1. **Problem Statement:** Evaluate the integral $$\int x^{\mu^{x^r}} \, dx$$ where $\mu$ and $r$ are constants. 2. **Understanding the problem:** This integral involves a variable exponent with a nested exponential function in the power. The integrand is $x$ raised to the power $\mu^{x^r}$. 3. **Formula and approach:** There is no elementary antiderivative for this function in terms of elementary functions because the exponent itself is an exponential function of $x^r$. Such integrals typically require special functions or numerical methods. 4. **Attempting substitution:** Let’s try substitution to simplify the exponent: Let $$u = x^r$$ Then $$du = r x^{r-1} dx \implies dx = \frac{du}{r x^{r-1}}$$ But since $x = u^{1/r}$, substituting back leads to a complicated expression involving $u$ and $x$. 5. **Conclusion:** The integral $$\int x^{\mu^{x^r}} \, dx$$ does not have a closed-form solution in elementary functions. It can be expressed in terms of special functions or evaluated numerically for specific values of $\mu$ and $r$. **Final answer:** The integral $$\int x^{\mu^{x^r}} \, dx$$ cannot be expressed in elementary closed form.