In high energy physics people compute scattering amplitudes, the goal is to estimate how many events of a particular kind a detector in a collider experiment will see at various energies. The predominant tool for doing that is to calculate contributions from so called Feynman diagrams, each diagram is a graphical representation of some fairly complicated integral, the integrant is some rational function of the particle momenta. The nominator has to be simplified by manipulating spinor identities and in the case of for example QCD one has to work out color factors. Just a few years before people were computing 1000s of those diagrams by hand for Quantum electro dynamics, often while cross checking each others results. For non-abelian Yang-Mills Theory (t'Hooft Veltman had published their landmark paper in 1972) this is more or less infeasible beyond first order.
The number of diagrams that have to be evaluated quickly explodes beyond the first order and in the case of QCD already at two loops would require herculean efforts to do by hand. In fact Schoonschip was one of the first computer algebra systems and developed for just this reason.
Beyond that areas like Supersymmetry and certain parts of General Relativity would be very painful to work in, if everything had to be done by hand.
