integration - Evaluating the definite integral $int_{-c}^{c} e^{-ax^2}cos^2(bx) ,mathrm{d}x$

How can one evaluate the following integral?

$$\int_{-c}^{c} e^{-ax^2}\cos^2(bx) \,\mathrm{d}x$$

Wolfram Alpha gives this. Is there not a more compact form?

If $\int_{-c}^{c} e^{-ax^2} \, \mathrm{d}x=k$, then can we express the first integral in terms of $k$? Thanks.