Indefinite integral of $\tan(x)e^{(tanx)}$.

Question

Indefinite integral of $$\int \tan (x) e^ {(\tan (x))} \, dx$$ I have tried using integration by parts but I couldn't integrate it.

Answer 1

Let $\tan(x)=y$ making $$I=\int \tan (x) e^ {\tan (x)} \, dx=\int \frac y{1+y^2} e^y \,dy=\int \frac y{(y+i)(y-i)} e^y \,dy$$

Now, partial fraction decomposition and some obvious changes of variable will take you to the definition of the exponential integral function.

$$I=\frac12 \left(\int\frac{e^y}{ y+i}dy+\int\frac{e^y}{ y-i}dy\right)$$

$$J=\int\frac{e^y}{ y+a}dy=e^{-a}\int\frac{e^{y+a}}{ y+a}dy=e^{-a}\int\frac{e^{t}}{ t}dt=e^{-a}\,\text{Ei}(t) $$ So, $$I=\frac12 \left(e^i\, \text{Ei}(\tan (x)-i)+e^{-i}\,\text{Ei}(\tan (x)+i) \right)$$