Let $\alpha=\{I_i:i\in I\}$ be a finite or countable partition of $[0,1]$ into intervals modulo endpoints, and let $T:[0,1]\to[0,1]$ be a nonsingular interval map such that, for each $i$, $T|_{I_i^\circ}$ is a $C^{1+\varepsilon}$ diffeomorphism from $I_i^\circ$ onto $(0,1)$ modulo endpoints. Let $\mu$ be an ergodic $T$-invariant probability measure equivalent to $\mathcal L^1$. Assume the following hypotheses hold.

1. The partition $\alpha$ is generating and has finite entropy $H_\mu(\alpha)<\infty$. 2. The function $\log |T'|$ belongs to $L^1(\mu)$. 3. For every $n$-cylinder $C\in\bigvee_{k=0}^{n-1}T^{-k}\alpha$, the map $T^n$ sends $C$ bijectively, modulo endpoints, onto $(0,1)$. 4. There is a constant $C_0\ge 1$ such that for every $n$-cylinder $C$ and all $x,y\in C$ whose orbit segments $x,T(x),\dots,T^{n-1}(x)$ and $y,T(y),\dots,T^{n-1}(y)$ lie in branch interiors,

\begin{align*} C_0^{-1}\le \frac{|(T^n)'(x)|}{|(T^n)'(y)|}\le C_0 \end{align*}

and

\begin{align*} C_0^{-1}\le \mu(C)\, |(T^n)'(x)|\le C_0. \end{align*}

Then

\begin{align*} h_\mu(T)=\int_0^1 \log |T'(x)|\,d\mu(x). \end{align*}