Algorithms for finding closed form approximations for integrals (with no closed form solutions)

Question

It is well known that many integrals have no closed form solutions, normally what you would do is solve them numerically.

My question is if there are algorithms that give you good closed form approximations instead.

Answer 1

You can write

$$f(x) = a_0x^{b_0} + a_1x^{b_1} + ... + o(x^{b_n})$$

So

$$\int f(x) dx = a_0\frac{x^{b_0+1}}{b_0+1} + a_1\frac{x^{b_1+1}}{b_1+1} + ... + o(x^{b_n+1})$$

(however be carefull with $b_k=-1$)