The ActionDispatch Show archive.org snapshot module of Rails gives you the helper method flash to access the flash messages in a response.

describe PostsController, type: :request do

  describe 'update' do

    it 'shows a success message on update' do
      post_record = create(:post)

      put "/posts/#{post_record.id}"

      # Same as @request.flash[:alert]
      expect(flash[:alert]).to eq('Post updated successfully.')
    end

  end


end

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