  We consider the Bayesian detection statistic for a targeted search for
continuous gravitational waves, known as the $\mathcal{B}$-statistic. This is a
Bayes factor between signal and noise hypotheses, produced by marginalizing
over the four amplitude parameters of the signal. We show that by
Taylor-expanding to first order in certain averaged combinations of antenna
patterns (elements of the parameter space metric), the marginalization integral
can be performed analytically, producing a closed-form approximation in terms
of confluent hypergeometric functions. We demonstrate using Monte Carlo
simulations that this approximation is as powerful as the full
$\mathcal{B}$-statistic, and outperforms the traditional maximum-likelihood
$\mathcal{F}$-statistic, for several observing scenarios which involve an
average over sidereal times. We also show that the approximation does not
perform well for a near-instantaneous observation, so the approximation is
suited to long-time continuous wave observations rather than transient modelled
signals such as compact binary inspiral.
