Integrating a Fraction using the product rule

[sid777]

Homework Statement

Find the integral of

This was the question. There is a way to do it by long division but I am confused with Long division. Instead I tried to do by the method below but I failed...

Homework Equations

None

The Attempt at a Solution

I thought maybe I could reduce the denominator to the power of -1 and then Integrate by Parts.
Like this

Is this correct? If no then what could I do to this integral. I would be very thankful if someone would post an answer...

 

[SteamKing]

Your attempt is valid, but you have created a more complex integral to solve. While there might be a way to solve this integral by dividing out to find the quotient, that too looks like it would create a messy integral to solve.

 

[MostlyHarmless]

Break it down into $$∫{\frac{x^3}{16xtanx+sinx}}dx+∫{\frac{56x^2sin^6x}{16xtanx+sinx}}dx$$
Then try long division. If you can't do long division with polynomials, I would recommend reading up on it. It creeps up quite a bit, and you're much better off knowing how to do it.

This is still going to be a nasty nasty integral, unless I'm missing something. I've not attempted it yet, but it seems like it might be easier that trying to to Integration by parts 20 times. Also With long division it may help to write ${\tan{x}}$ as ${\frac{\sin{x}}{\cos{x}}}$ and write the denominator as a fraction then simplify.