  Let $\mathscr{M}$ be a von Neumann algebra and $a$ be a self-adjoint operator
affiliated with $\mathscr{M}$. We define the notion of an "integral
symmetrically normed ideal" of $\mathscr{M}$ and introduce a space
$OC^{[k]}(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of functions $\mathbb{R} \to
\mathbb{C}$ such that the following result holds: for any integral
symmetrically normed ideal $\mathscr{I}$ of $\mathscr{M}$ and any $f \in
OC^{[k]}(\mathbb{R})$, the operator function $\mathscr{I}_{\mathrm{sa}} \ni b
\mapsto f(a+b)-f(a) \in \mathscr{I}$ is $k$-times continuously Fr\'{e}chet
differentiable, and the formula for its derivatives may be written in terms of
multiple operator integrals. Moreover, we prove that if $f \in
\dot{B}_1^{1,\infty}(\mathbb{R}) \cap \dot{B}_1^{k,\infty}(\mathbb{R})$ and
$f'$ is bounded, then $f \in OC^{[k]}(\mathbb{R})$. Finally, we prove that all
of the following ideals are integral symmetrically normed: $\mathscr{M}$
itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal
of compact operators, and -- when $\mathscr{M}$ is semifinite -- ideals induced
by fully symmetric spaces of measurable operators.
