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Fault detection and diagnosis is one of the most critical components of preventing accidents and ensuring the system safety of industrial processes. In this paper, we propose an integrated learning approach for jointly achieving fault detection and fault diagnosis of rare events in multivariate time series data. The proposed approach combines an autoencoder to detect a rare fault event and a long short-term memory (LSTM) network to classify different types of faults. The autoencoder is trained with offline normal data, which is then used as the anomaly detection. The predicted faulty data, captured by autoencoder, are put into the LSTM network to identify the types of faults. It basically combines the strong low-dimensional nonlinear representations of the autoencoder for the rare event detection and the strong time series learning ability of LSTM for the fault diagnosis. The proposed approach is compared with a deep convolutional neural network approach for fault detection and identification on the Tennessee Eastman process. Experimental results show that the combined approach accurately detects deviations from normal behaviour and identifies the types of faults within the useful time.