How do I simplify the calculation of this trace involving six gamma matrices?

[VIS123]

Trace of six gamma matrices

I need to calculate this expression:
$$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$
I know that I can express this as:
$$ Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5})=-4i(g^{\mu\nu}\epsilon^{\rho\sigma\alpha\beta}-g^{\mu\rho}\epsilon^{\nu\sigma\alpha\beta}+g^{\mu\sigma}\epsilon^{\nu\rho\alpha\beta}-g^{\mu\alpha}\epsilon^{\nu\rho\sigma\beta}+g^{\mu\beta}\epsilon^{\nu\rho\sigma\alpha}+g^{\nu\rho}\epsilon^{\mu\sigma\alpha\beta}-g^{\nu\sigma}\epsilon^{\mu\rho\alpha\beta}+g^{\nu\alpha}\epsilon^{\mu\rho\sigma\beta}-g^{\nu\beta}\epsilon^{\mu\rho\sigma\alpha}+g^{\rho\sigma}\epsilon^{\mu\nu\alpha\beta}-g^{\rho\alpha}\epsilon^{\mu\nu\sigma\beta}+g^{\rho\beta}\epsilon^{\mu\nu\sigma\alpha}+g^{\sigma\alpha}\epsilon^{\mu\nu\rho\beta}-g^{\sigma\beta}\epsilon^{\mu\nu\rho\alpha}+g^{\alpha\beta}\epsilon^{\mu\nu\rho\sigma}) $$
So, some of this terms are the same and some vanish. My question is how to show that:
I know that:
$$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5})=-4i(g^{\mu\nu}\epsilon^{\rho\sigma\alpha\beta}-g^{\mu\rho}\epsilon^{\nu\sigma\alpha\beta}+g^{\rho\nu}\epsilon^{\mu\sigma\alpha\beta}-g^{\alpha\beta}\epsilon^{\sigma\mu\nu\rho}+g^{\sigma\beta}\epsilon^{\alpha\mu\nu\rho}-g^{\sigma\alpha}\epsilon^{\beta\mu\nu\rho}) $$
So only six terms survive, but how?

 

[fzero]

I'm not sure where the expression in your intermediate step comes from so I would rather try to use it. There is an identity
$$ \gamma^\mu \gamma^\nu \gamma^\rho = \eta^{\mu\nu} \gamma^\rho + \eta^{\nu\rho} \gamma^\mu - \eta^{\mu\rho} \gamma^\nu - i \epsilon^{\sigma\mu\nu\rho} \gamma_\sigma\gamma^5,$$
that is proved in many places (including https://en.wikipedia.org/wiki/Gamma_matrices#Miscellaneous_identities). I would suggest using this for the two groups of 3 matrices. Before doing a lot of algebra, you will find that only the cross terms of the form $\text{tr}[\gamma^\mu\gamma^\nu(\gamma^5)^2]$ are non trivial. This should yield the 6 terms that you've written above without a lot of fuss.