I've tried integration by parts. I can integrate both factors:

$\int\cos(x)dx = \sin(x) + C_1$

$\int\cos(\frac a x) = a Si(\frac a x) + x \cos(\frac a x) + C_2$

However I'm stuck at this point. I think the way to proceed would be integration by parts, however that requires calculating the integral of $\int (u' \int v dx) dx$. However, I don't think there will be a nice closed form solution, no matter how many times I apply integration by parts, because I'll never be able to escape from $u$ being or containing $f(x)$, where $f$ is a periodic function, and $v$ being or containing $g(\frac a x)$, where $g$ is also a periodic function.

There's also some interesting ideas in Closed form of $\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$?, however I'm not sure how to adapt that solution to integrate from $0$ to $b$ instead of $0$ to $\infty$.

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    $\begingroup$ WolframAlpha was unable to find a closed-form solution $\endgroup$ – Victoria M Feb 18 at 19:52
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    $\begingroup$ Perhaps this integral won't even be decent at all. Here is maybe something similarly: math.stackexchange.com/q/3068201/515527 $\endgroup$ – Zacky Feb 18 at 23:19
  • $\begingroup$ I doubt you will find a closed form solution, but have you looked at $\cos(x)\cos\left(\frac a x\right)=\frac{1}{2}\cos\left(x+\frac{a}{x}\right)+\frac{1}{2}\cos\left(x-\frac{a}{x}\right)$? $\endgroup$ – John Wayland Bales Feb 20 at 22:36
  • $\begingroup$ @JohnWaylandBales I have looked into that. Unfortunately I'm not able to get much further with it. $\endgroup$ – Joel Feb 21 at 0:39

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