\begin{array}{rcl}
\displaystyle\frac{(3+x+\sin^2x)^{x+\sin^2x}-1}x & = & \displaystyle\frac{e^{(x+\sin^2x)\ln(3+x+\sin^2 x)}-1}x
\vspace{0.5em}\\
& = & \displaystyle
\frac{e^{(x+o(x))(\ln 3+o(1))}-1}{x}
\vspace{0.5em}\\
& = & \displaystyle
\frac{e^{x\ln 3+o(x)}-1}{x}
\vspace{0.5em}\\
& = & \displaystyle
\frac{1+x\ln 3+o(x)+o(x\ln 3+o(x))-1}x
\vspace{0.5em}\\
& = & \displaystyle
\frac{x\ln 3+o(x)}{x}
\vspace{0.5em}\\
& = & \displaystyle
\ln 3+o(1)
\end{array}