TITLE: Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$
QUESTION [18 upvotes]: Calculate 

$$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$

$\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 x}}dx = \int_{0}^{\frac{\pi}{4}}\frac{\pi}{2}-\int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{\cos 2x}{2\cos^2 x}}dx$
Using The formula $\displaystyle \tan^{-1}(x)+\cot^{-1}(x) = \frac{\pi}{2}\Rightarrow \tan^{-1}(x) = \frac{\pi}{2}-\cot^{-1}(x).$
Now Let $\displaystyle J = \int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{\cos 2x}{2\cos^2 x}}dx = \int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{\cos^2 x-\sin^2 x}{2\cos^2 x}}dx = \int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{1}{2}-\frac{\tan^2 x}{2}}dx$
Now How can I solve after that? Help required.
Thanks

REPLY [18 votes]: Proposition :

\begin{equation}
\int_0^{\Large\frac{\pi}{4}}\arctan\sqrt{\frac{\mu\cos2x}{\cos^2x}}\,dx=\frac{\pi}{2}\left[\arctan\sqrt{2\mu}-\arctan\sqrt{\frac{\mu}{\mu+1}}\right]\quad,\quad\text{for }\,\mu\ge0
\end{equation}

Proof :
Let
\begin{equation}
I(\mu)=\int_0^{\Large\frac{\pi}{4}}\arctan\sqrt{\frac{\mu\cos2x}{\cos^2x}}\,dx
\end{equation}
then
\begin{align}
I'(\mu)&=\partial_\mu\int_0^{\Large\frac{\pi}{4}}\arctan\sqrt{\frac{\mu\cos2x}{\cos^2x}}\,dx\\
&=\frac{1}{2}\int_0^{\Large\frac{\pi}{4}}\frac{\sqrt{\frac{\mu\cos2x}{\cos^2x}}}{\mu+\frac{\cos2x}{\cos^2x}\mu^2}\,dx\\
&=\frac{1}{2}\int_0^{\Large\frac{\pi}{4}}\frac{\sqrt{\mu(1-2\sin^2x)}}{\mu(1-\sin^2x)+(1-2\sin^2x)\mu^2}\cdot\cos x\,dx\\
&=\frac{1}{2\sqrt{2\mu}}\int_0^{\Large\frac{\pi}{2}}\frac{\sqrt{1-\sin^2\theta}}{\left(1-\frac{1}{2}\sin^2\theta\right)+(1-\sin^2\theta)\mu}\cdot\cos \theta\,d\theta\quad\Rightarrow\quad\sin^2\theta=2\sin^2x\\
&=\frac{1}{\sqrt{2\mu}}\int_0^{\Large\frac{\pi}{2}}\frac{\cos^2\theta}{\sin^2\theta+2(1+\mu)\cos^2\theta}\,d\theta\\
&=\frac{1}{\sqrt{2\mu}}\int_0^{\Large\frac{\pi}{2}}\frac{1}{\tan^2\theta+2(1+\mu)}\,d\theta\\
&=\frac{1}{\sqrt{2\mu}}\int_0^{\infty}\frac{1}{t^2+2(1+\mu)}\cdot\frac{1}{t^2+1}\,dt\quad\Rightarrow\quad t=\tan\theta\\
&=\frac{1}{\sqrt{2\mu}(1+2\mu)}\int_0^{\infty}\left[\frac{1}{t^2+1}-\frac{1}{t^2+2(1+\mu)}\right]\,dt\\
&=\frac{1}{\sqrt{2\mu}(1+2\mu)}\left[\frac{\pi}{2}-\frac{\pi}{2\sqrt{2(1+\mu)}}\right]\\
I(\mu)&=\int\frac{1}{\sqrt{2\mu}(1+2\mu)}\left[\frac{\pi}{2}-\frac{\pi}{2\sqrt{2(1+\mu)}}\right]\,d\mu\\
&=\frac{\pi}{2}\int\left[\frac{1}{\sqrt{2\mu}(1+2\mu)}-\frac{1}{\sqrt{2\mu}(1+2\mu)\sqrt{2(1+\mu)}}\right]\,d\mu\\
\end{align}
where
\begin{align}
\int\frac{1}{\sqrt{2\mu}(1+2\mu)}\,d\mu&=\int\frac{1}{1+y^2}\,dy\qquad\Rightarrow\quad y=\sqrt{2\mu}\\
&=\arctan y+C_1\\
&=\arctan\sqrt{2\mu}+C_1
\end{align}
and
\begin{align}
\int\frac{1}{\sqrt{2\mu}(1+2\mu)\sqrt{2(1+\mu)}}\,d\mu&=\int\frac{1}{(1+y^2)\sqrt{2+y^2}}\,dy\qquad\Rightarrow\quad y=\sqrt{2\mu}\\
\end{align}
Using
\begin{align}
\color{blue}{\int \frac{dx}{(x^2+1)\sqrt{x^2+a}}=\frac{1}{\sqrt{a-1}}\tan^{-1}\left(\frac{x\sqrt{a‌​-1}}{\sqrt{x^2+a}}\right)+C}
\end{align}
It can be derived by using substitution $x=\dfrac{1}{t}$ followed by $z=\sqrt{at^2+1}$.
Hence
\begin{align}
\int\frac{1}{\sqrt{2\mu}(1+2\mu)\sqrt{2(1+\mu)}}\,d\mu&=\arctan\sqrt{\frac{\mu}{\mu+1}}+C_2
\end{align}
then
\begin{equation}
I(\mu)=\frac{\pi}{2}\left[\arctan\sqrt{2\mu}-\arctan\sqrt{\frac{\mu}{\mu+1}}\right]+C
\end{equation}
For $\mu=0$, we have $I(0)=0$ implying $C=0$. Thus
\begin{equation}
I(\mu)=\int_0^{\Large\frac{\pi}{4}}\arctan\sqrt{\frac{\mu\cos2x}{\cos^2x}}\,dx=\frac{\pi}{2}\left[\arctan\sqrt{2\mu}-\arctan\sqrt{\frac{\mu}{\mu+1}}\right]\qquad\quad\square
\end{equation}

For $\mu=\frac{1}{2}$, we obtain
\begin{align}
I\left(\frac{1}{2}\right)&=\int_0^{\Large\frac{\pi}{4}}\arctan\sqrt{\frac{\cos2x}{2\cos^2x}}\,dx\\
&=\frac{\pi}{2}\left[\arctan 1-\arctan\left(\frac{1}{\sqrt{3}}\right)\right]\\
&=\frac{\pi}{2}\left[\frac{\pi}{4}-\frac{\pi}{6}\right]\\
&=\frac{\pi^2}{24}
\end{align}