We got reciprocated functions but how do we utilise them in exponential trigonometric functions?

Brute force: create a 50 term Taylor series expansion and then numerically integrate the series from $0$ to $\pi$ : 1.66504399816, The series is a mess so large that the $\LaTeX$ of the series cannot be included.

The answer apparently is not $\frac53$ as, even when working with 200 digit simulated floating point, the answer is:

$1.66504399815638159256367539767815492775949753925341850565583330192562\\ 8778785504346403548954543604444849101306137475440074843711581710218528\\ 7641270510528079508665973032975689767260095886296429335425740$