I regret damaging your book.
I regret that I've damaged your book.
I wonder if the second sentence is much better than the first ? Somehow I find it is not suitable to use the first as a ' gerund ' can be replaced by ' to+infinitive ', then it just sounds illogical.
eg I regret to damage your book. Or I regret to have damaged your book.
Both don't seem right to me from the logical point as I choose ' I regret that I've damaged your book ' myself.