Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$

Question

I'm trying to obtain a closed form expression for the following definite integral in $\mathbb{R}^2$:

$$ \int_{0}^{\pi} \int_{u = 0}^{u= A \sin\theta} f(\theta, u)\,\mathrm{d}u \,\mathrm{d}\theta ~,\qquad 0 < A<1,\qquad f(\theta,u)\equiv \bigl(1 - \cos 2\theta \bigr) \Bigl(1 + \cos\bigl(\frac{\theta + 2 \pi u}2 \bigr)\Bigr) $$

The region of integration is the "hump" under the sine curve with amplitude $A$.

Note that $f(\theta,u)$ is asymmetric (either mirror or rotational) within $0<\theta<\pi$. Nonetheless, it is rotationally symmetric $f(2\pi - \theta, 1- u) = f(\theta,u)$ if one considers twice the integral by extending the domain of $\theta$.

The figures below display the integrand as the yellow surface (front and back view). The sine hump region of integration is enclosed by the blue wall, and outside the region is dimmed. The top row is $f(\theta, u)$ itself, the middle row is for region $A = 9/10$, and the bottom row is $A = 1/4$.

My Questions

1. Can this deceivingly simple-looking integral be evaluated to a closed form as a function of $A$?
2. Regardless of (1), is this integral related to some "named" integral or special functions?

My Failed Attempts So Far

This is a 2-dim region so the basic two ways are "vertical first" and "horizontal first".

Doing vertical first $\int \mathrm{d}u$ for $u = 0$ to $u = A \sin\theta$ yields a nested sine $\sin\bigl( \frac{\theta}2 + A \pi \sin\theta\bigr)$. It doesn't take long for my Mathematica to declare that this cannot be evaluated (returning the input as is after 50 sec).

Doing horizontal first $\int \mathrm{d}\theta$ for $\theta = \theta_u$ to $u = \pi - \theta_u$, where $\theta_u \equiv \arcsin(\frac{u}A)$, produces many terms like $\sin\bigl( \pi u + \frac{\theta_u}2 \bigr)$ which I don't know how to handle. My Mathematica churned out something (after about 3 min) with many complex (imaginary) terms and a lot of Erf function (antiderivative of Gaussian). On the coding side, I tried ComplexExpand with FullSimplify etc but it just got worse (over 30 thousand terms) and less comprehensible.

So far I don't have any good ideas for change of variables. Any input would be appreciated.