The exact error probability for the M-PSK constellation is derived in "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" by JW Craig, a well-known technique, especially for fading channels. Eq. (6) in the reference denotes the probability of error as given below: $$ \mathrm{P}_{\mathrm{M}}=\frac{1}{\pi} \int_{0}^{\pi-\Psi} \exp \left[-\frac{\gamma_{3} \sin ^{2}(\Psi)}{\sin ^{2}(\theta)}\right] \mathrm{d} \theta$$$$ \mathrm{P}_{\mathrm{M}}=\frac{1}{\pi} \int_{0}^{\pi-\Psi} \exp \left[-\frac{\bar{\gamma} \sin ^{2}(\Psi)}{\sin ^{2}(\theta)}\right] \mathrm{d} \theta$$
where $\bar{\gamma}$ is the SNR and $\Psi= \pi/M$.