The existing literature on generalized trigonometric functions is scarce and it seems there isn't a comprehensive account of generalized trigonometric functions anywhere. Also there isn't a unified accepted notation and different authors use different notations.
The generalized sine and cosine functions $\sin_{pr}x$, $\ \cos_{pr}x$ are defined by the formulas
$$
x=\int_0^{\sin_{pr}x}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos_{pr}x=\sqrt[r]{1-(\sin_{pr}x)^r}.
$$For generic values of parameters $p,r$ it appears that these functions are not very interesting, but when these parameters have special values, then $\sin_{pr}x$ can be expressed algebraically through Jacobi elliptic functions, and as a consequence one can establish addition theorems analogous to addition theorems for elliptic functions.

Below I give a unified relatively simple discussion of several such cases. The first case $r=2,\ p=2$ is trivial. The second case $r=3,\ p=\frac{3}{2}$ is due to Cayley and Dixon. The third case $r=4,\ p=\frac{4}{3}$ has been considered relatively recently by Edmunds, Gurka, and Lang (Properties of generalized trigonometric functions, J. Approx. Theory 164 (2012), no. 1, 47-56.) I couldn't find the discussion of the fourth case $r=6,\ p=\frac{6}{5}$ in the literature, but it easily follows from the same considerations as in the first three cases. The parameter $r$ is denoted in analogy with Ramanujan's theory of elliptic functions to alternative bases (see "Ramanujan's notebooks, vol. 5" by Bruce Berndt). I don't know whether it is just a coincidence or there is a deeper connection, but $r=2,3,4,6$ are the four signatures for which such alternative theories of elliptic functions have been developed.

Let's consider the case when $\frac{1}{p}=1-\frac{1}{r}$. Then by a series of simple changes of variables one obtains
\begin{align}
\int_0^{u}\frac{dt}{(1-t^r)^{1-1/r}}=\frac{1}{r}\int_0^{u^r}\frac{dt}{(1-t)^{1-1/r}t^{1-1/r}}=\tag{1}\\
\frac{1}{r}\int_0^{u^r}\frac{dt}{(t-t^2)^{1-1/r}}=\frac{1}{r}\int_0^{u^r}\frac{dt}{\left(\frac{1}{4}-\left(\frac{1}{2}-t\right)^2\right)^{1-1/r}}=\\
\frac{2^{2-2/r}}{r}\int\limits_{1-2u^r}^{1}\frac{dt}{(1-t^2)^{1-1/r}}=\frac{2^{1-2/r}}{r}\int\limits_{(1-2u^r)^2}^{1}\frac{dt}{\sqrt{t}(1-t)^{1-1/r}}=\\
\frac{2^{1-2/r}}{r}\int\limits_0^{1-(1-2u^r)^2}\frac{t^{1/r-1}dt}{\sqrt{1-t}}=2^{1-2/r}\cdot\int\limits_0^\sqrt[r]{1-(1-2u^r)^2}\frac{dt}{\sqrt{1-t^r}}
\end{align}
When $r=2,3,4$ the last integral can be inverted in terms of elliptic functions. But the case $r=6$ also can be inverted, since
$$
\int\frac{dt}{\sqrt{1-t^6}}=-\frac{1}{2}\int\frac{d\left(\frac{1}{t^2}\right)}{\sqrt{\frac{1}{t^6}-1}}.
$$

As an illustration, according to Edmunds,Gurka, and Lang ($r=4$) one has
$$
\sqrt{1-\sqrt{\frac{1-\left(\sin_{\frac{4}{3},4}x\right)^2}{1+\left(\sin_{\frac{4}{3},4}x\right)^2}}}=\text{sn}\left(x,\frac{1}{\sqrt{2}}\right).
$$

Representation in terms of elliptic functions is possible also when $\frac{1}{p}\neq 1-\frac{1}{r}$. This can be seen by substitution $t\to \frac{at+b}{ct+d}$ in the integral (second integral in eq.$(1)$)
$$
\int\frac{dt}{(1-t)^{1-1/r}t^{1-1/r}}, \quad (r=2,3,4,6),
$$
and specifying parameters $a,b,c,d$ such that the resulting integral again has he form of an incomplete beta function.