In a research that I am conducting, my main focus of measuring a set-up has diverted to modelling the theoretical working of a magnetic field around a current-inducing solenoid (coil). Given a coil with length $L$, radius $R$, number of winds $N$ and point $P$ with coordinates $(a,b,c)$, the magnetic field on point $P$ is given by:
$$\vec{B}_z = \frac{\mu_0IR}{4\pi} \int\limits_0^{2\pi N} \frac{(c - k\theta)\cos(\theta) + k\sin(\theta) - \frac{bk}{R}}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
$$\vec{B}_y = \frac{\mu_0IR}{4\pi} \int\limits_0^{2\pi N} \frac{(c - k\theta)\sin(\theta) - k\cos(\theta) + \frac{ak}{R}}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
$$\vec{B}_z = \frac{\mu_0IR}{4\pi} \int\limits_0^{2\pi N} \frac{R - a\cos(\theta) - b\sin(\theta)}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
I have succesfully made a numerical model of the magnetic field. Although the presentation of the data still has to have some revisions, it already is adequate to put in my research paper for my study.
Anyway, what I now wish to know, is whether there is a way to analytically solve the integrals. WolframAlpha exceeds its computation time, and my own mathematical toolbox is not expansive enough to tackle this on my own. I have already been able to solve a simplified version of the $B_z$ integral when $a = b = 0$ using both $u$ and trig substitution. It then comes down to:
$$\vec{B}_z = \frac{\mu_0I}{4\pi} \int\limits_0^{2\pi N} \frac{R^2}{\sqrt{R^2+(c-k\theta)^2}^{~3~}} d\theta = \frac{\mu_0I}{4\pi k} \sin\left(\arctan\left(\frac{c+2\pi Nk}{R}\right)\right) - \frac{\mu_0I}{4\pi k} \sin\left(\arctan\left(\frac{c}{R}\right)\right)$$
The integrals I wish to solve, boil down to a general form where the operator can be split into multiple integrals in similar forms. So to ask it definitively, is there a way to solve an integral in the forms:
$$I = \int\limits_0^{2\pi N} \frac{\cos(\theta)}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
$$I = \int\limits_0^{2\pi N} \frac{\theta\cos(\theta)}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
$$I = \int\limits_0^{2\pi N} \frac{1}{\sqrt{(a-R\cos(\theta))^2 + (b-R\sin(\theta))^2 + (c-k\theta)^2}^{~3}} d\theta$$
And also $\cos(\theta)$ changed to $\sin(\theta)$? Let $a,b,c,k$ all be independent real variables and $k$ is always a positive real number.
I suspect the way to go about this is by using complex trig identities, and that the integral will yield an unwieldy or cumbersome result. However, I am still eager to know if there is an analytical solution. Thanks for your help in advance!