Transform $x^{\sin(x)}$ to expression with constant exponent?

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I am working on an exercice which asks to convert expressions into those basic blocs :

  • Standard operations ($+$, $-$, $*$, $/$)
  • $\exp$
  • $\ln$
  • $\sin$
  • $\cos$
  • powers (as long as the exposant is constant)

I have an issue with this expression :
$$\\X^{\sin(X)}$$

By developing sine, I get that :
$$X^{\frac{\exp(iX) - \exp(-iX)}{2i}}$$

I don't see at all how the exponent could be constant here?
Is there some way to write this expression to get a constant exponent?

Thank you.

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$$x^{\sin x} = \exp(\sin x\ln x).$$ Motivation: $$x^{\sin x} = \exp(\ln(x^{\sin x})).$$