<s>[INST] Integral of square of variable in expression in square root in denominator I'm having trouble with following integral
$$
\int \frac{1}{\sqrt{1+4x^2+4y^2}} dy
$$
that according to WolframAlpha is
$$
\frac{1}{2} \ln( \sqrt{1+4x^2+4y^2} + 2y) + c
$$
which I can verify but I can't figure out what substitution to apply or what functions to use for integration by parts in original integral. What would be your first step to attack this integral please? Thank you.
 [/INST] Notice, the standard formula $$\color{red}{\int\frac{dx}{\sqrt{a^2+x^2}}=\ln(x+\sqrt{a^2+x^2})+c}$$ For derivation, let $x=a\tan \theta \implies dx=a\sec^2\theta d\theta $ $$\int\frac{dx}{\sqrt{a^2+x^2}}=\int\frac{a\sec^2 \theta d\theta}{\sqrt{a^2+(a\tan\theta)^2}}=$$ $$=\int\frac{\sec^2 \theta d\theta}{\sec\theta}=\int\sec\theta d\theta=\ln(\sec\theta+\tan\theta)+c$$  $$=\ln(\sqrt{1+\tan^2\theta}+\tan\theta)+c$$ Now, substituting the value of $\tan \theta=\frac{x}{a}$ $$=\ln(x+\sqrt{a^2+x^2})+c$$ 
In the given integration, $x $ is treated as a constant. Now we have  $$\int \frac{1}{\sqrt{1+4x^2+4y^2}} dy $$ $$ =\int \frac{1}{\sqrt{(1+4x^2)+(2y)^2}} dy $$ Let, $2y=t \implies 2dy=dt\ or\ dy=\frac{dt}{2}$ $$ =\frac{1}{2}\int \frac{dt}{\sqrt{(1+4x^2)+t^2}}+c$$ $$ =\frac{1}{2}\ln(t+\sqrt{(1+4x^2)+t^2})+c $$  $$ =\frac{1}{2}\ln(2y+\sqrt{(1+4x^2)+(2y)^2})+c$$  $$ =\frac{1}{2}\ln(\sqrt{1+4x^2+4y^2}+2y)+c $$
</s>