Calculating $\int\frac{e^{at}\cos(at)}{\sin t}dt$.

Question

Calculate $$\int\frac{e^{at}\cos(at)}{\sin t}dt.$$

Which way should I follow to solve this? At least in approach. There is no complete integration. Laplace etc doesn't work either. Should I approach serial?

Answer 1

Even for $a=1$ there is no closed form solution and, as soon as $a \neq 0$, you face gaussian hypergeometric functions.

For a series solution, consider that, by composition of Taylor series $$\csc (t) \cos (a t)=\frac 1 t+\sum_{n=1}^\infty b_n t^n$$ where the first coefficients are $$\left\{-\frac{3 a^2+5}{6} ,0,\frac{15 \left(a^2-2\right) a^2+7}{360} ,0,\frac{31-21 a^2 \left(a^4-5 a^2+7\right)}{15120},\cdots\right\}$$ Integrating termwise, you will have $$\int\frac{e^{at}\cos(at)}{\sin (t)}dt=\text{Ei}(a t)+\sum_{n=1}^\infty b_n\int e^{at}\, t^n \,dt$$ and the last integrals correspond to the incomplete gamma function.