# Evaluating indefinite integral with any hint or solution

How can I evaluate this indefinite integral.

$$\int\frac{dx}{x\sqrt{x^3+x+1}}$$

any hit would be appreciated.

• It does seem to have a nice looking closed-form. Where is this from? – mrtaurho Nov 26 '18 at 17:38
• Some sort of elliptic integral? – Richard Martin Nov 26 '18 at 17:39

$$\int \dfrac{dx}{x \sqrt{(x-a)(x-b)(x-c)}} = {\frac {\pm 2\,i}{a \sqrt {a-c}}{\it EllipticPi} \left( {\frac {\sqrt {a-x}}{\sqrt {a-b}}},{\frac {a-b}{a}},{\frac {\sqrt {a-b}}{\sqrt {a-c}}} \right) }$$
(using Maple's notation). In your case you want to take $$a,b,c$$ to be the three roots of $$x^3+x+1$$ (one real root approximately $$-0.682327803828019$$, two complex roots approximately $$0.341163901914009693 \pm 1.16154139999725192\,i$$).