3.265 \(\int \frac{1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{8} \log (1-x)-\frac{1}{5} \log (2-x)+\frac{1}{12} \log (3-x)-\frac{1}{120} \log (x+3) \]

[Out]

Log[1 - x]/8 - Log[2 - x]/5 + Log[3 - x]/12 - Log[3 + x]/120

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Rubi [A]  time = 0.0199904, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2058} \[ \frac{1}{8} \log (1-x)-\frac{1}{5} \log (2-x)+\frac{1}{12} \log (3-x)-\frac{1}{120} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[(-18 + 27*x - 7*x^2 - 3*x^3 + x^4)^(-1),x]

[Out]

Log[1 - x]/8 - Log[2 - x]/5 + Log[3 - x]/12 - Log[3 + x]/120

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx &=\int \left (\frac{1}{12 (-3+x)}-\frac{1}{5 (-2+x)}+\frac{1}{8 (-1+x)}-\frac{1}{120 (3+x)}\right ) \, dx\\ &=\frac{1}{8} \log (1-x)-\frac{1}{5} \log (2-x)+\frac{1}{12} \log (3-x)-\frac{1}{120} \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.0062116, size = 39, normalized size = 1. \[ \frac{1}{8} \log (1-x)-\frac{1}{5} \log (2-x)+\frac{1}{12} \log (3-x)-\frac{1}{120} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(-18 + 27*x - 7*x^2 - 3*x^3 + x^4)^(-1),x]

[Out]

Log[1 - x]/8 - Log[2 - x]/5 + Log[3 - x]/12 - Log[3 + x]/120

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Maple [A]  time = 0.007, size = 26, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( x-1 \right ) }{8}}+{\frac{\ln \left ( -3+x \right ) }{12}}-{\frac{\ln \left ( 3+x \right ) }{120}}-{\frac{\ln \left ( -2+x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4-3*x^3-7*x^2+27*x-18),x)

[Out]

1/8*ln(x-1)+1/12*ln(-3+x)-1/120*ln(3+x)-1/5*ln(-2+x)

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Maxima [A]  time = 0.974703, size = 34, normalized size = 0.87 \begin{align*} -\frac{1}{120} \, \log \left (x + 3\right ) + \frac{1}{8} \, \log \left (x - 1\right ) - \frac{1}{5} \, \log \left (x - 2\right ) + \frac{1}{12} \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="maxima")

[Out]

-1/120*log(x + 3) + 1/8*log(x - 1) - 1/5*log(x - 2) + 1/12*log(x - 3)

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Fricas [A]  time = 1.26296, size = 96, normalized size = 2.46 \begin{align*} -\frac{1}{120} \, \log \left (x + 3\right ) + \frac{1}{8} \, \log \left (x - 1\right ) - \frac{1}{5} \, \log \left (x - 2\right ) + \frac{1}{12} \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="fricas")

[Out]

-1/120*log(x + 3) + 1/8*log(x - 1) - 1/5*log(x - 2) + 1/12*log(x - 3)

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Sympy [A]  time = 0.210924, size = 26, normalized size = 0.67 \begin{align*} \frac{\log{\left (x - 3 \right )}}{12} - \frac{\log{\left (x - 2 \right )}}{5} + \frac{\log{\left (x - 1 \right )}}{8} - \frac{\log{\left (x + 3 \right )}}{120} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4-3*x**3-7*x**2+27*x-18),x)

[Out]

log(x - 3)/12 - log(x - 2)/5 + log(x - 1)/8 - log(x + 3)/120

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Giac [A]  time = 1.1608, size = 39, normalized size = 1. \begin{align*} -\frac{1}{120} \, \log \left ({\left | x + 3 \right |}\right ) + \frac{1}{8} \, \log \left ({\left | x - 1 \right |}\right ) - \frac{1}{5} \, \log \left ({\left | x - 2 \right |}\right ) + \frac{1}{12} \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="giac")

[Out]

-1/120*log(abs(x + 3)) + 1/8*log(abs(x - 1)) - 1/5*log(abs(x - 2)) + 1/12*log(abs(x - 3))