Mutual information of $ \infty $-PAM
I got an answer here, 

https://math.stackexchange.com/questions/1554659/mutual-information-for-a-continuous-uniform-distribution

The $ \text{erf}(x) $ function approximates to 1 with an error less than 0.0001 for $ x \geq 3 $. I plotted $ f_{y} (y) $ and I saw that the most of probability is within a band denpending on $ \sqrt{ 3 \gamma } $. Moreover, taking the limit $ \lim_{ y\to\infty } g(y,\gamma) \log g(y,\gamma) = 0 $. Then, I computed

$$ h(y) = \frac{1}{2} \log \left( 24 \gamma \right) - \frac{ 1 }{ 2 \sqrt{ 3\gamma} } \frac{1}{2} \int_{ - \sqrt{ 3\gamma } - a }^{ \sqrt{ 3\gamma } + a } g(y,\gamma) \log_{2}( g(y,\gamma) ) dy $$

with

$$ g (y,\gamma) = \text{erf} \left( \frac{ y + \sqrt{3\gamma} }{ \sqrt{2} } \right) - \text{erf} \left( \frac{ y - \sqrt{3\gamma} }{ \sqrt{2} } \right) $$

using $ quadgk(\cdot) $ function in matlab.