Difficult integral.

[Vampire]

1. The problem statement, all variables and given/known data
The whole integral is a double integral, but I can't even perform the first integration.

The integrand is cosx*sqrt(3+(cosx)^2) dxdy
The x bounds are from arctan(y) to pi/4
the y bounds are from 0 to 1

2. Relevant equations
Converting rectangular to polar may help.


3. The attempt at a solution

I've tried parts, but it was a disaster. I also started converting it to polar, but that was also a disaster.

[Vampire]

I switched dx and dy and the bounds.

The new y bounds are from 0 to tanx
The new x bounds are from 0 to pi/4

The resulting integrand after successfully integrating in terms of y is

-sinx*sqrt(3 + (cosx)^2) dx

It's a little nicer, but still ugly.

Putting the integrand into the mathematica online integrator, I get this:

(-3*ArcSinh[Cos[x]/Sqrt[3]])/2 - (Cos[x]*Sqrt[3 + Cos[x]^2])/2

How do I get here? I tried by starting with parts, but it doesn't get me far.

[Vampire]

Solved with trig substitution and then parts.