Different result with Mathematica for $\int \frac{x}{x^2 + \frac{1}{4}}\ dx$?

Question

I am confused why Mathematica gives a different answer with this simple integral:

$$ \int \frac{x}{x^2 + \frac{1}{4}}\ dx = \frac{1}{2}\log(x^2 +\frac{1}{4}) + C. $$

Mathematica produces $$ \frac{1}{2}\log(4x^2 + 1). $$

Answer 1

It is the same set of solutions, note that $$\log(4x^2+1)=\log(4(x^2+1/4))=\log4+\log(x^2+1/4)$$ So $\frac12\log4$ can be added to the constant of integration.

Answer 2

The two solutions are the same after working with logarithms and throwing the constants in $C$

Note that $\log (4x)=\log4+\log x$