Can this odd-looking fraction be integrated?

[Sleek]

Homework Statement

\displaystyle \int{\frac{dx}{a^2+\left(x-\frac{1}{x} \right)^2}}

Homework Equations

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The Attempt at a Solution

This one looks a bit odd. Had the denominator been a^2 + x^2, it is in one of the standard forms, whose integral is \frac{1}{a} \atan{\frac{x}{a}}. But the denominator is in the form of a^2 + u^2 (where u is a function of x). I did try some manipulations, but to no avail. I tried putting x as sin(theta), but got something like cos(theta)d(theta)/(a^2+cos^4(theta)/sin^2(theta)), which seems even more complex. If someone can just point me into the direction to look, I'll attempt the solution.

Thank you,
Sleek.

 

[Gib Z]

Expand the brackets, simplify, multiply the entire integral by x^2/x^2, factor the denominator and partial fractions.

 

[Sleek]

Thanks for the quick reply, I'm currently here,

\displaystyle \int{\frac{x^2dx}{x^2(a^2-2)+x^4+1}

I don't see how I can factorize/simplify the denominator or the expression...?

Regards,
Sleek.

 

[Gib Z]

Well let a^2-2 =b and u=x^2. Now it resembles a nice quadratic equation =]