1. Simplify $\frac{\left(x^{3}+x\right) \sinh \left(4 \log _{e}(x)\right)}{\left(x^{4}+1\right) \sinh \left(\log _{e}(x)\right.}$ for $x>0$.

Step 1 ：Rewrite the expression using natural logarithm notation (ln) instead of log base e: \(\frac{\left(x^{3}+x\right) \sinh \left(4 \ln(x)\right)}{\left(x^{4}+1\right) \sinh \left(\ln(x)\right)}\) for \(x>0\)

Step 2 ：Use the property of logarithms: \(n\ln(x) = \ln(x^n)\)

Step 3 ：Use the property of hyperbolic functions: \(\sinh(a+b) = \sinh(a)\cosh(b) + \cosh(a)\sinh(b)\)

Step 4 ：\(\frac{x\left(x^{2}+1\right) \sinh \left(4 \ln(x)\right)}{\left(x^{4}+1\right) \sinh \left(\ln(x)\right)}\)

Step 5 ：\(\boxed{\frac{x\left(x^{2}+1\right) \sinh \left(4 \ln(x)\right)}{\left(x^{4}+1\right) \sinh \left(\ln(x)\right)}}\)