# Prove π/2∫0dx1+sin2(tanx)=π2√2(e2+3−2√2e2−3+2√2)\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

This integral result was calculated using Mathematica and
I like this integral. But I can’t solve it.

My idea:

Let
so

then I can’t proceed. Can you help me? Thank you.

First note that

Then using the identity

we have

Therefore,

Now rewrite the integral as

and let $a= 3 - 2 \sqrt{2}$.

Then

which implies