"busily engaged" is a correct and usable phrase in written English.
It is typically used to describe someone who is actively and enthusiastically involved in a task or activity. Example: The students were busily engaged in preparing for their upcoming science fair, eagerly testing their experiments and creating their display boards.
The Catholic Church is still busily engaged in covering up for pedophiles.
We also saw a picture of a welterweight boxer named Steve Belloise, busily engaged in needlework on a quilt.
This all happened in the mid-00s, when Rudolph was busily engaged with life in lots of other ways.
We have for many years been busily engaged in a grand project of Americanizing the world's understanding of mental health and illness.
China is busily engaged in a massive military build-up, is developing weapons of mass destruction and has threatened directly a neighbour, Taiwan, with invasion.
For the rest of his life El Greco continued to live in Toledo, busily engaged on commissions for the churches and monasteries there and in the province.
Since such sidelong and intimate music threatened to be lost in the wide open spaces of the Festival Hall, it was a tribute to Bley's hypnotic compositional touch and the subtleties and the sharp ears of Swallow and Sheppard that the set was so busily engaging.
On the wilder fringes of the genetic engineering community, there are true believers prattling of the gene-rich and the gene-poor - Huxley's alphas and epsilons - and busily engaging in schemes for genetic enhancement and - to go one better than Brave New World - for immortality.
Post-downturn it's hard to tell, but on a sunny day you can feel the heartbeat of a place where the lion's share of the populace seems engaged, busily trading, cleaning, repairing, building.
While my colleagues – Paul Bright, Kate Saunders, Jeanne Willis – and I have been engaged on our Milne "impoohsonations", illustrator Mark Burgess has been busily capturing the illustrative style of the original decorations by EH Shepard.
