Question: Can we find the maximum value of this expression in a symbolic form?

I have an expression and I want to find its maximum value.

expr:=sin(sqrt(3)*t)*cos(sqrt(3)*t)*(sqrt(3)*cos(sqrt(3)*t) - sin(sqrt(3)*t))/3

It is easy to find its maximum value in a numerical form.

Optimization:-Maximize(sin(sqrt(3)*t)*cos(sqrt(3)*t)*(sqrt(3)*cos(sqrt(3)*t) - sin(sqrt(3)*t))/3)

[0.324263244248023330, [t = 1.39084587050767]]

The images of the expression is as follows.

But does it exist an acceptable maximum value in symbolic form?  As the function maximize seems to take a lot of time, I don't see any hope so far. Perhaps the expression is indeed complex.

maximize(expr)# it runs long time.

We try to find the derivative of expr and get some points where thier derivatives are 0.

s:=[solve(diff(expr,t),t)]
evalf~(s) # Some solutions seem to have been left out.

[0.7607468963 + 0.*I, -0.4229534936 - 0.*I, 0.2668063857 + 0.*I]

ex:=convert(expr,exp):
s:=[solve(diff(ex,t)=0,t)]:
s1:=evalf~(s);# choose the 6th item: 1.390845877 + (4.655829150*10^(-9))*I
fexpr := unapply(ex, t);
evalf(fexpr(s1[6]));
fexpr (s[6]); # a very long expression that is not quite acceptable.

-I/6*(-sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44))/(6*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)) + 6*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)/sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44)))*(-sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44))/(12*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)) - 3*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)/sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44)))*(sqrt(3)*(-sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44))/(12*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)) - 3*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)/sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44))) + (-sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44))/(6*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)) + 6*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)/sqrt(-(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3)*((1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3)*sqrt(3)*I - 2*I*sqrt(3)*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) + (1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(2/3) + 36*I*sqrt(3) + 2*(1404*I*sqrt(3) - 1396 + 36*sqrt(-3018 - 3018*I*sqrt(3)))^(1/3) - 44)))*I/2)

An interesting problem is that an acceptably concise expression (although it is very subjective) for the maximum value may not exist mathematically. But knowing this in advance is difficult.

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