I think we can obtain a good looking reduction formula separately for even and odd numbers .The key is that can we obtain it for . For the integral is but for the result looks quite bad .... I am looking for a reduction formula for the integral which only consists of two integrals and it only reduces the power of the csc function .
I got bored so did a power series expansion which is reasonable for ...
Integrating this with for example gives you while .
These were the first 10 values for n=1..10. Guessing n=1 must be wrong then...