What is the integral of (1-x^2)/(3x-x^3) ?

The integral of (1-x^2)/(3x-x^3) is 1/3 ln|x|-1/6 ln|x+sqrt(3)|-1/6 ln|x-sqrt(3)|+1/2 ln|3-x^2|+C

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integral of (1-x^2)/(3x-x^3)

\int \frac{1 - x ^{2}}{3 x - x ^{3}} d x

\frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x + 3 - \frac{1}{6} ln x - 3 + \frac{1}{2} ln 3 - x^{2} + C

Solution steps

\int \frac{1 - x ^{2}}{3 x - x ^{3}} d x

Expand

\frac{1 - x ^{2}}{3 x - x ^{3}} : \frac{1}{3 x - x ^{3}} - \frac{x ^{2}}{3 x - x ^{3}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{3 x - x ^{3}} d x - \int \frac{x ^{2}}{3 x - x ^{3}} d x

\int \frac{1}{3 x - x ^{3}} d x = \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x + 3 - \frac{1}{6} ln x - 3

\int \frac{x ^{2}}{3 x - x ^{3}} d x = - \frac{1}{2} ln 3 - x^{2}

= \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x + 3 - \frac{1}{6} ln x - 3 - (- \frac{1}{2} ln 3 - x^{2})

Simplify

= \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x + 3 - \frac{1}{6} ln x - 3 + \frac{1}{2} ln 3 - x^{2}

Add a constant to the solution

= \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x + 3 - \frac{1}{6} ln x - 3 + \frac{1}{2} ln 3 - x^{2} + C

What is the integral of (1-x^2)/(3x-x^3) ?
The integral of (1-x^2)/(3x-x^3) is 1/3 ln|x|-1/6 ln|x+sqrt(3)|-1/6 ln|x-sqrt(3)|+1/2 ln|3-x^2|+C