An Integral With A Square Root In The Denominator

[Radek Vavra]

How would you integrate it?

$\int \frac{d \varphi}{\sqrt{1 + \frac{a^2 b^2 \sin^2 \alpha}{(a \sin \varphi + b \sin (\alpha - \varphi))^2}}}$

I know that solving it numerically would probably be easier, but I would prefer a closed form solution in this case.

 

[cmcraes]

Try putting it as ^-.5 and also remember that a, b and sin^2(alpha) are all constants with respect to d(phi).
With that in mind try using partial fractions and/or By parts. Tell if You solve it!

Also pay attention to the fact that every term in the square root is a perfect square!

 

[mathman]

It is a little work, but as a first step: asinφ + bsin(α-φ) = csin(φ+β) where c and β are constants depending on a,b, and α.