$$
\begin{align}
\int_0^1\log\left(\frac{1+\sqrt{1-x^2}}2\right)\mathrm{d}x
&=4\int_0^{\pi/4}\log(\cos(\theta))\cos(2\theta)\,\mathrm{d}\theta\\
&=4\int_0^{\pi/4}\left(-\log(2)-\sum_{k=1}^\infty(-1)^k\frac{\cos(2k\theta)}{k}\right)\cos(2\theta)\,\mathrm{d}\theta\\
&=2\int_0^{\pi/2}\left(-\log(2)-\sum_{k=1}^\infty(-1)^k\frac{\cos(k\theta)}{k}\right)\cos(\theta)\,\mathrm{d}\theta\\
&=-2\log(2)-\sum_{k=1}^\infty\frac{(-1)^k}{k}\int_0^{\pi/2}[\cos((k-1)\theta)+\cos((k+1)\theta)]\,\mathrm{d}\theta\\
If I choose two open sets $A$ and $B$ as depicted on Wikipedia here:
then I have an isomorphism between $H_n(A \cap B)$ and $H_n(A) \oplus H_n(B)$ because the two tubes in $A \cap B$ are disjoint.
OK, so far so good. Then I write down the Mayer-Vietoris sequence and try to compute $H_n(T^2)$ ...
