I think, until you can justify speeding it up (by offering the source code) in the context of solving all partial differential equations, differential equations, equations, over fields and modulus - you should trust that there is a reason it does that; that it is doing something somewhat complicated (needed in general solving) your math book does not show the need of. The knowlege of some of the math people at Wolfram is quite humbling I think.

You should limit your variables to be [Element] Reals if you don't want complex surprises and you will faster processing, often.

Nice, really nice! Tanks for sharing!

Greetings - I just checked to see if telling M that t is real helps - I am afraid that it does not:

In[1]:= Element[t, Reals]

Out[1]= t \[Element] Reals

In[2]:= LaplaceTransform[UnitStep[t - 1]*Cos[t], t, 
s] // AbsoluteTiming

Out[2]= {35.4615, (E^-s (s Cos[1] - Sin[1]))/(1 + s^2)}