A space $X$ that contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and whose dual is not weakly sequentially complete I want to find an example of a Banach space $X$ which contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and so that $X^*$ is not weakly sequentially complete.
Let $Y$ be a quasi-reflexive space which is not reflexive (for example, the James space). Then neither $Y$ nor $Y^*$ is weakly sequentially complete. Since $Y$ is quasi-reflexive, it does not contain copies of $\ell_1$. Now, set $X=C[0,1]\oplus Y$. Then $X^*$ is not weakly sequentially complete because $Y^*$ is not. Further, $X$ does not contain complemented copies of $\ell_1$.

Such artificial construction was necessary here because $C[0,1]^*$ is weakly sequentially complete.
A Banach space with an  unconditional basis containing a copy of $\ell_1$ contains a complemented copy of $ell_1.$  See JMAA342(2008) pp.83-88