$\cfrac{{\sin x}}{{\sin \left( {x - a} \right)}}$

Let $I = \int {\cfrac{{\sin x}}{{\sin \left( {x - a} \right)}}} dx = \int {\cfrac{{\sin \left( {\left( {x - a} \right) + a} \right)}}{{\sin \left( {x - a} \right)}}} dx$

$= \int {\cfrac{{\sin \left( {x - a} \right)\cos \,a + cos\left( {x - a} \right)\sin a}}{{\sin \left( {x - a} \right)}}dx}$

$= \cos a\int {\left( 1 \right)dx} + \sin a\int {\cot \left( {x - a} \right)dx}$

$= x\cos a + \sin a\log \left| {\sin \left( {x - a} \right)} \right| + C$