My initial thoughts are no, since all you have to do is code all known integral techniques, and have the computer brute force the result.
But would there be any integrals, or any integral techniques which would take less time to do by hand than by a program?
Apologies if this sounds like a silly question but I'm just now getting back into maths.
Yes, there are. An example is integrating Dirichlet's function over a closed interval. The value of this function depends on whether the input to the function is rational or irrational. This type of function can be evaluated exactly. But, for a computer, everything is a rational number due to finite precision.