# What was the largest ratio (result size)/(integrand size) you have seen?

Sometimes a definite or indefinite integral of a simple-looking one-liner integrand can give astonishingly huge result. What was the largest ratio of the size of shortest known closed-form result to the size of the corresponding integrand you have seen? I am particularly interested in unexpectedly large results, not those, for example, that occur from an intentionally large exponent in the integrand or otherwise obviously tailored for that purpose.

• What about $\int\cos^{2013}x\,dx$? – André Nicolas Sep 29 '13 at 18:35
• @AndréNicolas Good comment! I edited the question to try to exclude cases like this (although in somewhat vague way). – Vladimir Reshetnikov Sep 29 '13 at 18:48
• If you insist on writing $\int dx/(x^4+x+1)$ in terms of radicals as you did in math.stackexchange.com/questions/516263 (rather than leaving it in terms of the roots of the denominator) then the indefinite integral will look horrendous. Even worse if the denominator is a polynomial of higher degree with a solvable but complicated Galois group such as the solvable $168$-element subgroup of $A_8$ (obtained from the $ax+b$ group over ${\bf F}_8$ by adjoining the field automorphisms of ${\bf F}_8$). [This is not an answer because I haven't actually seen such a formula...] – Noam D. Elkies Oct 12 '13 at 1:13
• Not an integral, but I remember vividly my astonishment seeing a computer printout of the solution to solve(x^4 + ax^3 + bx^2 + c*x + d = 0, x) some 25 years ago, when Mathematica was released unto the world. It made me realize there are better ways to solve equations than learning formulas by heart. – Per Manne Oct 18 '13 at 12:27

## 2 Answers

$\int\ \ln(x+a) \cdot ln(x+b) \cdot ln(x+c)\ dx\ -$ It spews forth a formula almost the size of my entire screen, even with Full Simplify $^{and}/_{or}$ Function Expand activated . . .

$\int\ \sqrt{(x-a)(x-b)(x-c)(x-d)}\ dx\ -$ The same, only that this time its size is about seven full screens of resolution $1366\times768$, even with Simplify on.

The following example I learned from James Davenport (Cambridge UK): 