I am doing a project for my AP Calculus class where I have to do 5 questions, ones in the class in which they are the hardest I've ever done.
The first question is a complex integral in which I have the answer to, but must show a proof of solving to get the credit for.

The integral is Int(1/(1+x^4)) with respect to x. Using the partial fractions method I have narrowed down the integral to:

[[ (sqrt[2]/4)Int(x/(x^2+sqrt[2]x+1))
+ (1/2)Int(1/(x^2+sqrt[2]x+1))
- (sqrt[2]/4)Int(x/(x^2-sqrt[2]x+1))
- (1/2)Int(1/(x^2-sqrt[2]x+1)) ]]

These integrals have been split apart from the whole and are all with respect to x. The problem I am having is in solving the first integral of the 4. I have a TI-89 which will show me the answer to the integral.

The answer to Int(x/(x^2+sqrt[2]x+1)) = ln|x^2+sqrt[2]x+1|/2 + arctan(sqrt[2]x+1) + C. I cannot get the integral, when I solve to come to this answer. If I had this one solved, the rest would be easy. If anyone could show me the steps they you take in solving this integral, it would show me the error I made in my steps. Any help would be much Appreciated, thanks!