As I understand it, you want a sentence that has this meaning:
Unfortunately, the exact velocity cannot be calculated analytically without the use of computers
Although but can mean except, the but by collocation gives a strong suggestion that the following clause is a complete alternative to the previous clause, so your sentence is most likely to be interpreted as follows:
You cannot calculate the velocity analytically, but you can calculate it using computers.
Here are a couple of examples that shows but by used in this way:
Most sword fights are not won by such skill, but by brute strength - Excalibur: a novel. Bernard Cornwell, 1999
The mayor is chosen annually, not by the citizens as in other corporations, but by the majority of the other members of this court the Eighteenth century constitution
I therefore suggest that you use a sentence like the one at the start of this answer, or alternatively use one like this:
Unfortunately, the exact velocity cannot be calculated analytically except by computer.