How can I transfer mathematical expressions between Mathematica and Maple?

I assume you have Maple to use. If so, Simply open Maple and type the Mathematica command itself directly into Maple using the FromMma package built-into Maple, like this:

restart; 
with(MmaTranslator); #load the package
(*[FromMma, FromMmaNotebook, Mma, MmaToMaple]*)

and now can use it

FromMma(`Integrate[Cos[x],x]`);

One can also use Maple convert command with the FromMma option, like this:

convert(`Integrate[Cos[x],x]`, FromMma);

For your example:

FromMma(`EllipticF[Pi/6, 1/4]`);

You can also use a Mathematica computational expression, not just single commands, like this, and then use the resulting Maple command inside Maple:

r:=convert(`Table[i,{i,10}];`, FromMma);
(* r := [seq(i, i = 1 .. 10)] *)

Now run the result in Maple:

r;
(*[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]*)

see http://www.maplesoft.com/support/help/Maple/view.aspx?path=MmaTranslator for information on the MmaTranslator package.

For translating Maple back to Mathematica: The only program I know about that converts Maple to Mathematica is http://library.wolfram.com/infocenter/Conferences/5397

From Maple 9 Worksheets to Mathematica Notebooks

by Yves Papegay. However, I can't find the actual program or the software. You can try to contact the author on this. This was from The 2004 Wolfram Technology Conference.

You can open a whole Mathematica notebook in Maple using its Open... menu, and all expressions in the notebook will be converted to Maple representation:

Convert Maple expressions to Mathematica:

Through latex：

Through free form input：