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 B(B⁺→K⁺π⁺π⁻µ⁺µ⁻) = (4.36 ^+0.29_−0.27 (stat) ± 0.21 (syst) ± 0.18 (norm)) × 10⁻⁷, B(B⁺→φK⁺µ⁺µ⁻) = 0.82^+0.19_−0.17 (stat) ^+0.10_−0.04 (syst) ± 0.27 (norm)) × 10⁻⁷,  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^-.