Disprove that $ \int_0^1 \bigg((1-\sqrt{x})(1-\sqrt{1-x}\bigg)^n~dx $ can be written in the form $\frac{k\pi}{c}-\frac{P}{d}$

Question

I want to disprove that for $n=2,3,4,\cdot \cdot\cdot$

$$ \int_0^1 \bigg((1-\sqrt{x})(1-\sqrt{1-x})\bigg)^n~dx $$

can be put into a form $\frac{k\pi}{c}-\frac{P}{d}$ where $k,c,d$ are integers and $P$ is either prime or semiprime.

It's true for $n=2,3,4,5,6,7,8.$

Answer 1

Mathematica 11 tells that

$$\int_{0}^{1} \big( (1 - \sqrt{x})(1-\sqrt{1-x}) \big)^{13} \, \mathrm{d}x = \frac{1639293419725 \pi }{33554432}-\frac{9482034162443}{61779564} $$

in lowest terms, and

$$ P_{13} = 9482034162443 = 6073 \times 34963 \times 44657$$

is neither a prime nor a semiprime. This is indeed the smallest counter-example.