Integration problems of algebraic functions, including problems on elliptic integrals, had been the research focus of many mathematicians in the 19th century, such as Euler, Legendre, Abel, Jacobi etc. Abelian integrals are the generalization of elliptic integrals, which can be written as \[ \int R(x,y) dx, \] where $ R(x,y) $ is a rational function of $x$ and $y$ with the polynomial relation $P(x,y)=0$. Abel's addition theorem has played an important role in exploring the theory of elliptic integrals which says that \[ \int_{a}^{u}R(x,y)dx+\int_{a}^{v}R(x,y)dx=\int_{a}^{w} R(x,y)dx, \] where $a, u, v, w$ are on the Riemann surface corresponding to $R(x, y)$ such that $a$ is a fixed point and $w$ is an algebraic function of $u$ and $v$. A simple and classical example is \[ \int_{0}^{u}\dfrac{1}{\sqrt{1-x^2}}dx +\int_{0}^{v}\dfrac{1}{\sqrt{1-x^2}}dx =\int_{0}^{u\sqrt{1-v^2}+v\sqrt{1-u^2}}\dfrac{1}{\sqrt{1-x^2}} dx, \] which leads to the trigonometric formula: $ \sin(u+v)=\sin(u)\cos(v)+\cos(u)\sin(v) $. Usually the procedure of deriving Abel's addition formulae by hand is complicated and highly skillful. In this paper, we show that some algorithms in symbolic computation can be used to find the explicit forms of $w$ in a more efficient way.