First observations of the rare decays B⁺→ K⁺π⁺π^-μ⁺μ^-and B⁺→ φK⁺μ⁺μ^-

First observations of the rare decays B ⁺→ K ⁺π ⁺π ^-μ ⁺μ ^-and B ⁺→ φK ⁺μ ⁺μ ^-are presented using data corresponding to an integrated luminosity of 3.0 fb ^-1, collected by the LHCb experiment at centre-of-mass energies of 7 and 8TeV. The branching fractions of the decays are (Formula Presented.) where the uncertainties are statistical, systematic, and due to the uncertainty on the branching fractions of the normalisation modes. A measurement of the differential branching fraction in bins of the invariant mass squared of the dimuon system is also presented for the decay B ⁺→ K ⁺π ⁺π ^-μ ⁺μ ^-.