"intricately complicated" is correct and can be used in written English.
It is often used to describe something that is difficult to understand or has a complex structure. For example, "The tax laws in this country are intricate and complicated, making it difficult for citizens to keep up with the latest changes."
For all the arialike speeches in the play, Joe really went to work on understanding its intricately complicated relationships".
Much of the recording equipment used to create the movie's intricately complicated sound effects was, at the time, extremely cutting edge.
Diatoms possess intricately complicated nanopatterned silica outer shells, the so called frustules.
We should change the way we think about these scenarios and recognise the intricately complicated human processes entangled with a changing climate rather than being drawn into a racialised generalisations of climate induced mass migration.
No Knives in the Kitchens of this City is intricately plotted, chronologically complicated and a pleasure to read.
In this telling Julie's family took the land from John's generations ago, and though there is love between them and a common desire for freedom from their current circumstances, their relationship is intricately, and hopelessly, complicated by the shared history of oppression.
With a bit of study you see that what seems bewilderingly complicated is intricately logical.
The most compelling lesson of scientific research is that the body is far more complicated and intricately balanced than we could possibly imagine.
Consumers interact with a library of 3D graphics on an iPad to get the machine to form a number of chocolate designs – including complicated hexagons and intricately laced patterns.
On a simpler note, Emanuel Levy has said: "Whether viewers perceive The Prestige as intricately complex or just unnecessarily complicated would depend to a large degree on their willingness to suspend disbelief for two hours".
