# Indefinite Integral — exponential and arctan

Please solve the following indefinite integral: $$\int \exp\left({\tan^{-1}\left(1+ \frac{x}{x^2+1}\right)}\right)\mathrm{d}x$$

important I have the above integral as one of the types of integrals in my curriculum and tomorrow I have an important exam. So I request you to make haste in giving me the solution. I will be very grateful to you.

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Are you sure that you are expected to antidifferentiate $e^{\tan^{-1}\left(1+\frac{x}{x_2+1}\right)}$? – alex.jordan Mar 19 '13 at 3:32
Calling on my experience as a calculus teacher, this strikes me as the type of problem where you have actually been asked to find $\frac{d}{dx}\left(\int e^{\tan^{-1}\left(1+\frac{x}{1+x^2}\right)}dx\right)$, which can be done without antidifferentiating anything. – alex.jordan Mar 19 '13 at 3:35
Yes. Sir: This question was printed in a sample paper. – IcyFlame Mar 19 '13 at 3:53

I'm pretty sure that this integral doesn't have a closed-form expression in terms of elementary functions and operations. You could let $$f(x)=\int_0^x \exp\left({\tan^{-1}\left(1+ \frac{t}{t^2+1}\right)}\right)\mathrm{d}t+C$$ Then $$\frac{d}{dx} f(x)=\exp\left({\tan^{-1}\left(1+ \frac{x}{x^2+1}\right)}\right)$$I know that this has come way too late to be of any help to you in your exam.