itegration of 1/(x^3+1)dx

[jazzman]

I'm having trouble finding a good method of solving \(\displaystyle \int{\frac{1}{x^3+1}}\,dx\).
I tried integration by parts and substitution but haven't succeded in simplifying the expresion...
Please help.

 

[galactus]

Notice the sum of two cubes?.

\(\displaystyle \frac{1}{x^{3}+1}=\frac{1}{(x+1)(x^{2}-x+1)}\)

Now, try partial fractions.

\(\displaystyle \frac{Ax}{x^{2}-x+1}+\frac{B}{x^{2}-x+1}+\frac{C}{x+1}\)

 

[jazzman]

Notice the sum of two cubes?.

\(\displaystyle \frac{1}{x^{3}+1}=\frac{1}{(x+1)(x^{2}-x+1)}\)

Now, try partial fractions.

\(\displaystyle \frac{Ax}{x^{2}-x+1}+\frac{B}{x^{2}-x+1}+\frac{C}{x+1}\)

Thanks a lot!
Didn't think of treating it as an integration of a rational function (even though it is such... )