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Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$. $$\int 2^x \tan^9(x^2)\sec(x^2)dx$$

I also tried integrating by parts but it got more complicated.

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Do you have a reason to expect this has an elementary antiderivative? – Alex Becker Mar 5 '13 at 1:55
How does $u=\sec(x)$ relate to $\sec(x^2)$? – Maesumi Mar 5 '13 at 1:59
@AlexBecker it is one of the questions on my Calc 2 mock test that I could not answer. – Ddayne Mar 5 '13 at 2:01
@Infodayne I'm going to go ahead and guess that $2^x$ should be $2x$. – Alex Youcis Mar 5 '13 at 2:07
@AlexYoucis Ya im starting to think the test had a typo. If its 2x then I would sub u=x^2. It also could have been (x^2) = (2^x),,, trying to solve that right now. – Ddayne Mar 5 '13 at 2:12

This is not an answer to the posted question (the $2^x$ should probably be $2x$ as discussed in the comments). However, I hope my solution will be still be instructive to those who happen to come across this question anyway:

$$\int 2x\tan^9(x^2)\sec(x^2) \, dx$$ We first start by simplifying the argument in the tangent and secant terms. Let's make a $u$-substitution where $u = x^2 \implies du = 2x \, dx$. So we have $$ \int \tan^9(u)\sec(u) \, du$$ It takes some practice to see what another substitution should be. It has to do with the parity [even-ness vs. odd-ness] of both the powers of the tangent and secant. Click here for Pauls Online Notes, or here for some more general approaches to these types of integration problems along with examples. Anyway, the appropriate substitution is $v=\sec(u) \implies dv = \sec(u)\tan(u) \, du$ and we're left with $$\int \tan^9(u)\sec(u)\, du \\ = \int \tan(u)\tan^8(u)\sec(u)\,du \\ = \int \tan^8(u)\sec(u)\tan(u)\,du \\ = \int [\tan^2(u)]^4\sec(u)\tan(u)\,du \\ = \int [\sec^2(u)-1]^4 \sec(u)\tan(u)\,du \\ = \int [v^2-1]^4 \, dv$$ At this point it's just expanding into a polynomial, which is always welcome when it comes to integration: $$ \int [v^2-1]^4 \, dv \\ = \int [(v^2-1)(v^2-1)]^2\,dv \\ = \int [v^4-2v^2+1]^2\, dv \\ = \int (v^4-2v^2+1)(v^4-2v^2+1)\,dv \\ = \int (v^8 -2v^6+v^4-2v^6+4v^4-2v^2+v^4-2v^2+1)\, dv \\ = \int (v^8 -4v^6 + 6v^4-4v^2+1)\,dv \\ = \frac{1}{9}v^9-\frac{4}{7}v^7+\frac{6}{5}v^5-\frac{4}{3}v^3+v + C$$ But we know $v=\sec(u)$, and that $u=x^2$ from earlier. So we know $v=\sec(x^2)$. Let's back-substitute and replace all the $v$'s with $\sec(x^2)$ to get: $$ \frac{1}{9}\sec^9(x^2)-\frac{4}{7}\sec^7(x^2)+\frac{6}{5}\sec^5(x^2)-\frac{4}{3}\sec^3(x^2)+\sec(x^2)+C$$

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