Integrating the Gielis Superformula. The Gielis Superformula (https://en.wikipedia.org/wiki/Superformula) is a powerful expression capable of generating a vast number of geometric shapes that typically occur in nature. 
For $0<\theta< 2\pi$ , the superformula is given by $$ \left({\Bigl|\frac{\cos \frac{m\theta}{4}}{a}\Bigl|^p} +{\Bigl|\frac{\sin\frac{m\theta}{4}}{b}\Bigl|^q} \right)^{-r}  $$
where $m, p, q, r, a, b$ are positive rational fixed constants. I have been trying to determine the first integral of this expression in the limits $0$ to $2\pi$ such that
$$ \int_0^{2\pi} {\left({\Bigl|\frac{\cos \frac{m\theta}{4}}{a}\Bigl|^p} +{\Bigl|\frac{\sin\frac{m\theta}{4}}{b}\Bigl|^q} \right)^{-r}d\theta }$$
I have tried using the method of substitution to simplify this composite function but have not had any success. I get stuck trying to deal with the exponent $r$.
Do you have any suggestions to deal with this? Any help would be greatly appreciated. 
 A: I would caution you about the use of Gielis's formula. Put simply, it is NOT some form of higher mathematics. (See, for example, G.G Szpiro, "The Secret Life of Numbers," p. 157.) Gielis starts with the formula for a circle and then concocts a polar radius composed of six (!) free parameters, and so can create a large number of shapes. My principal objection to Gielis's formula is that it not amenable to mathematical analysis. For example, try to calculate the area under the curve. You can't. By contrast, the superellipse and its various generalizations are truly mathematical.
Let me introduce my own generalization, called superconics. The general form is given by
$$f(X) = b(1-c^2|X/a|^q)^{1/p}$$
or its canonical form
$$f(x) = (1-c^2|x|^q)^{1/p}$$
Here, $a$ and $b$ scale the $x$ and $y$ axes, resp. and $c^2=\pm1$. $c^2=1$ corresponds to elliptic and parabolic types and $c^2=-1$ corresponds to hyperbolic types. (More generally, $c^2$ can vary smoothly between.)
The following integral gives the area under the curve, the centroid, moments, and moments of inertia of all the superconics.
$$\int_{-a}^a X^nf^m(X) dX = a^{n+1}b^m\int_{-1}^1 x^nf^m(x) dx$$
First we notice that $a$ and $b$ are superfluous and that we can concentrate on the canonical equation. The results can be scaled appropriately for any $a$ and $b$ afterward according to the factor $a^{n+1}b^m$. Next, and most important, this equation can be solved exactly in terms of known functions, specifically, the Gauss hypergeometric function and the incomplete beta function
$$\int_{-1}^1 x^nf^m(x) dx = \frac{2}{n+1} {_2F_1}(-mp,\frac{n+1}{q};1+\frac{n+1}{q};c^2)$$
$$ = \frac{2}{q} (c^2)^{-\frac{n+1}{q}} B(\frac{n+1}{q},mp+1,c^2)$$
Further simplifications accrue when $c^2=1$. To wit, for $m=1, n=0$ we get the area
$$\int_{-1}^1 f(x) dx = \Psi(p,q) = 2\frac{\Gamma(p+1)\Gamma(1+1/q)}{\Gamma(1+1/q)}  = \Psi(1/q,1/p)$$
More generally, for arbitrary $m$ and $n$ we substitute $p\to mp$ and $q\to q/(n+1)$. All of the integrals for $c^2=1$ can be expressed solely in terms of the parameter $\Psi$. This is true for bodies of revolution as well as other three-dimensional bodies composed of superconics profiles.
Now, in addition to the above intrinsic equation for the superconics, we can derive the following $parametric$ equation for superconics in the complex plane:
$$z=|\text{sin}^{2/q}(u)|\text{sgn}(\text{sin}(u))+i|(1-c^2\text{sin}(u))^p|\text{sgn}(\text{cos}(u))$$
where $u = [-\pi/2,\pi/2]$ for the upper half plane and $u = [-3\pi/2,\pi/2]$ for the full plane, e.g., a closed curve.
I have the feeling that this is what people actually have in mind when they seek the polar form of the superellipse.
You can find some illustrations and animations of superconics here: http://web.calstatela.edu/curvebank/superconicncb/superconicncb.htm. There is even an animation of a smooth transition from a sphere to a hyperboloid of one and two sheets using only superconics (with variable $c^2$). Here is a small version of that animation.

In the spirit of full disclosure, this is the same response as used here: Polar form of generalized superellipse
A: In relation to Gielis' Formula: 
http://euro-math-soc.eu/review/geometrical-beauty-plants
It may not be generating simple answers, but is a generalization of Pythagoras, with relations to various fields.
Yours,
Johan Gielis 
