos(x)^2),x) and get nothing but the input.

    Martin> When loading abs_integrate, the result is an error:

    Martin> sign: argument cannot be imaginary; found %i
    Martin> #0:
    Martin>
intfudu(exp=sqrt(6*%e^-(2*%i*x)+%e^-(4*%i*x)+1)*%e^(%i*x),%voi=x)(partition.
    Martin> mac line 95)
    Martin> #1:
extra_integrate(q=sqrt(6*%e^-(2*%i*x)+%e^-(4*%i*x)+1)*%e^(%i*x),x=x)
    Martin> -- an error. To debug this try: debugmode(true);

    Martin> Maple gives:

    Martin> {sqrt(sin(x)^2)*EllipticE(cos(x),i)}/sin(x)

    Martin> Can this be done in Maxima?

As you've discovered, Maxima cannot do this integral.  Maxima
currently has no support for elliptic integrals.

I've done a little work on this, but have never finished it. It
definitely won't recognize this integral, but it might recognize the
equivalent integrate(sqrt(1+x^2)/sqrt(1-x^2),x). I didn't try it,
however.

Ray



Example. Compute integrate(sqrt(1+cos(x)^2),x).

(%i1) f:sqrt(1+cos(x)^2);
(%o1) sqrt(cos(x)^2+1)
After reading maxima documentation about elliptic integrals
I conclude that the answer is

(%i2) F:elliptic_e(x-%pi/2, -1);
(%o2) elliptic_e(x-%pi/2,-1)

Test:
(%i3) diff(F,x);
(%o3) sqrt(cos(x)^2+1)

Only the derivative with respect to x is known by Maxima.
Serious problems obtained in the calculation of the numerical values of F.

best

Aleksas D