Evaluate $\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$

Question

The following question was asked in JEE 2010.

Evaluate $\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$

One method is to open the bracket in the numerator, divide by denominator and then integrate the result.

Is there any other way?

I tried $x=\tan\theta$ but couldn't conclude.

I also tried the property $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$ but in vain.