there is a closed form solution for $I_n$ for integer $n$, for example, for $n=0$:

$$I_0=(\lambda (c_3+\lambda))^{-1}$$
$$\qquad\times\left[e^{i c_1 \tau} \left((c_3+\lambda) \, _1F_2\left(\frac{\lambda}{2 c_3};\frac{1}{2},\frac{\lambda}{2 c_3}+1;-\frac{1}{4} c_2^2 \tau^2\right)-i c_2 \lambda \tau \, _1F_2\left(\frac{\lambda}{2 c_3}+\frac{1}{2};\frac{3}{2},\frac{\lambda}{2 c_3}+\frac{3}{2};-\frac{1}{4} c_2^2 \tau^2\right)\right)\right].$$