We present new measurements of the power spectra of the cosmic infrared background (CIB) anisotropies using the $Planck$ 2015 full-mission High frequency instrument data at 353, 545 and 857 GHz over 20 000 deg$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$. We use techniques similar to those applied for the cosmological analysis of $Planck$, subtracting dust emission at the power spectrum level. Our analysis gives stable solutions for the CIB power spectra with increasing sky coverage up to about 50 per cent of the sky. These spectra agree well with H i-cleaned spectra from $Planck$ measured on much smaller areas of sky with low Galactic dust emission. At 545 and 857 GHz, our CIB spectra agree well with those measured from $Herschel$ data. We find that the CIB spectra at ℓ ≳ 500 are well fitted by a power-law model for the clustered CIB, with a shallow index γ = 0.53 ± 0.02. This is consistent with the CIB results at 217 GHz from the cosmological parameter analysis of $Planck$. We show that a linear combination of the 545 and 857 GHz $Planck$ maps is dominated by the CIB fluctuations at multipoles ℓ ≳ 300.300.