Is My Approach Correct? Solving the Integral of tan^4x-sec^4x

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The integral of tan^4(x) - sec^4(x) can be approached by separating it into two distinct integrals: ∫tan^4(x)dx and ∫sec^4(x)dx. The user initially derived the solution (tan^3(x))/3 - 2tan(x) - x - (tan^2(x))/2, which does not equate to the simpler solution of x - 2tan(x). It is recommended to simplify tan^4(x) - sec^4(x) before attempting integration for a more straightforward solution.

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I looked online and it gave a really simple method to solve it and it ended up with the solution
x-2tan(x)
I took a really weird approach and I'm not sure if my answer is right, and I'm hoping if someone could take the same approach and confirm if I did it correctly.
I basically separated the integral into 2 separate integrals and got a solution.
∫tan(^4)xdx-∫sec(^4)xdx

When I was done, I got the answer of
(tan(^3)x)/3-2tanx-x-(tan(^2)x)/2
 
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You can always differentiate your answer and see if you get the original integrand.
However, the answer you got isn't equivalent to the answer x - 2tan x.
 
You might want to try simplifying tan^4x-sec^4x first.
 

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