Integral $\int_{-\pi/2}^{\pi/2}\cos^2(x)\cos(a+b\tan(x))dx$

Evaluate $$\int_{-\pi/2}^{\pi/2}\cos^2(x)\cos(a+b\tan(x))\mathrm dx$$

I tried the following ideas:

$u=a+b\tan(x)$, $du=b\sec^2(x)dx$

$$\frac{1}{b}\int \cos^4(x)\cos(u)du$$

using $1+\tan^2(x)=\sec^2(x)$

$$b^3\int \frac{\cos(u)du}{(u-a)^4}$$

I am stuck at this point. Any help?

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