How to get rid of all complex numbers and functions?

Question

For almost esthetical reasons, I always have been fascinated by infinite sums of series and I always wonder if theyhave a closed form. A couple of days ago, I found on MSE a question related to the convergence of $$S=\Sigma_{n=0}^{\infty}{\frac{n^2 x}{1+n^4 x^2}}$$ where $x$ is a real number. So, I gave the problem to one of my former colleagues who ran Mathematica for this summation.

He sent me back the following (surprizing result) $$\frac{\sqrt[4]{-1} \pi \left(\cot \left(\frac{(-1)^{3/4} \pi }{\sqrt{x}}\right)+i \cot \left(\frac{\sqrt[4]{-1} \pi }{\sqrt{x}}\right)\right)}{4 \sqrt{x}}$$ telling that he did not find any way to make it simpler. For sure, the result is a real. I tried to simplify manually, expanding the $cot$ functions and ... arrived nowhere after pages of manual calculations (being almost blind does not make things easier).

Could any one tell me what should be the appropriate syntax to be used in order to remove all the complex numbers from this expression ? Any help will be really appreciated.

Answer 1

You can do :

expr = (-1)^(1/4) π (Cot[(-1)^(3/4) π / Sqrt[x]] + 
        I Cot[(-1)^(1/4) π / Sqrt[x]])/(4 Sqrt[x]);

FullSimplify[ComplexExpand[expr, TargetFunctions -> {Re, Im}], Assumptions -> {x > 0}]
(* (π (Sin[(Sqrt[2] π)/Sqrt[x]] - Sinh[(Sqrt[2] π)/Sqrt[x]])) / 
   (2 Sqrt[2] Sqrt[x] (Cos[(Sqrt[2] π)/Sqrt[x]] - Cosh[(Sqrt[2] π)/Sqrt[x]])) *)