Maclaurin series expansion

I want to calculate the Maclaurin series for $\displaystyle log(1+x^2)$ by using a previous result, that $\displaystyle log(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + ...$

The following is not correct, but I don't know why, nor how to do it the right way

let $\displaystyle u=x^2$. Then

$\displaystyle log(1+x^2) = log(1+u) = u - \frac{1}{2}u^2 + \frac{1}{3}u^3 - \frac{1}{4}u^4 + ... = x^2 - \frac{1}{2}x^4 + \frac{1}{3}x^6 - \frac{1}{4}x^8 + ...$

Thanks for any help.