<s>[INST] Doubts in solving the integral $\int \frac{1}{\sqrt{1+\sin2x}}dx$ I tried to solve the integral $\int \frac{1}{\sqrt{1+\sin2x}}dx$ by using $1+\sin2x=(\sin x+\cos x)^2$ but got stuck. So I referred the solution in my book which is given below:
$$
I=\int \frac{1}{\sqrt{1+\sin2x}}dx
$$
$$
= \int \frac{1}{\sqrt{1-\cos(\frac \pi 2+2x)}}dx\tag1\\
= \int \frac{1}{\sqrt{2\sin^2(\frac \pi 4+x)}}dx\\
=\frac 1 {\sqrt2} \int \csc\left(\frac \pi 4+x\right)dx\\
= \color{red}{\frac 1 {\sqrt2} \log\left|\tan\left(\frac \pi 8+\frac x 2\right)\right|+C}(*)
$$
I wondered what if I replaced $\sin 2x$ by $\cos\left(\frac \pi 2 -2x\right)$ instead of $-\cos\left(\frac \pi 2 +2x\right)$ in step $(1)$. So I proceeded as follows:
$$
I=\int \frac{1}{\sqrt{1+\sin2x}}dx
$$
$$
= \int \frac{1}{\sqrt{1+\cos(\frac \pi 2-2x)}}dx\\
= \int \frac{1}{\sqrt{2\cos^2(\frac \pi 4-x)}}dx\\
=\frac 1 {\sqrt2} \int \sec\left(\frac \pi 4-x\right)dx\\
= \color{red}{\frac 1 {\sqrt2} \log\left|\tan\left(\frac {3\pi} 8-\frac x 2\right)\right|+C}(**)
$$
But I got a different result. Could you please explain the reason for this anomaly? Is it wrong to do a different replacement in step $(1)$? I think it shouldn't make any difference.
Further, could you please explain how to think we must be doing the replacement instead of using $1+\sin2x=(\sin x+\cos x)^2$ to solve this integral? I got this idea only after looking the solution.

*Using $\int \csc x dx=\log\left|\tan\left(x/2\right)\right|+C$
**Using $\int \sec x dx=\log\left|\tan\left(\frac \pi 4 +\frac x 2\right)\right|+C$
 [/INST] $\sqrt{1+sin2x}=\sqrt{(sinx+cosx)^{2}}=\sqrt{2}\left| sin(x+\frac{\pi}{4}) \right|\\
z=x+\frac{\pi}{4}\Rightarrow dz=dx\\
\displaystyle\int\frac{dx}{\sqrt{1+sin2x}}=\frac{1}{\sqrt{2}}\int\frac{dz}{\left| sinz \right|}\\
1)sinz\gt 0\\
\displaystyle\int\frac{dz}{\left| sinz \right|}=\int\frac{sinzdz}{sin^{2}z}=\int\frac{sinzdz}{1-cos^{2}z}\\
(y=cosz)\to \displaystyle \int\frac{dz}{\left| sinz \right|}=-\int\frac{dy}{1-y^{2}}=-\frac{1}{2}\int(\frac{1}{1+y}+\frac{1}{1-y})dy=-ln\sqrt{1-y^{2}}+c\\
\displaystyle\int\frac{dx}{\sqrt{1+sin2x}}=-\frac{1}{\sqrt{2}}ln\left| sin(x+\frac{\pi}{4}) \right|+c\\
2)sinz\lt 0\\
\displaystyle\int\frac{dz}{\left| sinz \right|}=-\int\frac{dz}{sinz}=\frac{1}{\sqrt{2}}ln\left| sin(x+\frac{\pi}{4}) \right|+c$
</s>