Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$ 
Prove the following integral
  $$I=\int\limits_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\dfrac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$$

This integral result was calculated using Mathematica and 
I like this integral. But I can't solve it. 
My idea:
Let $$\tan{x}=t\Longrightarrow dx=\dfrac{1}{1+t^2}dt$$
so
$$I=\int\limits_{0}^{\infty}\dfrac{dt}{1+\sin^2{t}}\cdot \dfrac{1}{1+t^2}$$
then I can't proceed. Can you help me? Thank you.

First note that
$$ \begin{align} \int_{0}^{\pi /2} \frac{1}{1+ \sin^{2} ( \tan x)} \ dx &=  \int_{0}^{\infty} \frac{1}{(1+\sin^{2} t)(1+t^{2})} \ dt \\ &= 2 \int_{0}^{\infty} \frac{1}{(3 - \cos 2t)(1+t^{2})} \ dt \end{align}$$
Then using the identity $$\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}} , \ \ |a|<1$$
we have
$$1 + 2 \sum_{k=1}^{\infty} a^{k} \cos(kx) = 1 + 2 \left(\frac{1-a \cos x}{1-2a \cos x +a^{2}} -1\right) = \frac{1-a^{2}}{1-2 a \cos x + a^{2}}$$
Therefore,
$$ \begin{align} \int_{0}^{\infty} \frac{1}{(1-2a \cos 2x +a^{2})(1+x^{2})} \ dx &= \frac{1}{1-a^{2}} \int_{0}^{\infty} \Big(1+2 \sum_{k=1}^{\infty} a^{k} \cos(2kx) \Big) \ \frac{1}{1+x^{2}} \ dx \\ &= \frac{\pi}{2} \frac{1}{1-a^{2}} + \frac{2}{1-a^{2}} \sum_{k=1}^{\infty} a^{k} \int_{0}^{\infty} \frac{\cos(2kx)}{1+x^{2}} \ dx \\  &= \frac{\pi}{2} \frac{1}{1-a^{2}} + \frac{\pi}{1-a^{2}} \sum_{k=1}^{\infty} \Big(\frac{a}{e^{2}} \Big)^{k} \\ &= \frac{\pi}{2} \frac{1}{1-a^{2}} + \frac{\pi}{1-a^{2}} \frac{a/e^{2}}{1-a^/e^{2}} \\ &= \frac{\pi}{2} \frac{1}{1-a^{2}} \Big(1+ \frac{2a}{e^{2}-a} \Big) = \frac{\pi}{2} \frac{1}{1-a^{2}} \frac{e^{2}+a}{e^{2}-a} \end{align}$$
Now rewrite the integral as
$$ \frac{1}{1+a^{2}} \int_{0}^{\infty} \frac{1}{(1 - \frac{2a}{1+a^{2}} \cos 2x)(1+x^{2})} \ dx$$
and let $a= 3 - 2 \sqrt{2}$.
Then
$$ \begin{align} \frac{1}{1+a^{2}} \int_{0}^{\infty} \frac{1}{(1- \frac{1}{3} \cos 2x)(1+x^{2})} \ dx  &= \frac{3}{1+a^{2}} \int_{0}^{\infty} \frac{1}{(3 - \cos 2x)(1+x^{2})} \ dx \\ &= \frac{\pi}{2} \frac{1}{1-a^{2}} \frac{e^{2} + 3 - 2 \sqrt{2}}{e^{2} - 3 + 2 \sqrt{2}} \end{align}$$
which implies
$$ \begin{align} \int_{0}^{\pi /2} \frac{1}{1+\sin^{2} (\tan x)} \ dx &= \frac{\pi}{3} \frac{1+a^{2}}{1-a^{2}}\frac{e^{2} + 3 - 2 \sqrt{2}}{e^{2} - 3 + 2 \sqrt{2}} \\ &= \frac{\pi}{3} \frac{3}{2 \sqrt{2}}\frac{e^{2} + 3 - 2 \sqrt{2}}{e^{2} - 3 + 2 \sqrt{2}} \\ &= \frac{\pi}{2 \sqrt{2}}\frac{e^{2} + 3 - 2 \sqrt{2}}{e^{2} - 3 + 2 \sqrt{2}}  \end{align}$$
