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[integrators] upgrade gradient function and exponential integrators #642

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Run time

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Job	Run time
test_linux (3.9)	49s
test_linux (3.10)	48s
test_macos (3.9)	2m 10s
test_linux (3.11)	48s
test_windows (3.9)	2m 12s
test_macos (3.10)	1m 28s
test_windows (3.10)	1m 45s
test_macos (3.11)	1m 11s
test_windows (3.11)	2m 8s
	13m 19s