The problem of predicting integrals of stochastic processes is considered. Linear estimators have been constructed by means of samples at N discrete times for processes having a fixed Hölderian regularity s > 0 in quadratic mean. It is known that the rate of convergence of the mean squared error is of order N-(2s+1). In the class of analytic processes Hp, p ≥ 1, we show that among all estimators, the linear ones are optimal. Moreover, using optimal coefficient estimators derived through the inversion of the covariance matrix, the corresponding maximal error has lower and upper bounds with exponential rates. Optimal simple nonparametric estimators with optimal sampling designs are constructed in H² and H∞ and have also bounds with exponential rates.