How do I integrate $\int\frac1{ \sqrt{x^n + 1}}dx$

Question

$$\int\frac1{ \sqrt{x^n + 1}}dx$$

This question popped up in my mind after we took trigonometric substitution in calc II. I tried solving it until I got $${\sinh}^{{\frac{2}{{n}}-{1}}}{\left(\theta\right)}$$ and then it got ugly afterwards.

I came up with some reduction formula, but then the values did not really work out when I tested them out with constants and limits.

Moreover, a quick search on the internet showed me that such type of questions are solved with complex analysis, Mobius transformation etc.

I think this integral is solvable for all infinite integras and finite rationals, but again I am not sure.

Answer 1

The antiderivative for general $n \neq 0$ is expressible in terms of a hypergeometric function or, just as well, the incomplete beta function: $$\color{#df0000}{\boxed{\int \frac{dx}{\sqrt{x^n + 1}} = x \cdot{}_2 F_1\left(\frac{1}{2}, \frac{1}{n}; 1 + \frac{1}{n}; -x^n\right) + C}} .$$ For most values of $n$ this expression cannot be written as a closed expression in elementary functions, but for certain special rational values it can. We can read off from the result of your hyperbolic trigonometric solution that there is a closed form for $n = \frac{1}{m}$ and $n = \frac{2}{m}$, $m \in \Bbb Z \setminus \{ 0 \}$.

If $n$ is nonzero and rational, say, $n = \frac{a}{b}$ for $a, b \in \Bbb Z \setminus \{0\}$. Then, the substitution $$x = u^b, \qquad dx = b u^{b - 1} \,du$$ transforms the integral to $$b \int \frac{u^{b - 1} \,du}{\sqrt{u^a + 1}} .$$

• For $a = 1$ (so that $n = \frac{1}{b}$), the substitution $u = v^2 - 1$, $du = 2 v \,dv$, transforms the integral to the polynomial integral $$2 b \int (v^2 - 1)^{b - 1} \,dv .$$
• For $a = 2$, (so that $n = \frac{2}{b}$), the substitution $u = \sinh t$ transforms the integral to the hyperbolic integral $$b \int \sinh^{b - 1} t \,dt .$$
• For $a = 3, 4$ (so that $n = \frac{3}{b}$ or $n = \frac{4}{b}$), the integral is $$b \int \frac{u^{b - 1} \,du}{\sqrt{u^3 + 1}} \qquad \textrm{or} \qquad b \int \frac{u^{b - 1} \,du}{\sqrt{u^4 + 1}} ,$$ which can be expressed in terms of elliptic functions. For example, $$\int \frac{dx}{\sqrt{x^4 + 1}} = e^{-\pi i / 4} F(e^{\pi i / 4} u, i) + C ,$$ where $F$ is the incomplete elliptic integral of the first kind.

Of course, if $n = 0$, the integrand is constant.