Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.

Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^*_{\rm sing}(X_{\mathbb C}, \mathbb{Z}_p)$ using a variant of the de Rham complex. However, the obvious integral version of de Rham cohomology, gives the completely wrong answer even for $X = \mathbb A^1$.

This is surprising to me, because (if I am reading the literature correctly, as a non-expert) the de Rham complex of $X_{\mathbb Z_p}$ computes the crystalline cohomology of $X_{\mathbb F_p}$. And crystalline cohomology is often claimed to be the "correct" $p$-adic replacement for the etale cohomology groups $H^*_{et}(X_{\overline{\mathbb F_p}}, \mathbb Z_l)$. However, it seems that crystalline cohomology/de Rham cohomology groups of affine space are "wrong", in the moral sense of not lining up with what I would naively expect.

Is there a correction/explanation for this discrepancy? Is there a variant of de Rham/crystalline cohomology which computes the "right" cohomology for $\mathbb A^1$? Morally, is there a reason why we would want integral crystalline cohomology to carry so much extraneous torsion? Is there something I'm missing?