calculation and analysis – How to calculate the integral of a given function by a generic series of powers?

I have a function that can not be written through known functions, but it remains a son of God and deserves to be an integral part of it. $ [0, 1] $ calculated.
How can I do this in Mathematica?

I hope there is a way to do it.

Just for illustration, let's say that this function is (where $ n $ is an integer):

$$
int_ {0} ^ {1} mathrm sum _ {k = 1} ^ { infty} frac {(- 1) ^ ku ^ {2k} (2k) ^ {2k}} {(2k)! ^ 2} cot { pi u} sin {2 pi nu} , from
$$

It's part of my complicated formula that I'm working on.

Edit:

The sum of the integrals (instead of integrating the sum) one by one is not an option, because the individual integrals take an indeterminate time to calculate. The given example is close to the real integral, so it should be long to calculate, say, 50 iterations. Plus $ n $Mathematica takes longer to calculate individual integrals.