Vtyjkyuk

I need integral result for following integral:
$$int_0^2pifracd theta sqrt1-k^2cos( theta )$$

It will be useful in an electromagnetic simulator. It is obtained as the medium 1/distance from one ring to a point that is separated to the axis.
Actually due the objects where split in a 3-d mesh of 24k pieces so to solve the problem its takes 3 hours due inverse of 24k x 24k arrays (that reach the 16GB limit of RAM).

I have look for at definite integral tables but was not found
It would help an APPROXIMATE solution that reduces the 3-D problem to 2D.
It is said it is a first kind elliptic integral, but I do not know how to place in an elliptic expression and what lib can be used with C++. Boost is included in actual C++11 but I do not know if the elliptic integral lib is included in it

In the notation here, we get a complete elliptic integral of the first kind viz. $$int_0^2pifracdthetasqrt1-k^2costheta=2int_0^pifracdthetasqrt1-k^2costheta=frac4sqrt1-k^2int_0^pi/2fracdphisqrt1-frac2k^2k^2-1sin^2phi=frac4sqrt1-k^2Kleft(sqrtfrac2k^2k^2-1right).$$As discussed therein, the easiest fast accurate calculation for a given $k$ uses the aithmetic-geometric mean.

The response is:
$$displaystyleint_0^2pifracd theta sqrt1-k^2cos( theta )=2displaystyleint_0^pifracd theta sqrt1-k^2cos( theta )$$

$$displaystyleint_0^pi/2fracdt sqrt1-k^2cos(2t)=4displaystyleint_0^pi/2fracdt sqrt1-k^2(1-2sin^2(t))=4displaystyleint_0^pi/2fracdt sqrt(1-k^2)+2k^2sin^2(t)$$
That is an elliptic integral of the first order.

Result can be found here:
Forum math

Where
$$theta=2t$$

I used boost and worked. Elliptic integrals are included in in C++17, so it is not needed install boost:

#include<iostream>
#include <cmath>


const double PI = 3.1415926535897932384626433832795;
const double PI2 = 2.0 * PI;

using namespace std;


int main()

 double flimit = PI, k = 0.25, f;
 f = ellint_1(k, flimit);
 double expected = 1.596242222131783510149;
 cout << "Ellint error for k,F=" << k << " & " << flimit << " = " << fabs(f - expected) << endl;

 cout << endl << "=== END ===" << endl; getchar();
 return 1;