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We present a new closed-form formula for the ergodic capacity of multiple-input multiple-output (MIMO) wireless channels. Assuming independent and identically distributed (i.i.d.) Rayleigh flat-fading between antenna pairs and equal power allocation to each of the transmit antennas, the channel capacity is expressed in closed form as finite sums of the exponential integrals which are the special cases of the complementary incomplete gamma function. Using the well-known asymptotic behavior of the MIMO capacity, we also give a simple approximate expression for the channel capacity. Numerical results show that the approximation is quite accurate for the entire range of average signal-to-noise ratios.