I have an integral, and it looks simple enough for me to believe it could have an analytic solution; however I am unable to find it. I was trying "Gradshteyn and Ryzhik's Table of Integrals, Series, and Products", but it didn't help me either.
Maybe somebody of you can give me a hint:
$$f_{\pm}(x)=\int_0^{\theta_0} e^{-\frac{2 \theta^2}{\sigma^2}} \cdot \theta \cdot \Big( 1 + \cos\big(A\cdot[ x\pm B\cdot\theta]^2\big)\Big) d\theta$$
where $A,B,\sigma \in \mathbb{R}_+$ are constants, $\theta_0>\sigma$ (for instance, $\theta_0=4\sigma$) and $x \in (\mathbb{R}_+ \cup 0)$ is the argument of the function I want to find.
EDIT: I missed a square inside the cosine, sorry!
The following decomposition $$\int_0^{\theta_0} e^{-\frac{2 \theta^2}{\sigma^2}} \cdot \theta \cdot \Big( 1 + \cos(A\cdot x\pm B\cdot\theta)\Big) d\theta = \int_0^{\theta_0} e^{-\frac{2 \theta^2}{\sigma^2}} \cdot \theta d\theta+ Re\int_0^{\theta_0} e^{-\frac{2 \theta^2}{\sigma^2}} e^{iA(x\pm B\theta)^2} \theta d\theta $$ could help ? Each of the integrands of the right hand side has an explicit formula.