Hey JiriV and welcome to the forums.
I'm not too familiar with the legendre polynomial myself but I think I might be able to offer a few suggestions.
One suggestion is to expand the results of the integral and simplify. Another suggestion is to use the properties of the projection via an integral transform and a correct basis. For more of the specifics on this check out the following link and scroll down to the Gram-Schmidt process for getting the bases:
You might actually be better off doing the projection or using properties of the derivative in conjunction with the projection, but you would have to do a bit of investigation on your part.