  Quantum circuits for mathematical functions such as division are necessary to
use quantum computers for scientific computing. Quantum circuits based on
Clifford+T gates can easily be made fault-tolerant but the T gate is very
costly to implement. The small number of qubits available in existing quantum
computers adds another constraint on quantum circuits. As a result, reducing
T-count and qubit cost have become important optimization goals. The design of
quantum circuits for integer division has caught the attention of researchers
and designs have been proposed in the literature. However, these designs suffer
from excessive T gate and qubit costs. Many of these designs also produce
significant garbage output resulting in additional qubit and T gate costs to
eliminate these outputs. In this work, we propose two quantum integer division
circuits. The first proposed quantum integer division circuit is based on the
restoring division algorithm and the second proposed design implements the
non-restoring division algorithm. Both proposed designs are optimized in terms
of T-count, T-depth and qubits. Both proposed quantum circuit designs are based
on (i) a quantum subtractor, (ii) a quantum adder-subtractor circuit, and (iii)
a novel quantum conditional addition circuit. Our proposed restoring division
circuit achieves average T-count savings from $79.03 \%$ to $91.69 \%$ compared
to the existing works. Our proposed non-restoring division circuit achieves
average T-count savings from $49.75 \%$ to $90.37 \%$ compared to the existing
works. Further, both our proposed designs have linear T-depth.
