$$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\sin\left(A(\eta-B)^3\right)$$
$$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\cos\left(A(\eta-B)^3\right)$$
where $a,b,A>0$ and $B\in \mathbb{R}$.
Can anyone help me with this? it seems to me there has to be a closed expression. I can find it for the case $B=0$, but not for any real $B$. For example, I know that:
$$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\cos\left(A\eta^3\right)=\frac16\left[\text{Si}(2A\,b^3)-\text{Ci}(2A\,a^3)\right]$$
Where $\text{Ci}$, $\text{Si}$ are the cosine and sine integrals.
Do you know how to get closed expressions for the integrals above when $B\neq0$? Thanks a lot in advance!