I got this to solve on an interview (time limit 60 mins + 5 other math questions)
$$ \int[arccsc[1+\sin^2 (x)] + \arctan \frac{\phi \cos(x)}{1-\phi \sin(x)}-arccot\frac{\cos(x)}{\phi-\sin(x)}]dx $$
I finished my school (software engineer) 20 years ago, but as per my knowledge indefinite integration of nested trigonometric functions doesn't have a known solution. But my answer no solution was 'false' by interviewer.
Now at home, I tried with substitution+per partes methods (e.g.arccsc t dt) I got this answer for first nested one ( $\int[arccsc[1+\sin^2 (x)]dx$ ), but I am pretty sure it is wrong: $$(1+\sin^2(x))*arccsc(1+\sin^2(x))+\ln\left|\sqrt{(1+\sin^2(x))^2}+\sqrt{(1+\sin^2(x))^2-1}\right| + C $$ Does it have a solution?
I just found that the sum of the last two terms is in fact equal to x. Hence if your answer for the first term is correct, just add $\frac{x^2}{2}$.
Let the second term be a and the third term be b. Then we have $$tan a = \frac{\phi cosx}{1-\phi sinx}$$ and $$cot b = \frac{cosx}{\phi -sinx}$$
Use the formula sin(a - b) = sina . Cosb - Cosa . Sina
We can get sin(a - b) = sin x, that means a - b = x.