Formatting Expression to Standard Forms in Maxima - formatting

Question

I'm a newbie to Maxima and I'm finding the tool very useful. One of the more common use-cases for me is the validation of expressions given in academic texts. Take this as an example:
This is a transfer function written in a pretty standard format. Notice how the squared term in the denominator appears by itself. Fair enough.
So, I go about building this expression in pieces and my expectation is to compare results at the end. It helps then if the expressions are displayed using a common format. So, I'm trying to get my result to have the same normalized format as the expression in the book.
In this case, I do the following:
(%i10) F(s):=(1+s*tau_2)/(1+s*(tau_1+tau_2));
(%o10) F(s):=(1+s*tau_2)/(1+s*(tau_1+tau_2))
(%i13) H(s):=(K_0*K_d*F(s))/(s+((K_0*K_d*F(s)/N)));
(%o13) H(s):=(K_0*K_d*F(s))/(s+(K_0*K_d*F(s))/N)
(%i81) H_1(s):=ratsimp(H(s),s);
(%o81) H_1(s):=ratsimp(H(s),s)
(%i82) H_1(s);
(%o82) (K_0*K_d*N*s*tau_2+K_0*K_d*N)/(s^2*(N*tau_2+N*tau_1)+s*(K_0*K_d*tau_2+N)+K_0*K_d)
Which is very close. But, I've tried a few different ways of dividing across the expression by N*(tau_1 + tau_2) to leave s^2 without a coefficient and none worked. Is there an easy way to solve this?
UPDATE:
Some options I've tried:
--> divthru(e,d):=map(lambda([u], multthru(u,d)),e); /* map applies function f() to each subpart of expr*/
(%o164) divthru(e,d):=map(lambda([u],multthru(u,d)),e)
--> divthru(H_1(s), 1/(N*tau_2+N*tau_1));
(%o165) (K_0*K_d*N*s*tau_2+K_0*K_d*N)/(s^2*(N*tau_2+N*tau_1)+s*(K_0*K_d*tau_2+N)+K_0*K_d)
(%i35) matchdeclare([A,B,C,D],all);
(%o35) done
(%i26) defmatch(isDefPoly, A/(s^2*B+s*C+D), x);
(%o26) isDefPoly
(%i27) isDefPoly(H_1(s),s);
(%o27) [A=K_0*K_d*N*s*tau_2+K_0*K_d*N,D=K_0*K_d,B=N*tau_2+N*tau_1,C=K_0*K_d*tau_2+N,x=s]
(%i36) tellsimpafter(A/(s^2*B+s*C+D), (A/B)/(s^2+(s*C)/B+D/B));
(%o36) [\*rule1,simptimes]

Answer 1

ratsimp tries to put expressions in a standardized format which makes further operations more straightforward, and it is difficult to convince it to rearrange terms in some other way. For example, for an expression like (a + b/c)/(d + e/f), ratsimp will try to rearrange it to something which is one sum of terms over another set of terms.
I think there are some add-on functions which rearrange terms in different ways, but I'm not sure there is anything to achieve the form you have shown.
One thing that will help in a small way. My advice is to write subscripted variables as x[k] instead of x_k. That can help the TeX formatting code recognize the variable name, e.g. tau, and format it as a Greek letter. Also, you can enter Unicode characters (from any language, multibyte characters included) as variable names.

Answer 2

I worked on it some more and maybe there are some ideas here which can help. Here is a function to divide above and below. I'll apply it to Hs as shown in the problem statement.
(%i1) divabovebelow (e, r) := (multthru (1/e, num(r)))/(multthru (1/e, denom(r))) $
(%i2) Hs: (K_0*K_d*N*s*tau_2+K_0*K_d*N)/(s^2*(N*tau_2+N*tau_1)+s*(K_0*K_d*tau_2+N)+K_0*K_d);
K_0 K_d N s tau_2 + K_0 K_d N
(%o2) --------------------------------------------------------
2
s (N tau_2 + N tau_1) + s (K_0 K_d tau_2 + N) + K_0 K_d
(%i3) divabovebelow (N*tau_2 + N*tau_1, Hs);
K_0 K_d N s tau_2 K_0 K_d N
----------------- + -----------------
N tau_2 + N tau_1 N tau_2 + N tau_1
(%o3) ----------------------------------------------
s (K_0 K_d tau_2 + N) K_0 K_d 2
--------------------- + ----------------- + s
N tau_2 + N tau_1 N tau_2 + N tau_1
Here are a couple more functions that could be helpful. polynomial_in rearranges terms to kind of isolate one variable from the others. nicer_rational tries to make a rational function look nicer, via polynomial_in.
(%i4) Hs1: ratsimp (Hs);
K_0 K_d N s tau_2 + K_0 K_d N
(%o4) -----------------------------------------------------
2 2
(N s + K_0 K_d s) tau_2 + N s tau_1 + N s + K_0 K_d
(%i5) polynomial_in (var, e) := apply ("+", makelist (var^k * ratcoef (e, var, k), k, 0, hipow (e, var))) $
(%i6) nicer_rational (var, e) := (polynomial_in (var, num(e)))/(polynomial_in (var, denom(e))) $
(%i7) nicer_rational (s, Hs1);
K_0 K_d N s tau_2 + K_0 K_d N
(%o7) --------------------------------------------------------
2
s (N tau_2 + N tau_1) + s (K_0 K_d tau_2 + N) + K_0 K_d
These yield the same eventual result as ratsimp above, but maybe they make more sense since they are devised for the kind of expressions you're working with.
I was thinking about some additional functions to get closer to the textbook form, but it's a little fragile, since Maxima wants to reorganize expressions into its standard form, so for better or worse, one can get either a nice display, or an expression for working with. Hope this helps anyway.