Interesting Integral

I am currently preparing for IITJEE 2017.After class, my teacher gave me a sheet with a few indefinite integral questions.We got most of the questions but we couldn't solve one question.Nobody could solve it, which is when I decided to discuss about this problem online.The question is,

Integrate: $\int$ $\frac{ln(1+x^{2})}{\sqrt{1-x^{2}}}$ $dx$

I substituted $x$ = $\sin \theta$ but I finally got an expression like

$\int$ $\ln(1+\sin^{2} \theta)$ $d \theta$

Then I converted $\sin^{2} \theta$ to $1-\cos^{2} \theta$ and applied the formula $a^{2}-b^{2}$ = $(a-b)(a+b)$

At last I got , $\int$ $ln(\sqrt{2}-\cos \theta)$ $d \theta$ $+$ $\int$ $ln(\sqrt{2}+\cos \theta)$ $d \theta$

After this I got stuck.I tried many methods but I couldn't get a perfect solution.

I usually go to classes in morning.So if anyone replies during that time I won't be able to respond.I come home by 8pm. After that I can reply.Please post the solution if anyone has got a different method than mine.

Easy Math Editor

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Math	Appears as
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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$