Integrate $\int \frac{1}{1+\arctan(x)}dx$

calculus trigonometry integration

There exists no solution in terms of standard mathematical functions. This can be proven using Risch Algorithm. This is not something I recommend you do by hand.

The most easy way to check if an integral has a solution is by asking Wolfram Alpha which uses this algorithm or a better one to guarantee a solution is it exists. Otherwise, it displays:

(no result found in terms of standard mathematical functions)

To back up my claim:

For indefinite integrals, an extended version of the Risch algorithm is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral functions, polylogarithms, and other related functions.


I'm pretty sure this doesn't have a closed form. A change of variable and integration by parts got me to: $$\int \frac{dx}{1 + \arctan(x)} = \frac{x}{1 + \arctan(x)} + \int \frac{\tan(y)}{(1 + y)^2}dy$$ WolframAlpha says the right-most integral has no closed form, and I can't think of anything useful to do with it. It might have some cute definite integral values?

Related

Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?
How to know if you are "tough enough" to study Algebraic Topology [closed]
Why is that the extended real line $\mathbb{\overline R}$ do not enjoy widespread use as $\mathbb{R}$?
Proof that the dimension of a matrix row space is equal to the dimension of its column space
Prove or disprove: $\sum_{k=1}^{\infty}\frac{\sqrt{k+1}-\sqrt{k}}{k\sqrt{k}}$ is convergent
If $C$ is the Cantor set, then $C+C=[0,2]$.
Self study Control Theory
Is the "$\pi$ is wrong"-movement serious or a joke? [closed]
Integral $ \int_{0}^{\infty} \ln x\left[\ln \left( \frac{x+1}{2} \right) - \frac{1}{x+1} - \psi \left( \frac{x+1}{2} \right) \right] \mathrm{d}x $
Prove that:$\int_{0}^{1}{x+x^2+\cdots+x^{2n}-2nx\over (1+x)\ln{x}}dx=\ln{\left[\left({2\over \pi}\right)^n(2n)!!\right]}$
Is there a term for a magma with identity (only)?
Compute engine became inaccessible after a reboot