Fresnel Integral multiplied with cosine term. $$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$
Can anybody tell me, how to solve this integral ?
 I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent. 

In almost the same spirit as user71352's answer, the antiderivative of $$I=\int \sin(\alpha-\beta x^2)\cos(x)\, dx$$ can be written as $$2 \sqrt{\frac{2\beta}{\pi}} I=$$ $$\sin \left(\alpha +\frac{1}{4 \beta }\right) \left(C\left(\frac{2 x \beta -1}{\sqrt{2
   \pi\beta} }\right)+C\left(\frac{2 x \beta +1}{\sqrt{2 \pi \beta} }\right)\right)-\cos \left(\alpha +\frac{1}{4 \beta }\right) \left(S\left(\frac{2
   x \beta -1}{\sqrt{2 \pi \beta} }\right)+S\left(\frac{2 x \beta +1}{\sqrt{2
   \pi \beta} }\right)\right)$$ where $C$ and $S$ are the Fresnel integrals already mentioned by user71352.
