How can I prove the following?

$$1-x+x^2+x^5-x^7-x^{12}+x^{15}-x^{22}-x^{26}+x^{35}-x^{40}+\dots \\= \prod_{i=1}^{\infty} [(1 - x^{8 i - 7}) (1 + x^{8 i - 6}) (1 + x^{8 i - 5}) (1 + x^{8 i - 4}) (1 + x^{8 i - 3}) (1 + x^{8 i - 2}) (1 - x^{8 i - 1}) (1 - x^{8 i})]$$

It doesn't seem to follow from the Triple Product formula and I haven't been able to come up with a combinatorial proof.