# Does Wolfram Alpha sometimes fail?

### Does Wolfram Alpha sometimes fail?

Hi all guys,
I had the following indefinite integral to solve:
$$\int\frac{a^{2}cos^{2}\left(x\right)}{1+a^{2}cos^{2}\left(x\right)}dx$$

The result I found is (ask for the step-by-step I did):
$$\int\frac{a^{2}cos^{2}\left(x\right)}{1+a^{2}cos^{2}\left(x\right)}dx = x+\frac{\arctan\left(\frac{\tan^{-1}\left(x\right)}{\sqrt{a^{2}+1}}\right)}{\sqrt{a^{2}+1}}$$

The wolfram alpha result is

$$\int\frac{a^{2}cos^{2}\left(x\right)}{1+a^{2}cos^{2}\left(x\right)}dx = x-\frac{\tan^{-1}\left(\sqrt{a^{2}+1}\tan\left(x\right)\right)}{\sqrt{a^{2}+1}}$$

Which one is correct?

Thank you all for your help!
frank91

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### Re: Does Wolfram Alpha sometimes fail?

Tangent is not a one-to-one function so "arctan" has different branches. I suspect they differ only in which branch of "arctan" you choose.
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