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

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

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 0.89 \[ \frac {-3 x^2+x+1}{2 (x-1) x^2}-\frac {7}{4} \log (x-1)+2 \log (x)-\frac {1}{4} \log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x - 3*x^2)/(2*(-1 + x)*x^2) - (7*Log[-1 + x])/4 + 2*Log[x] - Log[1 + x]/4

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fricas [A]  time = 0.93, size = 65, normalized size = 1.41 \[ -\frac {6 \, x^{2} + {\left (x^{3} - x^{2}\right )} \log \left (x + 1\right ) + 7 \, {\left (x^{3} - x^{2}\right )} \log \left (x - 1\right ) - 8 \, {\left (x^{3} - x^{2}\right )} \log \relax (x) - 2 \, x - 2}{4 \, {\left (x^{3} - x^{2}\right )}} \]

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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giac [A]  time = 1.03, size = 40, normalized size = 0.87 \[ -\frac {3 \, x^{2} - x - 1}{2 \, {\left (x - 1\right )} x^{2}} - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {7}{4} \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]

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))

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maple [A]  time = 0.04, size = 35, normalized size = 0.76

method	result	size
default	\(-\frac {1}{2 x^{2}}-\frac {1}{x}+2 \ln \relax (x )-\frac {1}{2 \left (-1+x \right )}-\frac {7 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{4}\)	\(35\)
norman	\(\frac {\frac {1}{2}-\frac {3}{2} x^{2}+\frac {1}{2} x}{x^{2} \left (-1+x \right )}+2 \ln \relax (x )-\frac {7 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{4}\)	\(37\)
risch	\(\frac {\frac {1}{2}-\frac {3}{2} x^{2}+\frac {1}{2} x}{x^{2} \left (-1+x \right )}+2 \ln \relax (x )-\frac {7 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{4}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 40, normalized size = 0.87 \[ -\frac {3 \, x^{2} - x - 1}{2 \, {\left (x^{3} - x^{2}\right )}} - \frac {1}{4} \, \log \left (x + 1\right ) - \frac {7}{4} \, \log \left (x - 1\right ) + 2 \, \log \relax (x) \]

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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mupad [B]  time = 0.18, size = 40, normalized size = 0.87 \[ 2\,\ln \relax (x)-\frac {\ln \left (x+1\right )}{4}-\frac {7\,\ln \left (x-1\right )}{4}-\frac {-\frac {3\,x^2}{2}+\frac {x}{2}+\frac {1}{2}}{x^2-x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3 - x^4 - x^5 + x^6),x)

[Out]

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

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sympy [A]  time = 0.17, size = 37, normalized size = 0.80 \[ 2 \log {\relax (x )} - \frac {7 \log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} + \frac {- 3 x^{2} + x + 1}{2 x^{3} - 2 x^{2}} \]

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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