1\int_{0}^{\frac{\pi}{2}}{\frac{\left(\sqrt{1+x^2} \right)\cos\left( {x+\arctan{x}\right)\text{dx}}}{x^2+\cos^2x}}
may be we can do this
$Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{cosx-xsinx}{x^2+cos^2x}dx$
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1708$ Original Question is $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{cosx+xsinx}{x^2+cos^2x}dx\\\\ $Sorry for my blunder typing........
1\int \frac{\left( \cos x +x \sin x\right)\text{dx}}{\cos^2 x\left(1+\left( \frac{x}{\cos x}\right) ^2\right)}
\text{Now put }\frac{x}{\cos x}=t
1\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{cosx-xsinx}{x^2+cos^2x}dx = \pi$