A cumbersome indefinite integral Does the indefinite integral 
$$ \int\left(\text{arccsc}[1+\sin^2x] +\arctan\dfrac {a \cos x}{1-a\sin x}-\text{arccot}\dfrac{\cos x}{a-\sin x }\right)dx$$ 
has any closed form expression ? ( here "$[]$" means greatest integer function )  

Note that $0 \le \sin^2 x \le 1$, and the extreme values are attained only at isolated points, so $\lfloor 1 + \sin^2 x \rfloor = 1$ for all intents and purposes here.
On the others I'd look for some of the standard trigonometric substitutions.
