I want to calculate characteristic function of this probability density function (PDF) of $X$. The distribution of $X$ is given by the following equation:
$$ f_{X}(x)=\frac{\beta}{\alpha\Gamma(1/\beta)}\ x \ e^{-\left( |x^{2}-\mu| / \alpha \right)^{\beta}} $$
where $\alpha, \beta \ $ and $ \mu \ $ are constant and lager than zero. Random variable $ x $ also larger than zero.
I tried calculation of characteristic function of $X$. But that is so complicated. How can I get this characteristic function?