Assume that $G$ is a compact Lie group, and $M$ is a smooth $G$manifold. We consider the equivariant cohomology of $M$, i.e. $H^{*}_{G}(M)$. The famous theorem of Cartan said that if $M$ is compact then the twisted de Rham complex $\Omega^{*}_{G}(M)$ computes the equivariant cohomology of $M$, i.e. $$H^{*}_{G}(M)\simeq H^{*}(\Omega^{*}_{G}(M),d_{eq}).$$ So if $M$ is not compact is the above result aslo true ?

Yes, the result is true without the compactness assumption. For a proof that describes explicitly this isomorphism, and does not rely on spectral sequences, see this old note. 

