Integral [-pi/2, pi/2] (sin(x)+cos(x))/(sin^4(x)-4) dx

[Noismaker]

Hallo,
An integral has the following text:

Through the goniometric substitution:

we reach this result:

From this result how is it possible to proceed? Obviously the exercise is in R.
Can I use some other replacement?

Can a different goniometric substitution be possible?

 

[HallsofIvy]

That's a rational function. The denominator can be written as a product of linear and quadratic functions and then you can use "partial fractions".