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[integrators] upgrade gradient function and exponential integrators #641

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Run time

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Job	Run time
test_windows (3.9)	2m 8s
test_macos (3.9)	1m 12s
test_linux (3.9)	47s
test_windows (3.10)	1m 41s
test_macos (3.10)	1m 8s
test_windows (3.11)	1m 43s
test_linux (3.10)	48s
test_macos (3.11)	1m 7s
test_linux (3.11)	48s
	11m 22s