Certain integration technique

Question

What technique to follow when integration functions in the form: $$\sin ax\over \sin bx$$ $$\cos ax\over \cos bx$$ $$\sin ax \over \cos bx$$

I do believe that all these forms should have a similar technique to follow.

Answer 1

This is standard technique: if both $a$ and $b$ are even, then you can transform your integrand into a rational function by the substitution $t = \tan x$. If at least one of $a$ and $b$ is odd, then the substitution changes a little bit into $t = \tan \frac x 2$. Mind you, though: the rational function that you get might not be easily integrable!

Answer 2

This is not an answer but it is too long for a comment

Let us consider the case $$I=\int\frac{\sin(x)}{\sin(4x)}\,dx$$ Using the multiple angle formula $$\sin(4x)=4\sin(x)\cos(x)-3\sin^3(x)\cos(x)$$ which simplifies a little as Alex M. explained.

Now, use the tangent half-angle substitution $t=\tan(\frac x 2)$ and, after simplifications, we get $$I=-\int \frac{\left(t^2+1\right)^2}{2 \left(t^6-7 t^4+7 t^2-1\right)}\,dt$$ which, fortunately, decomposes as $$I=\int \Big(-\frac{1}{2 \left(t^2-2 t-1\right)}-\frac{1}{2 \left(t^2+2 t-1\right)}-\frac{1}{4 (t+1)}+\frac{1}{4 (t-1)} \Big)\, dt$$ which can be done.