How do u integrate (tanx)^0.5/{ (sinx)^0.5 + (cosx)^0.5}?

Question

The first correct ans gets 10 points

Answer 1

There is no elementary expression for this integral.
It can be reduced to a combination of elliptic integrals.
Let's do that much.
We want to find
∫ (tanx)^0.5 dx /{ (sinx)^0.5 + (cosx)^0.5}
Let u = tan x
x = arctan u
dx = du/(1+ u²)
sin x = u/ √(1+u²), cos x = 1/√(1+u²)
So the integral becomes
∫ √u du/ (1+u²)( √u+1)/(√1+u²) )
= ∫√u √(1+u²) du / (1+u²) * (1 + √u).
This looks like a mess, but if we let u = t², du = 2tdt
it becomes
2*∫ t²/(1+t+t^4+t^5) * √(1+t^4) dt.
This is the integral of a rational function times
the square root of a quartic polynomial, so it
is definitely elliptic.
I ran it through the Wolfram integrator
and it produced a very complicated
answer of elliptic integrals of the first and third kinds.
Hope these calculations are correct. Will check them!

Answer 2

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