Discrete Logarithm problem with inverse Let $\mathbb G$ be a cyclic group of order $q$. The Discrete Logarithm Problem (DLP) is, given $g, g^x \in \mathbb G$, to compute $x \in \mathbb Z_q $.

I'm interested to know if there is a known variant of DLP as follows:


  Given $g, g^x, g^{x^{-1}\bmod q} \in \mathbb G$, compute $x \in \mathbb Z_q$


It is clear that this variant would not be harder than the DLP, since a DLP solver could be used to solve this problem simply by ignoring the third input.
This problem seem to be equivalent to a particular instance of the $k$-Strong Discrete Logarithm Problem.

Let $\mathbb G$ be a cyclic group of order $q$. The $k$-Strong Discrete Logarithm Problem ($k$-DLP) is, given $h, h^x, h^{x^2}, .., h^{x^k} \in \mathbb G$, to compute $x \in \mathbb Z_q $. Taking a simple change of variable ($h^x = g$), one obtains an equivalent version of the problem, which is as follows:


  Given $g^{x^{-1}\bmod q}, g, g^x, ..., g^{x^{k-1}} \in \mathbb G$, compute $x \in \mathbb Z_q $


It can be seen then that the problem in the question is the case of $k=2$.

The $k$-Strong Discrete Logarithm Problem is discussed in Correlated-Input Secure Hash Functions by Goyal, O'Neill, and Rao, for instance.