mathematica and maple have *extensive* on-line help systems with numerous
examples.  use them.  for reference, i've listed some commands you'll need or
might want to use.

also note that the percent symbol "%" stands in for the previous answer output
by mathematica or maple.  mathematica allows you to access the whole history of
outputs using the percent symbol followed by the number of the output desired
(e.g. "%17" gives the seventeenth output).  in maple, either use assignments to
keep track of objects you want to save, or use the "history" command.


for Mathematica:

=  -  assignment
==  -  equality

DSolve

Simplify
Collect
Factor
Expand

D
Integrate

Normal - cut off HOTs in series
Apart - perform partial fractions

Timing



for Maple:

:=  -  assignment
=  -  equality

dsolve

simplify
collect
factor
expand

D  *and*  diff  -  see the help files for when to use which
integrate

convert( expr, polynom )  -  cut off HOTs in series
convert( expr, parfrac )  -  perform partial fractions

#to time how long it takes to evaluate some set of expressions, use:
startTime := time();
#expressions to be timed
time() - startTime;
#there is also the command "timing" but it does not account for simplying the
#output