3.106 \(\int \frac{1}{x^3-x^4-x^5+x^6} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{2 x^2}+\frac{1}{2 (1-x)}-\frac{1}{x}-\frac{7}{4} \log (1-x)+2 \log (x)-\frac{1}{4} \log (x+1) \]

[Out]

1/(2*(1 - x)) - 1/(2*x^2) - x^(-1) - (7*Log[1 - x])/4 + 2*Log[x] - Log[1 + x]/4

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Rubi [A]  time = 0.0224983, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2058} \[ -\frac{1}{2 x^2}+\frac{1}{2 (1-x)}-\frac{1}{x}-\frac{7}{4} \log (1-x)+2 \log (x)-\frac{1}{4} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(x^3 - x^4 - x^5 + x^6)^(-1),x]

[Out]

1/(2*(1 - x)) - 1/(2*x^2) - x^(-1) - (7*Log[1 - x])/4 + 2*Log[x] - Log[1 + x]/4

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}{x^3-x^4-x^5+x^6} \, dx &=\int \left (\frac{1}{2 (-1+x)^2}-\frac{7}{4 (-1+x)}+\frac{1}{x^3}+\frac{1}{x^2}+\frac{2}{x}-\frac{1}{4 (1+x)}\right ) \, dx\\ &=\frac{1}{2 (1-x)}-\frac{1}{2 x^2}-\frac{1}{x}-\frac{7}{4} \log (1-x)+2 \log (x)-\frac{1}{4} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0155976, size = 40, normalized size = 0.87 \[ \frac{1}{4} \left (-\frac{2}{x^2}-\frac{2}{x-1}-\frac{4}{x}-7 \log (1-x)+8 \log (x)-\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3 - x^4 - x^5 + x^6)^(-1),x]

[Out]

(-2/(-1 + x) - 2/x^2 - 4/x - 7*Log[1 - x] + 8*Log[x] - Log[1 + x])/4

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Maple [A]  time = 0.008, size = 35, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{x}^{-1}+2\,\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{1}{2\,x-2}}-{\frac{7\,\ln \left ( -1+x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6-x^5-x^4+x^3),x)

[Out]

-1/2/x^2-1/x+2*ln(x)-1/4*ln(1+x)-1/2/(-1+x)-7/4*ln(-1+x)

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Maxima [A]  time = 0.930141, size = 54, normalized size = 1.17 \begin{align*} -\frac{3 \, x^{2} - x - 1}{2 \,{\left (x^{3} - x^{2}\right )}} - \frac{1}{4} \, \log \left (x + 1\right ) - \frac{7}{4} \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-x^5-x^4+x^3),x, algorithm="maxima")

[Out]

-1/2*(3*x^2 - x - 1)/(x^3 - x^2) - 1/4*log(x + 1) - 7/4*log(x - 1) + 2*log(x)

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Fricas [A]  time = 2.11719, size = 150, normalized size = 3.26 \begin{align*} -\frac{6 \, x^{2} +{\left (x^{3} - x^{2}\right )} \log \left (x + 1\right ) + 7 \,{\left (x^{3} - x^{2}\right )} \log \left (x - 1\right ) - 8 \,{\left (x^{3} - x^{2}\right )} \log \left (x\right ) - 2 \, x - 2}{4 \,{\left (x^{3} - x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-x^5-x^4+x^3),x, algorithm="fricas")

[Out]

-1/4*(6*x^2 + (x^3 - x^2)*log(x + 1) + 7*(x^3 - x^2)*log(x - 1) - 8*(x^3 - x^2)*log(x) - 2*x - 2)/(x^3 - x^2)

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Sympy [A]  time = 0.14559, size = 37, normalized size = 0.8 \begin{align*} 2 \log{\left (x \right )} - \frac{7 \log{\left (x - 1 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{4} - \frac{3 x^{2} - x - 1}{2 x^{3} - 2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**6-x**5-x**4+x**3),x)

[Out]

2*log(x) - 7*log(x - 1)/4 - log(x + 1)/4 - (3*x**2 - x - 1)/(2*x**3 - 2*x**2)

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Giac [A]  time = 1.05111, size = 54, normalized size = 1.17 \begin{align*} -\frac{3 \, x^{2} - x - 1}{2 \,{\left (x - 1\right )} x^{2}} - \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{7}{4} \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-x^5-x^4+x^3),x, algorithm="giac")

[Out]

-1/2*(3*x^2 - x - 1)/((x - 1)*x^2) - 1/4*log(abs(x + 1)) - 7/4*log(abs(x - 1)) + 2*log(abs(x))