When is ${\large\int}\frac{dx}{\left(1+x^a\right)^a}$ an elementary function?

Solution 1:

I can give partial answers to your question :

1) $a\in \mathbb{Z}$

Every rational function has elementary antiderivative.

2) $1/a\in\mathbb{Z}$.

Let $y=x^{1/n}$. Then the integral becomes $$ n\int y^{n-1}(1+y)^{-1/n}dy$$ which can be solved by $n-1$ times of integration by parts.

3) $a-1/a=n\in\mathbb{Z}$

Let $u=x^{a}$. Then the integral becomes $$ \int u^{a-n-1}(1+u)^{-a}du=\int u^{-n-1}\left(\frac{1+u}{u}\right)^{a}du$$

If $n\geq 1$, use substitution $t=(u+1)/u$ which gives $$ -\int t^{a}(t-1)^{n-1} dt$$ and if $n\leq -1$, use $t=u/(u+1)$ then $$ \int t^{n-1-a}(1-t)^{-n-1} dt$$ where both can be solved by integration by parts.

4) For $a \in \mathbb{Q}$, Chebyshev's theorem states that if $a, b\in\mathbb{R}$ and $p, q, r\in \mathbb{Q}$, then $\int x^{p}(a+bx^{r})^{q} dx$ can be expressed as elementary function iff at least one of $(p+1)/r, q, (p+1)/r+q$ is integer. Applying for our case, only for $a\in \mathbb{Z}$ or $1/a\in \mathbb{Z}$, $F_{a}(x)$ can be expressed as elementary function.