How to calculate the following integral ?$$\int_{0}^{\frac{\pi}{2}}\int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{x}}\frac{1}{\sqrt x}\cdot\frac{1}{1+(\tan t^2)^{\sqrt 2}}\text dt\text dx$$
And it equals to the following integral:$$\int^{\frac{\pi}{2}}_0\frac{1}{1+(\tan x)^{\sqrt 2}}\text dx=\int_{0}^{+\infty}\frac{1}{(1+x^{\sqrt 2})(1+x^2)}\text dx$$ But I have no idea about dealing with exponential $\sqrt 2$.
