Integrating by partial fraction.

[azatkgz]

Homework Statement

$$\int \frac{dx}{(x+1)(x^2+1)(x^3+1)}$$

The Attempt at a Solution

I tried to solve it in 3 ways.
1)$$\frac{A}{(x+1)}+\frac{B}{(x^2+1)}+\frac{C}{(x^3+1)}$$

2)$$\frac{A}{(x+1)}+\frac{B+Dx}{(x^2+1)}+\frac{C+Ex}{(x^3+1)}$$

3)$$\frac{A}{(x+1)}+\frac{B+Dx}{(x^2+1)}+\frac{C+Ex+Fx^2}{(x^3+1)}$$
But it gives me nonsense.

 

[Dick]

(x+1) divides (x^3+1). You can write the partial fraction with just linear and quadratic denominators.

 

[azatkgz]

Then should I solve in this way?

$$\frac{A}{(x+1)}+\frac{B+Dx}{(x^2+1)}+\frac{C}{(x+1)^2}+\frac{F+Ex}{(x^2-x+1)}$$

 

[Dick]

That looks ok to me.

 

[rocomath]

damn this prob is going to be long