  Recently, there is a considerable study on gapped symmetric phases of bosons
that do not break any symmetry. Even without symmetry breaking, the bosons can
still be in many exotic new states of matter, such as symmetry-protected
trivial (SPT) phases which are short-range entangled and symmetry-enriched
topological (SET) phases which are long-range entangled. It is well-known that
non-interacting fermionic topological insulators are SPT states protected by
time-reversal symmetry and U(1) fermion number conservation symmetry. In this
paper, we construct three-dimensional exotic phases of bosons with
time-reversal symmetry and boson number conservation U(1) symmetry by means of
fermionic projective construction. We first construct an algebraic bosonic
insulator which is a symmetric bosonic state with an emergent U(1) gapless
gauge field. We then obtain many gapped bosonic states that do not break the
time-reversal symmetry and boson number conservation via proper dyon
condensations. We identify the constructed states by calculating the allowed
electric and magnetic charges of their excitations, as well as the statistics
and the symmetric transformation properties of those excitations. This allows
us to show that our constructed states can be trivial SPT states (i.e. trivial
Mott insulators of bosons with symmetry), non-trivial SPT states (i.e. bosonic
topological insulators) and SET states (i.e. fractional bosonic topological
insulators). In non-trivial SPT states, the elementary monopole (carrying zero
electric charge but unit magnetic charge) and elementary dyon (carrying both
unit electric charge and unit magnetic charge) are fermionic and bosonic,
respectively. In SET states, intrinsic excitations may carry fractional charge.
