3.117 \(\int \frac{x}{2-3 x+x^3} \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{3 (1-x)}+\frac{2}{9} \log (1-x)-\frac{2}{9} \log (x+2) \]

[Out]

1/(3*(1 - x)) + (2*Log[1 - x])/9 - (2*Log[2 + x])/9

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Rubi [A]  time = 0.0165808, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2074} \[ \frac{1}{3 (1-x)}+\frac{2}{9} \log (1-x)-\frac{2}{9} \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[x/(2 - 3*x + x^3),x]

[Out]

1/(3*(1 - x)) + (2*Log[1 - x])/9 - (2*Log[2 + x])/9

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin{align*} \int \frac{x}{2-3 x+x^3} \, dx &=\int \left (\frac{1}{3 (-1+x)^2}+\frac{2}{9 (-1+x)}-\frac{2}{9 (2+x)}\right ) \, dx\\ &=\frac{1}{3 (1-x)}+\frac{2}{9} \log (1-x)-\frac{2}{9} \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0075359, size = 28, normalized size = 0.93 \[ -\frac{1}{3 (x-1)}+\frac{2}{9} \log (1-x)-\frac{2}{9} \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2 - 3*x + x^3),x]

[Out]

-1/(3*(-1 + x)) + (2*Log[1 - x])/9 - (2*Log[2 + x])/9

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Maple [A]  time = 0.007, size = 21, normalized size = 0.7 \begin{align*} -{\frac{2\,\ln \left ( 2+x \right ) }{9}}-{\frac{1}{-3+3\,x}}+{\frac{2\,\ln \left ( -1+x \right ) }{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^3-3*x+2),x)

[Out]

-2/9*ln(2+x)-1/3/(-1+x)+2/9*ln(-1+x)

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Maxima [A]  time = 0.966816, size = 27, normalized size = 0.9 \begin{align*} -\frac{1}{3 \,{\left (x - 1\right )}} - \frac{2}{9} \, \log \left (x + 2\right ) + \frac{2}{9} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3-3*x+2),x, algorithm="maxima")

[Out]

-1/3/(x - 1) - 2/9*log(x + 2) + 2/9*log(x - 1)

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Fricas [A]  time = 0.779541, size = 86, normalized size = 2.87 \begin{align*} -\frac{2 \,{\left (x - 1\right )} \log \left (x + 2\right ) - 2 \,{\left (x - 1\right )} \log \left (x - 1\right ) + 3}{9 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3-3*x+2),x, algorithm="fricas")

[Out]

-1/9*(2*(x - 1)*log(x + 2) - 2*(x - 1)*log(x - 1) + 3)/(x - 1)

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Sympy [A]  time = 0.098044, size = 22, normalized size = 0.73 \begin{align*} \frac{2 \log{\left (x - 1 \right )}}{9} - \frac{2 \log{\left (x + 2 \right )}}{9} - \frac{1}{3 x - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**3-3*x+2),x)

[Out]

2*log(x - 1)/9 - 2*log(x + 2)/9 - 1/(3*x - 3)

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Giac [A]  time = 1.0652, size = 30, normalized size = 1. \begin{align*} -\frac{1}{3 \,{\left (x - 1\right )}} - \frac{2}{9} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{2}{9} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3-3*x+2),x, algorithm="giac")

[Out]

-1/3/(x - 1) - 2/9*log(abs(x + 2)) + 2/9*log(abs(x - 1))