Evaluating the Definite Integral $\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$ 
How can I find this integral
  $$I=\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$$

Any trick that could compute the definite integral is acceptable. However, it will be more challenging to find a primitive.
Any hint or help is appreciated.

My Work
I just wrote the integral as
$$I=\int_{0}^{1}\frac{\sin 2 \pi x}{1+x^2}dx$$
Then, I decided to introduce
$$J(\alpha)=\int_{0}^{1}\frac{\sin \alpha x}{1+x^2}dx$$
and solve a more general problem. Hence, we would have $I=J(2\pi)$ as a special result. But I don't know how to go further. I tried integration by parts and substitutions but to no avail!

I don't think you will get a simple answer judging by Mathematica's output:
Integrate[2 Sin[Pi x] Cos[Pi x]/(1 + x^2), {x, 0, 1}]
produces
Cosh[\[Pi]] (CosIntegral[(-2 + 2 I) \[Pi]] - 
    2 CosIntegral[2 I \[Pi]] + 
    CosIntegral[(2 + 2 I) \[Pi]]) Sinh[\[Pi]] + 
 1/2 Cosh[2 \[Pi]] (2 SinhIntegral[2 \[Pi]] + 
    I (SinIntegral[(-2 + 2 I) \[Pi]] + SinIntegral[(2 + 2 I) \[Pi]]))
Numerically,
NIntegrate[2 Sin[Pi x] Cos[Pi x]/(1 + x^2), {x, 0, 1}]
produces
0.0926961.
