On series identities of Gosper and integrals of Ramanujan theta function $\psi (q)$

We prove some Lambert series which were stated by Gosper without proof or reference. As an application, we shall evaluate integrals involving Ramanujan theta function $\psi (q)$. Furthermore, motivated by Ramanujan’s identities for $q{\psi }^4(q^2)$ and $\text{fr}{\psi }^3(q)\psi (q^3)$, we shall evaluate the squares of $q{\psi }^4(q^2)$ and $\text{fr}{\psi }^3(q)\psi (q^3)$ in terms of Lambert series.