code: def test_integrability_tensor_derivative_vec(self, n_reps, atol):
      # TODO: need to be reviewed
      # TODO: can this be moved up?
      base_point = self.data_generator.random_point()
      horizontal_vec_x = self._get_horizontal_vec(base_point)
      horizontal_vec_y = self._get_horizontal_vec(base_point)
      nabla_x_y = self._get_nabla_x_y(
          horizontal_vec_x, horizontal_vec_y, base_point
      )
  
      tangent_vec_e = self.data_generator.random_tangent_vec(base_point)
      nabla_x_e = self.data_generator.random_tangent_vec(base_point)
  
      (
          nabla_x_a_y_e,
          a_y_e,
      ) = self.bundle.integrability_tensor_derivative(
          horizontal_vec_x,
          horizontal_vec_y,
          nabla_x_y,
          tangent_vec_e,
          nabla_x_e,
          base_point,
      )
  
      vec_data = generate_vectorization_data(
          data=[
              dict(
                  horizontal_vec_x=horizontal_vec_x,
                  horizontal_vec_y=horizontal_vec_y,
                  nabla_x_y=nabla_x_y,
                  tangent_vec_e=tangent_vec_e,
                  nabla_x_e=nabla_x_e,
                  base_point=base_point,
                  expected_nabla_x_a_y_e=nabla_x_a_y_e,
                  expected_a_y_e=a_y_e,
                  atol=atol,
              )
          ],
          arg_names=[
              "horizontal_vec_x",
              "horizontal_vec_y",
              "nabla_x_y",
              "tangent_vec_e",
              "nabla_x_e",
              "base_point",
          ],
          expected_name=["expected_nabla_x_a_y_e", "expected_a_y_e"],
          n_reps=n_reps,
          vectorization_type="basic",
      )
      self._test_vectorization(vec_data)
  
imports: import pytest
  import geomstats.backend
  from geomstats.geometry.matrices import Matrices
  from geomstats.test.random import KendalShapeRandomDataGenerator, get_random_times
  from geomstats.test.vectorization import generate_vectorization_data
  from geomstats.test_cases.geometry.base import LevelSetTestCase
  from geomstats.test_cases.geometry.fiber_bundle import FiberBundleTestCase
  from geomstats.test_cases.geometry.quotient_metric import QuotientMetricTestCase
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 12 for calling the PreShapeSpaceBundleTestCase obj's underlying member method code with example usage: obj.test_integrability_tensor_derivative_vec(n_reps, atol) without return types