I am trying to solve the integral:
$$I=\int { \sin(ct) \over \cos(t) + \tan(t) }dt$$
where $c$ is a real constant. No matter which trigonometric identity I use to transform this integral, I can't get it to a solvable form. I even tried using the Feynman's technique, but that didn't help either. Any suggestion on how to solve this integral would be appreciated.
$$\int { \sin(ct) \over \cos(t) + \tan(t) }dt=\int {\sin(t) \sin(ct) \over 1+\sin (t)-\sin ^2(t) }dt$$ If $c$ is an integer, use $$ \sin(nt) = \sum_{k\text{ odd}} (-1)^\frac{k-1}{2} {n \choose k}\cos^{n-k} (t) \sin^k (t) $$ and using $x=\sin(t)$ , $\cos^2(t)=1-\sin^2(t)$, you will face simple integrals in $x$.
For example, $$I_3=\int \frac{x^2 \left(4 x^2-3\right)}{\sqrt{1-x^2} \left(x^2-x-1\right)}\,dx$$ $$I_4=\int \frac{4 x^2 \sqrt{1-x^2} \left(2 x^2-1\right)}{x^2-x-1}\,dx$$
Remember that $$x^2-x-1=\left(x-\frac{1}{2} \left(1-\sqrt{5}\right)\right)\left(x-\frac{1}{2} \left(1+\sqrt{5}\right)\right)$$ will help for partial fraction decomposition.