How Do You Integrate \(\frac{\cos^2 x}{(1+\epsilon\cos x)^3}\) in Trigonometry?

[RobertT]

$$\int \frac{\cos^2 x}{(1+\epsilon\cos x)^3}\,dx$$

Where, $$\epsilon > 0$$ is a real number constant.

 

[RobertT]

One perhaps rather interesting note is that $$\int \frac{\sin x\cos x}{(1+\epsilon\cos x)^3}\,dx$$ is easy to calculate

 

[Nebuchadnezza]

My calculator gives me the answer as:

$$\int {\frac{{\cos {{\left( x \right)}^2}}}{{{{\left( {1 + n\cos \left( x \right)} \right)}^2}}}dx} = \frac{1}{2}\left( {\frac{{\sin \left( x \right)\left( { - 4{n^2}\cos \left( x \right) - 3n + \cos \left( x \right)} \right)}}{{{{\left( {{n^2} - 1} \right)}^2}{{\left( {n\cos \left( x \right) + 1} \right)}^2}}} + \frac{{2\left( {2{n^2} + 1} \right)\tanh \left( {\frac{{\left( {n - 1} \right)\tan \left( {\frac{x}{2}} \right)}}{{\sqrt {{n^2} - 1} }}} \right)}}{{{{\left( {{n^2} - 1} \right)}^{\frac{5}{2}}}}}} \right) + C$$

Perhaps some fancy trig substitution will do the trick along with some partial fractions =)

 

[RobertT]

My calculator gives me the answer as:

$$\int {\frac{{\cos {{\left( x \right)}^2}}}{{{{\left( {1 + n\cos \left( x \right)} \right)}^2}}}dx} = \frac{1}{2}\left( {\frac{{\sin \left( x \right)\left( { - 4{n^2}\cos \left( x \right) - 3n + \cos \left( x \right)} \right)}}{{{{\left( {{n^2} - 1} \right)}^2}{{\left( {n\cos \left( x \right) + 1} \right)}^2}}} + \frac{{2\left( {2{n^2} + 1} \right)\tanh \left( {\frac{{\left( {n - 1} \right)\tan \left( {\frac{x}{2}} \right)}}{{\sqrt {{n^2} - 1} }}} \right)}}{{{{\left( {{n^2} - 1} \right)}^{\frac{5}{2}}}}}} \right) + C$$

Perhaps some fancy trig substitution will do the trick along with some partial fractions =)

Wow... what magic calculator did you use? @@..

Anyway thanks a tonne for the answer but you get the wrong one...

notice the correct denominator is of power 3 and not 2

 

[Nebuchadnezza]

A small typo from my side, but the answer is still correct.

And here is my "Magical" calculator. Maple 13 gave me a tad uglier answer so I decided to use this one instead:

http://www.wolframalpha.com/input/?i=integrate+cos%28x%29^2%2F%281%2Bn*cos%28x%29%29^3

Just out of curiosity, what is this integral for?

 

[RobertT]

A small typo from my side, but the answer is still correct.

And here is my "Magical" calculator. Maple 13 gave me a tad uglier answer so I decided to use this one instead:

http://www.wolframalpha.com/input/?i=integrate+cos%28x%29^2%2F%281%2Bn*cos%28x%29%29^3

Just out of curiosity, what is this integral for?

It's one of the terms that popped out in an equation when I was calculating some fluid film force applied to a bearing system.

Thanks again for the answer