The following integrals involve `x^3+a^3`
NOTE: For formulas involving `x^3-a^3`, replace `a` by `-a`.
1) `int 1/(x^3+a^3) dx = 1/(6a^2)ln ((x+a)^2/(x^2-ax+a^2))+1/(a^2sqrt3) tan^-1((2x-a)/(asqrt3))`
2) `int x/(x^3+a^3) dx = 1/(6a)ln ((x^2-ax+a^2)/(x+a)^2)+1/(asqrt3) tan^-1((2x-a)/(asqrt3))`
3) `int x^2/(x^3+a^3) dx = 1/3 ln (x^3+a^3)`
4) `int 1/(x(x^3+a^3)) dx = 1/(3a^3) ln ((x^3)/(x^3+a^3))`
5) `int 1/(x^2(x^3+a^3)) dx = -1/(a^3x)-1/(6a^4) ln((x^2-ax+a^2)/(x+a)^2)-1/(a^4sqrt3) tan^-1((2x-a)/(asqrt3))`
6) `int 1/(x^3+a^3)^2 dx = x/(3a^2(x^3+a^3))+1/(9a^5) ln ((x+a)^2/(x^2-ax+a^2))+2/(3a^5sqrt3) tan^-1((2x-a)/(asqrt3))`
7) `int x/(x^3+a^3)^2 dx = x^2/(3a^3(x^3+a^3))+1/(18a^4) ln ((x^2-ax+a^2)/(x+a)^2)+1/(3a^4sqrt3) tan^-1((2x-a)/(asqrt3))`
8) `int x^2/(x^3+a^3)^2 dx = -1/(3(x^3+a^3)`
9) `int 1/(x(x^3+a^3)^2) dx = 1/(3a^3(x^3+a^3))+1/(3a^6) ln (x^3/(x^3+a^3))`
10) `int 1/(x^2(x^3+a^3)^2) dx = -1/(a^6x)-x^2/(3a^6(x^3+a^3))-4/(3a^6) int x/(x^3+a^3) dx` **See integral number 2 above.
11) `int x^m/(x^3+a^3) dx = x^(m-2)/(m-2)-a^3 int x^(m-3)/(x^3+a^3) dx`
12) `int 1/(x^n(x^3+a^3)) dx = (-1)/(a^3(n-1)x^(n-1))-1/a^3 int 1/(x^(n-3)(x^3+a^3)) dx`