Consider a unital \(*\) -algebra denoted as \(\mathfrak {A}\) over the complex field \(\mathbb {C}\) . For any \(\omega _1, \omega _2 \in \mathfrak {A}\) , we define product of \(\omega _1\circ \omega _2=\omega _1\omega _2+\omega _2\omega _1\) , known as the Jordan product, and \(\omega _1\bullet \omega _2=\omega _1\omega _2^*+\omega _2\omega _1^*\) , referred to as the bi-skew Jordan product. This article demonstrates that if a mapping \(\mho \) : \(\mathfrak {A} \rightarrow \mathfrak {A}\) (not necessarily linear) satisfies \(\mho (({\omega _1} \bullet {\omega _2}) \circ \omega _3)=(\mho (\omega _1) \bullet \omega _2) \circ \omega _3+(\omega _1 \bullet \mho (\omega _2)) \circ \omega _3+(\omega _1 \bullet \omega _2) \circ \mho (\omega _3)\) for all \(\omega _1, \omega _2, \omega _3 \in \mathfrak {A}\) , then \(\mho \) is additive. Furthermore, if \(\mho (I)\) is self-adjoint, then \(\mho \) is a \(*\) -derivation. As applications, we apply our main result to some special classes of unital \(*\) -algebras, such as prime \(*\) -algebras and factor von Neumann algebras.