Most  coulditbe(f(z)=0) assuming real;  for not very simple f(z) are anyway (and inevitable) FAIL.

@acer  If we must start (for some reason) with Digits = 15, then method=_d01ajc  will not work, and the nonNAG methods are really slow. 

@sija  Your problem is to solve equations of the form  Int( f(t), t=-4 .. x) = a, f(t) and a being given.

The function F(x) = Int( f(t), t=-4 .. x)  is the solution of the ode  F'(t) = f(t), F(-4) = 0. So, F can be obtained by dsolve (numerically very efficient).

Finally, fsolve(F - a, -10..10)  finds the wanted approximate solution (in the interval -10..10)).
 

@Exiu 

convert(f, radical);

@Christian Wolinski 

coulditbe(exp(z)=0);        #? true
coulditbe(abs(z)+2+z=0);    #? true
solve(abs(z)+2+z=0);        #ok NULL

f can be represented this way iff 0 is not a pole for f (i.e. 0 is not a root of denom(f)).

@Andiguys  I have already mentionrd that Maximize does not accept double inequalities a <= b <=c; use a<=b, b <= c.
So, correct C2:=....

@Andiguys  You do not have a constraint plane. Using
solve(C1 union C3 union C3);
it results
dom := tau1=0..lambda, lambda= 0.0224489798..1.022448980;

Then:
plot3d(TRC(tau1, lambda), dom, orientation=[165, 75,0]);

@Andiguys  Use:

plot3d(TRC(tau1, lambda), tau1 = 0 ..1, lambda = 0 .. 1, orientation=[165, 75,0]);

@Andiguys Replace the execution group C3c := ...   with

temp := (-2*R0er*d^2*lambda + 2*d^2*g1*lambda + Aer*d - Am*d + 2*Clr*lambda + Rer*d - Rm*d)/(2*(-R0er*d^2 - R0m*d^2 + d^2*g1 + d*g2 + Clm + Clr));
C3c := {0 <= temp, temp <= 1};

Use the same intervals for tau1 and lambda in Minimize and plot!

@ecterrab Please note that the formula is valid for all z in C, not just |z|<Pi/2 (unlike the power series one).

@ecterrab  The expansion comes from Mittag-Leffler theorem  Mittag-Leffler's theorem - Wikipedia.
The singularities are exactly those of tan, so your objection is not justified.
But of course, FunctionAdvisor cannot list all the possible formulae.
 

@nm This is obviously a bug. Actually a regression bug, inexistent in Maple 2023 (and an elementary one, the expression being continuous w.r.t. x).
Unfortunately, today is much more difficult to find an explanation than 20 (or even 10)  years ago! 

@nm  OK, then I suggest to start some other similar questions such as about:

e:=-tanh(sqrt(2)*(a*x+b)):
limit(e,x=0): expand(%);

and

e:=-tanh(sqrt(2)*(a*x+b)):
limit(e,x=0): convert(%, exp): %;

etc.

I did not delete it, but why don't you continue your previous question having exactly the same limit?
limit not evaluating - MaplePrimes