A torus of major radius $b$ and minor radius $a$ can be parametrized by $\mathbf{r}:[0,2\pi]^2\to\mathbb{R}^3$, $\mathbf{r}(u,v)$ $=(b+a\cos v)\cos u\mathbf{i}+(b+a\cos v)\sin u\mathbf{j}+a\sin v\mathbf{k}$. Given a point $\mathbf{x}$ outside the surface of the torus, I would like to calculate the following integral: $$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{2\pi} \int_{0}^{2\pi}\int_0^{2\pi}\frac{IN \partial_v\mathbf{r}(u,v) \times(\mathbf{x}-\mathbf{r}(u,v) )}{2\pi\|\mathbf{x}-\mathbf{r}(u,v)\|^3}dudv $$where $\mu_0, I$ and $N$ are constant. A physical interpretation of the expression is as the magnetic field generated by a toroidal solenoid of $N$ loops flown through by a current of intensity $I$ but reduced to an ideal solenoid with a surface distribution of current.
We can chose $\mathbf{x}=x\mathbf{i}+z\mathbf{k}$ without loss of generality. I calculate $$ \partial_v\mathbf{r}(u,v) \times(\mathbf{x}-\mathbf{r}(u,v))=(-az\sin v\sin u+a^2\sin u+ab\cos v\sin u)\mathbf{i}$$$$+(ax\cos v-ab\cos v\cos u-a^2\cos u+az\sin v\cos u)\mathbf{j}+ax\sin v \cos u\mathbf{k}$$and $$\|\mathbf{x}-\mathbf{r}(u,v)\|^2=x^2-2x(b+a\cos v)\cos u+(b+a\cos v)^2+z^2-2az\sin v+a^2\sin v.$$Please correct me if my calculation are wrong.
By reasoning on the $u$ symmetry, I would say that the $x$ and $z$ component of the integral are zero, while I calculate the $y$ component $B_2(\mathbf{x})$ as$$\frac{\mu_0 IN}{8\pi^2}\int_0^{2\pi}\int_0^{2\pi}\frac{ax\cos v-ab\cos u\cos v-a^2\cos u+az\cos u\sin v}{(x^2-2x(b+a\cos v)\cos u+(b+a\cos v)^2+z^2-2az\sin v+a^2\sin v)^{3/2}}dudv.$$ Is everything correct until this point? If it is, how can we calculate this integral? I suspect that complex analysis, of which I have some little elements, may help, but I am not sure how... I heartily thank you for any answer!
P.S.: By using intuitive arguments and Ampère's circuital law, which I have never seen proved for surface currents, my textbook of physics shows that, for $x\in(b-a,b+a)$, $\mathbf{B}(x\mathbf{i})=-\frac{\mu_0 NI}{2\pi x}\mathbf{j}$, while the field is assumed to be zero outside the torus. I would not be amazed if the integral gave us these same values...