Fullsimplify not working perfectly

Here I am using the below code for a simplification.

DB = \[Sigma]*\[Sigma]*\[Phi]*\[Phi]     BB = Sqrt[DB] // FullSimplify // PowerExpand     g = 1/DB         \[Alpha]I = a\[Phi]     \[Alpha] = \[Alpha]I - 0.5*BB*D[BB, \[Phi]]     \[Alpha]s = \[Alpha] - ((1/2)*BB*D[BB, \[Phi]])     UG = 0.375 \[Sigma]^2     Phi = UG - ((1/2)*(g*\[Alpha]s*\[Alpha]s)) // FullSimplify //        PowerExpand 

But I am getting a result without cancelling the common term $ $ \phi $ $ Below is what I am getting. But I need to cancel the common terms. Here $ $ a, \sigma $ $ are constants. $ $ \phi$ $ is a variable

Result I got:

res

FullSimplify of expressions with Mathematical Constants

Writing:

{c1, c2, c3, c4, c5} = N[{Tan[E], Sin[E], Tanh[E], E, Sinh[E]}];  a = (c1 + c3) / 2; b = Sqrt[c5^2 - (c2 - c4)^2] / 2; c = 0; d = (c2 + c4) / 2; e = (c1 - c3) (c2 - c4) / (4 b); f = c5 Sqrt[c5^2 - (c1 - c3)^2 - (c2 - c4)^2] / (4 b); x = a + b Cos[t] + c Sin[t]; y = d + e Cos[t] + f Sin[t];  xmin = Minimize[{x, 0 <= t <= 2π}, t][[1]]; xmax = Maximize[{x, 0 <= t <= 2π}, t][[1]]; FullSimplify[(xmin + xmax) / 2 == a]  ymin = Minimize[{y, 0 <= t <= 2π}, t][[1]]; ymax = Maximize[{y, 0 <= t <= 2π}, t][[1]]; FullSimplify[(ymin + ymax) / 2 == d] 

we get:

True

True

which is what is desired. On the other hand, by making a simple change:

{c1, c2, c3, c4, c5} = {Tan[E], Sin[E], Tanh[E], E, Sinh[E]}; 

we get:

True

that is, in the second case there is no answer. So by defining the constants in this other way:

SetAttributes[c1, Constant] NumericQ[c1] = True; N[c1, prec___] := N[Tan[E], prec]  SetAttributes[c2, Constant] NumericQ[c2] = True; N[c2, prec___] := N[Sin[E], prec]  SetAttributes[c3, Constant] NumericQ[c3] = True; N[c3, prec___] := N[Tanh[E], prec]  SetAttributes[c4, Constant] NumericQ[c4] = True; N[c4, prec___] := N[E, prec]  SetAttributes[c5, Constant] NumericQ[c5] = True; N[c5, prec___] := N[Sinh[E], prec] 

we get:

True

Minimize::infeas: There are no values of {t} for which the constraints 0<=t<=2π are satisfied and the objective function […] is real-valued.

Maximize::infeas: There are no values of {t} for which the constraints 0<=t<=2π are satisfied and the objective function […] is real-valued.

Infinity::indet: Indeterminate expression -∞+∞ encountered.

Indeterminate == (c2 + c4) / 2

where, apparently, another problem arises. How to solve it all?

FullSimplify does not work well (require answer in terms of a particular variable)

(new to Mathematica)

I am trying to simplify the Schwarzian derivative of f w.r.t. u.(In[10]). Here f is prime of Weierstrass function (In[8]) which has some constraints (In[9])

In[8]:= f[u_] := D[g[u], u]  In[9]:= f[u]^2 == 4 g[u]^3 - a*g[u] - b  Out[9]= Derivative[1][g][u]^2 == -b - a g[u] + 4 g[u]^3  In[10]:= FullSimplify[f'''[u]/f'[u] - 1.5 f''[u]^2/f'[u]^2] 

gives the output (for some reason Out[10] is of this form on copying)

( \!\(\*SuperscriptBox[\(g\),  TagBox[ RowBox[{"(", "4", ")"}], Derivative], MultilineFunction->None]\)[u] (g^\[Prime]\[Prime])[u] - 1.5  \!\(\*SuperscriptBox[\(g\),  TagBox[ RowBox[{"(", "3", ")"}], Derivative], MultilineFunction->None]\)[u]^2)/(g^\[Prime]\[Prime])[u]^2 

The output is in terms of primes of g[u] and the expression is not simplified. Any help is appriciated. I would prefer if the answer was a function of g[u].

enter image description here

difference between Simplify and FullSimplify

I try to make the following formula as simple as possible.

(\[Theta]*Subscript[\[Alpha], 1]*(2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] +        Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] + \[Theta]^2*Subscript[\[Alpha], 1]^2])))/   ((1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] +         \[Theta]^2*Subscript[\[Alpha], 1]^2]) + \[Theta]*Subscript[\[Alpha], 1]*     (2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] +           \[Theta]^2*Subscript[\[Alpha], 1]^2]))) 

So. I use Simplify and get the result.

(\[Theta]*Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/   (\[Theta]^2*Subscript[\[Alpha], 1]^2 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*      (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]*Subscript[\[Alpha], 1]*     (5 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) 

This is the thing I can calculate by hands as well. However, if I useFullSimplifythen it gives

(4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] +     Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]) 

which is more simple. However, I have a fundamental suspicion about the result from FullSimplify. Thus, I check these two are indentical.

sol3 = (\[Theta]*     Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] +         Sqrt[\[Theta]*         Subscript[\[Alpha],           1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/      (\[Theta]^2*Subscript[\[Alpha], 1]^2 +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*             (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]*      Subscript[\[Alpha], 1]*           (5 +         Sqrt[\[Theta]*          Subscript[\[Alpha],            1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) 

and

sol4 = (4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])    {sol3, sol4} // RootReduce     SameQ @@ (Sort /@ %)  

which gives False. I don’t know how could we get FullSimplify result from the Simplify result.