\int is a "variable-sized symbol", so that it comes in two sizes: one for in-line formulas and one for display-formulas. The same is true for \sum; it just so happens that the text-style integral is almost the same height as the display-style sum symbol (in Lamport's LaTeX User's Guide and Reference Manual, both are about 4 mm tall, the display-style sum style a bit more, while the text-style sum symbol is only about 3 mm tall; the display-style integral, by contrast, is a bit more than 7 mm tall). So all you want is the integral sign to be displayed in text-mode, but only the integral sign (so the sum symbol is not changed in size).
One way to accomplish this is to define a macro \smallint as \def\smallint{\begingroup\textstyle \int\endgroup}. Upon testing it, it shows just a shade smaller than a displaystyle summation (if you put limits on the integral but not on the summation, they look about the same); if you have limits on the sum and the integral, then both done in \textstyle looks better, but then you need to use the \limits command after \sum so that the limits occur above and below the sigma, instead of right after. Thus, if you have limits on the summation, I would suggest
\begin{equation}
\sum_{i}^j A_{ij}\begingroup\int\endgroup_{-\infty}^{+\infty}\cos x\,dx
\end{equation}
or
\begin{equation}
\mathop{\textstyle\sum}\limits_i^j
A_{ij}\begingroup\textstyle\int\endgroup_{-\infty}^{+\infty}\cos x\,dx
\end{equation}
(or using a macro for the small integral sign). (Note by the way the thin space before the dx in the integral).
