Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: part II

The non-elementary integrals ${Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha>\beta+1$ and ${Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,$ $\beta\ge1,$ $\alpha>2\beta+1$, where $\{\beta,\alpha\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral ${Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx,$ $\beta\ge1,$ $\alpha>\beta+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.