I'm studing Vector Calculus and my teacher gave to me a bunch of problems as homework, one of the problems was to evaluate $\int_{c}^{}\vec{F}\cdot d\vec{r}$ with $\vec{F}(x,y,z)= \left \langle yz e^{x},zx e^{y},xy e^{z} \right \rangle$ and $\vec{r}(t)=\left \langle \sin{t},\cos{t},\tan{t} \right \rangle$ for $0\leq t\leq \pi/4$.
I know that $\vec{F}(r(t))=\left \langle \sin{t}\; e^{\sin{t}},\sin{t}\;\tan{t}\; e^{cost},\sin{t}\; \cos{t} e^{\tan{t}} \right \rangle$ and $d\vec{r}(t)=\left \langle \cos{t},-\sin{t}, \sec^{2}{t} \right \rangle dt$ therefore
$\int_{c}^{}\vec{F} \cdot d\vec{r}=\int_{0}^{\pi/4}\left \langle \sin{t}\; e^{\sin{t}},\sin{t}\;\tan{t}\; e^{cost},\sin{t}\; \cos{t} e^{\tan{t}} \right \rangle \cdot \left \langle \cos{t},-\sin{t}, \sec^{2}{t} \right \rangle dt$
then $\int_{c}^{}\vec{F} \cdot d\vec{r}=\int_{0}^{\pi/4}(\cos{t}\sin{t}e^{\sin{t}}-\sin^{2}{t}\tan{t}\; e^{\cos{t}}+ \tan{t}\; e^{\tan{t}})dt$, and here comes my question.
Is it posible to do this last integral analytically? I mean, the first two integral are a simple substitution and with a proper substitution the last one became the exponential integral. As far as I know, $\int_{}^{} \tan{t}\; e^{\tan{t}}dt$ can't be done analytically, however my teacher told me it was posible. I thought I could do it by the Laplace Transform method, but I couldn't figure out how. I do not know if there is something that I am not considering at the moment or if I do not fully understand some part of the theory behind all.