Can $\int \frac{\sec x \ \mathrm{d}x}{\sqrt{\sin(x+2A)+\sin(A)}}$ be evaluated using elementary functions?

Question

This question recently popped up on one of my tests and I still have no idea on how to even begin.

I tried assuming $\sin(x+2A)+\sin(A)=t^2$ which did not help at all.

Then I went on to multiply and divide by $\tan x$ to create a derivative on the numerator, still does not help.

I even tried to expand $\sin(x+2A)$ which too does not seem to follow.

I have also tried Approach$0$ and Wolfram Alpha but they do not seem to help.

and many more...

Now I am stumped I have no idea as to how to even begin. I have tried everything that I could think of.

Note that we are only familiar with basics of integration(no contour integrals ; no special functions etc..)

$\bullet~\textbf{Question:}~$

"Can the above integral be evaluated in elementary functions?" If yes then how? If not , can we prove that it cannot be solved using elementary funcions?

Answer 1

Let $$A=\dfrac\pi2-2b,\quad x=4b-\dfrac\pi2-y,\tag1$$ then $$I=\int\dfrac{\text dy}{\cos(y-4b)\sqrt{\cos y+\cos2b}} =\int\dfrac{\text dy}{(\cos4b\cos y+\sin4b\sin y)\sqrt{\cos y+\cos2b}}$$ $$=\int\dfrac{(\cos4b\cos y-\sin4b\sin y)\,\text dy}{(\cos^2 4b\cos^2 y-\sin^2 4b\sin^2y)\sqrt{\cos y+\cos2b}}=I_1\cos4b-I_2\sin4b,$$ where $$I_1=\int\dfrac{\cos y\,\text dy}{(\cos^2y-\sin^24b)\sqrt{\cos y+\cos2b}}, \tag2$$ $$I_2=\int\dfrac{\sin y\,\text dy}{(\cos^2y-\sin^2 4b) )\sqrt{\cos y+\cos2b}}.\tag3$$ Integral $\,I_2\,$ can be expressed via elementary functions (see the WA result below).

At the same time, the first integral in the common case can be expressed via elliptic integral of the third kind and cannot be expressed via elementary functions.

However, for $\;b=\dfrac\pi2 k,\;A=\dfrac\pi2+\pi n,\; k,n\in\mathbb Z\;$ there are exceptions

Answer 2

Not elementary at all

$$I=\int \frac{\sec (x)}{\sqrt{\sin(x+2a)+\sin(a)}}\,dx$$ $$\sin(x+2a)+\sin(a)=\sin (a)+\sin (2 a) \cos (x)+\cos (2 a) \sin (x)$$ Using the tangent half-angle substitution $x=2 \tan^{-1}(t)$ $$\sin(x+2a)+\sin(a)=\frac{(\sin (a)+\sin (2 a))+2 t \cos (2 a)+t^2 (\sin (a)-\sin (2 a))}{1+t^2}$$ $$(\sin (a)+\sin (2 a))+2 t \cos (2 a)+t^2 (\sin (a)-\sin (2 a))=$$ $$(\sin (a)-\sin (2 a))\left(t+\tan \left(\frac{3 a}{2}\right)\right) \left(t-\cot \left(\frac{a}{2}\right)\right)$$ Let $$\alpha=\sin (a)+\sin (2 a)\qquad \beta=\tan \left(\frac{3 a}{2}\right) \qquad \gamma=\cot \left(\frac{a}{2}\right)$$

$$I=\frac 2 \alpha \int \frac {\sqrt{1+t^2}} {(1-t^2)\sqrt{(t+\beta) (t-\gamma) }}\,dt$$

The result is given in terms of elliptic integrals (have a look here).

Edit

$$a=\frac \pi 4 \qquad \qquad \implies \qquad I=2 \sqrt{2-\sqrt{2}} \,\,\Pi \left(2;\frac{x}{2}|4-2 \sqrt{2}\right)$$