An intriguing pattern in Ramanujan's theory of elliptic functions that stops? I. Define the ff integrals,
$$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(\frac13\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$
$$K_4(k)=\int_0^{\pi/2}\frac{\cos\left(\frac12\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac14,\tfrac34,1,\,k^2\right)}$$
$$K_6(k)=\int_0^{\pi/2}\frac{\cos\left(\frac23\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac16,\tfrac56,1,\,k^2\right)}$$
These are Ramanujan's theory of elliptic functions for alternative bases of signature $2,3,4,6$, respectively. There are only 4 signatures.
II. Then, using Wolfram, I observed the closed-forms of the ff definite integrals,
$$\int_0^1 K_2(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac12,\tfrac12;1,\tfrac32;1\right)}=2G$$
$$\int_0^1 K_3(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac13,\tfrac23;1,\tfrac32;1\right)}=\tfrac{3\sqrt3}2\, \ln2$$
$$\int_0^1 K_4(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac14,\tfrac34;1,\tfrac32;1\right)}=2\ln(1+\sqrt2)$$
$$\int_0^1 K_6(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac16,\tfrac56;1,\tfrac32;1\right)}=\tfrac{3\sqrt3}4\, \ln(2+\sqrt{3})$$
where $G$ is Catalan's constant. (Curiously, other than the first, Wolfram didn't recognize the closed-form of those hypergeometrics. I had to use the Inverse Symbolic Calculator.)

III. Questions


*

*Does the generalized hypergeometric function,
$$H(n)=\,_3F_2\left(\tfrac12,\tfrac1n,\tfrac{n-1}{n};1,\tfrac32;1\right)$$ have a closed form only for $n=2,3,4,6$? (I tried $n=5,7,8$, etc, and it doesn't seem to have a "neat" form using elementary functions.)

*If so, is it connected to why there are only 4 signatures of alternative bases?

 A: Some speculative clues have appeared after a bit of detective work...
First the integral we are interested in with the associated hypergeometric function and infinite series.
$$I_n=\int_0^1 K_n(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac{1}{n},\tfrac{n-1}{n};1,\tfrac32;1\right)}= \frac{ \pi}{2}\times\sum _{k=0}^{\infty } \frac{\prod _{j=0}^{k-1} \left(j+\frac{1}{n}\right) \prod _{j=0}^{k-1} \left(j+\frac{n-1}{n}\right)}{(2 k+1) (k!)^2}$$
Simplifying the infinite series a little I found that
$$I_n=\frac{ \pi}{2}\,\sum _{k=0}^{\infty } \frac{\prod _{j=1}^k \left(j^2-\frac{1}{n^2}\right)}{(k n+1)(2 k+1) (k!)^2  }$$
Now some interesting links appear to your integral if we study the much simpler sum
$$S_n=\sum _{k=0}^{\infty } \frac{(-1)^k}{(k n+1)( 2k+1)}$$
we find from Mathematica that 
$$S_2=G$$
$$S_3=\pi  \left(\frac{1}{\sqrt{3}}-\frac{1}{2}\right)+\log (2)$$
$$S_4=\frac{1}{4} \pi  \left(\sqrt{2}-1\right)+\frac{\log \left(\sqrt{2}+1\right)}{\sqrt{2}}$$
$$S_6=\frac{1}{8} \left(\pi +2 \sqrt{3} \log \left(\sqrt{3}+2\right)\right)$$ $$S_8=\frac{1}{12} \pi  \left(\sqrt{2}+1\right)+\frac{\log (2)}{3}+\frac{\log \left(\sqrt{2}+1\right)}{3 \sqrt{2}}$$
These are all the shortest and simplest closed forms between $n=2$ and $n=12$. 
For $I_2$, $I_3$, $I_4$ and $I_6$ that you found closed forms for, the respective sums have one term with the same principal constant and have a maximum of 3 terms. The next simplest sum I found is $S_8$ with four terms.
Have fun.
