  We study the cohomology of Lie superalgebras for the full complex of forms:
superforms, pseudoforms and integral forms. We use the technique of spectral
sequences to abstractly compute the Chevalley-Eilenberg cohomology. We first
focus on the superalgebra $\mathfrak{osp}(2|2)$ and show that there exist
non-empty cohomology spaces among pseudoforms related to sub-superalgebras. We
then extend some classical theorems by Koszul, as to include pseudoforms and
integral forms. Further, we conjecture that the algebraic Poincar\'e duality
extends to Lie superalgebras, as long as all the complexes of forms are taken
into account and we prove that this holds true for $\mathfrak{osp}(2|2)$. We
finally construct the cohomology representatives explicitly by using a
distributional realisation of pseudoforms and integral forms. On one hand,
these results show that the cohomology of Lie superalgebras is actually larger
than expected, whereas one restricts to superforms only; on the other hand, we
show the emergence of completely new cohomology classes represented by
pseudoforms. These classes realise as integral form classes of
sub-superstructures.
