# Stopping Simplify and FullSimplify expressing real terms as complex

I am working with real functions but FullSimplify and Simplify express them as complex terms. I assume this is because the LeafCount is smaller if some real terms are expressed as complex terms.

I have been looking for an ExcludedForms option for Simplify but it does not seem to have one for making complex expressions become real.

Here is an example I am working on:

      rhs = -((2 L Sin[
t ω] (Cos[α] - Cosh[α] +
Sin[α] Sinh[α]) F)/(α (Sin[α] -
Sinh[α])));
sol = q[t] /.
First@DSolve[{q''[t] + ωn^2 q[t] == 1/Subscript[M, n] rhs,
q[0] == 0, q'[0] == 0}, q[t], t];
sol2 = FullSimplify[sol]


With output

   (2 I F L (ωn Sin[t ω] - ω Sin[
t ωn]) (Sin[(1/2 + I/2) α] -
Sinh[(1/2 + I/
2) α])^2)/(α ωn (-ω + ωn) (\
ω + ωn) (Sin[α] - Sinh[α]) Subscript[M, n]
)


Note the complex I's in the output

If I take the complex part and do

ComplexExpand[
I (Sin[(1/2 + I/2) α] -
Sinh[(1/2 + I/2) α])^2 ] // Simplify


Then I get this real expression

2 (Cosh[α/2] Sin[α/2] - Cos[α/2] Sinh[α/2])^2


However, if I do ComplexExpand on the whole expression

ComplexExpand[sol2]


I get this

and I have lost my simple form.

Is there an option for Simplify and FullSimplify that suppresses the use of complex forms? Of course if the expression were complex one would need to see the complex form.

Thanks

• @chris How would you write the ComplexityFunction to avoid expressing real terms as complex? The Map approach is an improvement; thanks.
– Hugh
Oct 16, 2020 at 16:40
• @chris ComplexExpand /@ sol2 is much better. Thank you.
– Hugh
Oct 16, 2020 at 16:44

## 1 Answer

You can either write

ComplexExpand /@ sol2


or

FullSimplify[ComplexExpand /@ sol2, Trig -> False]


whichever you find most appropriate for your purpose.