Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$.

The derivative of the exponential map can be expressed in Terms of Jacobi Fields. Is there any slick way to express the second derivative $$ \nabla d \exp_x|_X: T_x M \times T_x M \longrightarrow T_{\exp_x(X)}M$$ in terms of integrals over curvature or Jacobi Fields? What about its Taylor expansion in $X \in T_x M$ about $0$?

(Just to clarify what I mean with the above expression: We can pullback the tangent bundle $TM$ and the Levi-Civita connection to $T_x M$. Look at the vector bundle $T^*T_x M \otimes \exp_x^* TM \rightarrow T_x M$. The first factor is naturally isomorphic to $T^*_x M$ and is just flat; for the second one, we use the pullback of the Levi-Civita Connection. Hence we have a vector bundle with connection, and $d \exp_x$ is a section of it. Hence we can form $\nabla$ of it, which will be a section of $T^*_x M \times T^*_x M \otimes \exp_x^* TM$; that is the quantity I am interested in.)