A question on exponentialize and demoivre



> What should I do, please, for the transformation of an expression
> containing exponentials to hyperbolic sines and cosines?
...
> I would prefer to use an existing Maxima command (such
> as demoivre) avoiding to prepare my own simplification
> rules for the second case (that finally leading to hyperbolic
> functions).

The simplest way to do this is probably with a change of variable from
x->%i*x, then demoivre, then the reverse transformation.  This even
catches cases where the hyperbolic functions are not completely obvious,
e.g.:

                                2 x           - 2 x
                              %E        x   %E        1
(D13)                         ----- - %E  + ------- + -
                                4              4      2

(C14) subst(%i*x,x,%);
                           2 %I x              - 2 %I x
                         %E           %I x   %E           1
(D14)                    -------- - %E     + ---------- + -
                            4                    4        2

(C15) demoivre(%);

      %I SIN(2 x) + COS(2 x)   COS(2 x) - %I SIN(2 x)
1
(D15) ---------------------- + ---------------------- - %I SIN(x) -
COS(x) + -
                4                        4
2

(C16) subst(x/%i,x,%);

        SINH(2 x) + COSH(2 x)   COSH(2 x) - SINH(2 x)
1
(D16)   --------------------- + --------------------- - SINH(x) -
COSH(x) + -
                  4                       4
2
...

(C19) ratsimp(d16);
                        COSH(2 x) - 2 SINH(x) - 2 COSH(x) + 1
(D19)                   -------------------------------------
                                          2