This paper considers the stability analysis of nonlinear Lur'e type systems where the nonlinearity is both (locally) sector and slope restricted. Convex conditions for verifying stability, computing outer estimates of reachable sets and upper bounds on the induced L2 gain in a local or global domain are proposed. The conditions use a Lyapunov function that is quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity. Numerical examples outline the benefits of the proposed approach.