Analytical solutions to integrals While calculating the magnetic field at an arbitrary point due to a charged sphere rotating about the z-axis, I came upon the following integral,
$$\iiint\frac{x^3\cos^2{\theta}\left[\left(c-x\sin{\theta}\right)\left(\cos{\phi}\hat{x}+\sin{\phi}\hat{y}\right)+\left(x\cos{\theta}-b\sin{\phi}-a\cos{\phi}\right)\hat{z}\right]}{\left[a^2+b^2+c^2-2x\left(a\cos{\theta}\cos{\phi}+b\cos{\theta}\sin{\phi}+c\sin{\theta}\right)+x^2\right]^{\frac{3}{2}}}dx\,d\theta\,d\phi$$
Here $a\hat{x}+b\hat{y}+c\hat{z}$ is the position vector of the arbitrary point at which the field is calculated.
I tried all the possible substitutions I know but none seem to work. By the look of it I do not think an analytic solution even exists. I even tried to look for numerical methods to solve integrals on the web but did not find anything useful.
I would like to know how to numerically calculate such integrals or what software/website to use since Wolframalpha did not turn out to be very useful. 
Also is there a way to tell just by looking at an equation whether an analytic solution exists or not?
I also do not understand why the solution to such a simple problem turn out to be this hard to compute? By simple I refer to the fact that this problem basically deals with classical physics and contains none of the heavy mathematical principles of QFT or GR and also there is a nice spherical symmetry associated to it.
 A: The problem can actually be solved analytically, although it is not exactly straightforward. An important part to obtain the solution, is choosing a suitable reference frame that simplifies the computation. 
Since the problem is symmetric under rotations about the $z$-axis, you can choose the point $(a,b,c)$ to lie in the $xz$-plane, effectively eliminating $b$ from the integral. With that in mind and considering the current of the rotating charges about the $z$-axis, Ampere's law will tell you that the magnetic field will have no azimuthal component, and hence that for $b=0$ there will be no $\hat{y}$-contribution in the magnetic field. Also, using spherical rather than cartesian coordinates is helpful.
Assuming the result you provided is correct, it is not impossible to perform the integral, but it is not convenient. A simpler approach is to compute the vector potential $\vec{A}(r,\theta,\phi)$ of a spherical shell with constant charge density. Use that result to integrate over the full sphere, and finally use $\vec{B}=\nabla \times \vec{A}$ to obtain the result.
The derivation of a rotating homogeneously charged spherical shell can be found for instance in "Introduction to electrodynamics" by David J. Griffiths, Example 5.11 ( I am sure you can find the book).
The result for a rotating homogeneously charged sphere of radius $R$, charge $Q$, and angular velocity $\omega$ is
$$
\vec{B}(r,\theta)= \frac{\mu_0}{4 \pi} \frac{Q R^2 \omega}{5 r^3} (2 \cos \theta ~\hat{r} + \sin \theta ~ \hat{\theta})
$$
with $r \geq R$.
Is there a way to tell wether there exists an analytic solution? The short answer is NO, you can only try to find one. In all honesty very few problems in physics have an analytic solution, but those that have are the ones you will typically encounter in all the courses and textbooks and basic courses. It is a misconception to assume that problems in classical physics are easier to solve. Consider for instance the "simple" problem of 3 point masses, which in general can not be solved analytically. 
