integral evaluation of an exponential let be the function 
$$ e^{-a|x|^{b}} $$
with $ a,b $ positive numbers bigger than zero
then how could i evaluate this 2 integrals ?
$$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$
here 'c' can be either positive or negative or even pure complex (Fourier transform)
also how i would evaluate the Fourier cosine trasnform
$$ \int_{0}^{\infty}dxe^{-a|x|^{b}}cos(cx)$$
thanks in advance if possible give a hinto of course i know tht i could expand the function in powers of $ |x| $ but if possible i would like a closed answer thanks.

If you really interested in finding a closed form formula, you can have it in terms of the Fox $H$-function. However, I am adopting the following convention (see section 3 & 5) of the $H$-function which follows from the Mellin-Barnes integrals. 
$$\int_{-\infty}^{\infty}e^{-a|x|^{b}}e^{cx} dx= \frac{1}{ac}H^{1,1}_{1,1} \left[ \frac{c}{a^b} \left| \begin{matrix} ( 1 , \frac{1}{b} ) \\ ( 1 , 1 )  \end{matrix} \right. \right] -\frac{1}{ac}H^{1,1}_{1,1} \left[ \frac{-c}{a^b} \left| \begin{matrix} ( 1 , \frac{1}{b} ) \\ ( 1 , 1 )  \end{matrix} \right. \right] \,.$$
Offcourse there are existence conditions for the above formula. The above formula can be simplified in terms of less general functions depending on $b$. Just try it for some special values of $b$.
