<s>[INST] Very curious sequence of integrals $I_n=\int_0^1 \frac {(x(1-x))^{4n}} {1+x^2}\mathrm dx$ I was studying the behaviour of very curious sequence of integrals 
$$I_n=\int_0^1 \frac {(x(1-x))^{4n}} {1+x^2} \,\mathrm dx$$ which gives a very beautiful result for $n=1$; I tried to calculate for different values of $n$ but every time what I get is a $4^{n-1}$ times $\pi$ along with a fraction that in the denominator has almost a product a consecutive primes, Can we generalize this pattern? Any help would be appreciated!
Here are few calculations:
$$
I_1=22/7-\pi
$$
$$
I_2=-\frac {2^2 \cdot 43\cdot 1097} {3\cdot 5\cdot 7\cdot 11 \cdot 13} +4\pi
$$
$$
I_3=\frac {13\cdot 31\cdot 13912991} {3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23}-16\pi
$$
 [/INST] Well, I guess I can't compete with this research cited by Norbert, but maybe this will be of any help.
One can try to expand $(1-x)^n = \sum_{k=0}^n {n \choose k}(-x)^k$, so 
$$\int_0^1 \frac {(x(1-x))^{4n}} {1+x^2} \,\mathrm dx= \sum_{k=0}^n {n \choose k}(-1)^k\int_0^1 \frac {x^{4n+k}} {1+x^2} \,\mathrm dx$$
Then one can change variable so  $$\int_0^1 \frac {x^{4n+k}} {1+x^2} \,\mathrm dx=\frac{1}{2}\int_0^1 \frac {t^{2n+\frac{k-1}{2}}} {1+t} \,\mathrm dt$$
The last integral (in the indefinite form) is:
$$\int_0^1 \frac {t^{2n+\frac{k-1}{2}}} {1+t} \,\mathrm dt=\frac{2 t^{\frac{1}{2} (k+4 n+1)} \, _2F_1\left(1,\frac{1}{2} (k+4 n+1);\frac{1}{2} (k+4 n+3);-t\right)}{k+4 n+1}$$
Plugging the limits will give:
$$\int_0^1 \frac {(x(1-x))^{4n}} {1+x^2} \,\mathrm dx=\frac{1}{4}\sum_{k=0}^n {n \choose k}(-1)^k\left(\psi ^{(0)}\left(\frac{k}{4}+n+\frac{3}{4}\right)-\psi ^{(0)}\left(\frac{k}{4}+n+\frac{1}{4}\right)\right)$$
where $\psi ^{(0)}\left(\frac{k}{4}+n+\frac{1}{4}\right)$ is the $0$-derivative of the digamma function $\psi(z)$.
</s>