calculus - How to Integrate $int_0^pi ln(1+alpha cos(x)) ,mathrm{d}x$

I've been trying to learn how to integrate by differentiation under the integral. I've made good progress on some problems, but I seem to not be able to get an answer for

$$f(\alpha)=\int_0^\pi \ln(1+\alpha \cos(x)) \,\mathrm{d}x$$

I've managed to get as far as

$$f'(\alpha)=\int_0^\pi \frac{\cos(x)}{1+ \alpha \cos(x)} \,\mathrm{d}x$$

But this seems like a ridiculous integral to try and integrate by elementary methods, indeed an integral calculator returns

$$\dfrac{x}{a}+\dfrac{\ln\left(\left|\left(a-1\right)\tan\left(\frac{x}{2}\right)-\sqrt{a^2-1}\right|\right)-\ln\left(\left|\left(a-1\right)\tan\left(\frac{x}{2}\right)+\sqrt{a^2-1}\right|\right)}{a\sqrt{a^2-1}}$$

Hopefully someone can advise on whether I've already made a mistake in my working, or whether I've just completely misunderstood the method.