3.271 \(\int \frac {1}{x^4-2 x^5+2 x^6-2 x^7+x^8} \, dx\)

Optimal. Leaf size=53 \[ -\frac {1}{3 x^3}-\frac {1}{x^2}+\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{2 (1-x)}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x) \]

[Out]

1/2/(1-x)-1/3/x^3-1/x^2-2/x-5/2*ln(1-x)+2*ln(x)+1/4*ln(x^2+1)

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2058, 260} \[ -\frac {1}{x^2}-\frac {1}{3 x^3}+\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{2 (1-x)}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8)^(-1),x]

[Out]

1/(2*(1 - x)) - 1/(3*x^3) - x^(-2) - 2/x - (5*Log[1 - x])/2 + 2*Log[x] + Log[1 + x^2]/4

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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^4-2 x^5+2 x^6-2 x^7+x^8} \, dx &=\int \left (\frac {1}{2 (-1+x)^2}-\frac {5}{2 (-1+x)}+\frac {1}{x^4}+\frac {2}{x^3}+\frac {2}{x^2}+\frac {2}{x}+\frac {x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2 (1-x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{2 (1-x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{4} \log \left (1+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 0.96 \[ -\frac {1}{3 x^3}-\frac {1}{x^2}+\frac {1}{4} \log \left (x^2+1\right )-\frac {1}{2 (x-1)}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8)^(-1),x]

[Out]

-1/2*1/(-1 + x) - 1/(3*x^3) - x^(-2) - 2/x - (5*Log[1 - x])/2 + 2*Log[x] + Log[1 + x^2]/4

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fricas [A]  time = 0.41, size = 73, normalized size = 1.38 \[ -\frac {30 \, x^{3} - 12 \, x^{2} - 3 \, {\left (x^{4} - x^{3}\right )} \log \left (x^{2} + 1\right ) + 30 \, {\left (x^{4} - x^{3}\right )} \log \left (x - 1\right ) - 24 \, {\left (x^{4} - x^{3}\right )} \log \relax (x) - 8 \, x - 4}{12 \, {\left (x^{4} - x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-2*x^7+2*x^6-2*x^5+x^4),x, algorithm="fricas")

[Out]

-1/12*(30*x^3 - 12*x^2 - 3*(x^4 - x^3)*log(x^2 + 1) + 30*(x^4 - x^3)*log(x - 1) - 24*(x^4 - x^3)*log(x) - 8*x
- 4)/(x^4 - x^3)

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giac [A]  time = 1.12, size = 46, normalized size = 0.87 \[ -\frac {15 \, x^{3} - 6 \, x^{2} - 4 \, x - 2}{6 \, {\left (x - 1\right )} x^{3}} + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {5}{2} \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-2*x^7+2*x^6-2*x^5+x^4),x, algorithm="giac")

[Out]

-1/6*(15*x^3 - 6*x^2 - 4*x - 2)/((x - 1)*x^3) + 1/4*log(x^2 + 1) - 5/2*log(abs(x - 1)) + 2*log(abs(x))

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maple [A]  time = 0.01, size = 42, normalized size = 0.79 \[ 2 \ln \relax (x )-\frac {5 \ln \left (x -1\right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}-\frac {2}{x}-\frac {1}{x^{2}}-\frac {1}{3 x^{3}}-\frac {1}{2 \left (x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8-2*x^7+2*x^6-2*x^5+x^4),x)

[Out]

-1/2/(x-1)-5/2*ln(x-1)+1/4*ln(x^2+1)-1/3/x^3-1/x^2-2/x+2*ln(x)

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maxima [A]  time = 0.96, size = 47, normalized size = 0.89 \[ -\frac {15 \, x^{3} - 6 \, x^{2} - 4 \, x - 2}{6 \, {\left (x^{4} - x^{3}\right )}} + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {5}{2} \, \log \left (x - 1\right ) + 2 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-2*x^7+2*x^6-2*x^5+x^4),x, algorithm="maxima")

[Out]

-1/6*(15*x^3 - 6*x^2 - 4*x - 2)/(x^4 - x^3) + 1/4*log(x^2 + 1) - 5/2*log(x - 1) + 2*log(x)

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mupad [B]  time = 0.06, size = 45, normalized size = 0.85 \[ \frac {\ln \left (x^2+1\right )}{4}-\frac {5\,\ln \left (x-1\right )}{2}+2\,\ln \relax (x)-\frac {-\frac {5\,x^3}{2}+x^2+\frac {2\,x}{3}+\frac {1}{3}}{x^3-x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8),x)

[Out]

log(x^2 + 1)/4 - (5*log(x - 1))/2 + 2*log(x) - ((2*x)/3 + x^2 - (5*x^3)/2 + 1/3)/(x^3 - x^4)

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sympy [A]  time = 0.18, size = 46, normalized size = 0.87 \[ 2 \log {\relax (x )} - \frac {5 \log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {- 15 x^{3} + 6 x^{2} + 4 x + 2}{6 x^{4} - 6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**8-2*x**7+2*x**6-2*x**5+x**4),x)

[Out]

2*log(x) - 5*log(x - 1)/2 + log(x**2 + 1)/4 + (-15*x**3 + 6*x**2 + 4*x + 2)/(6*x**4 - 6*x**3)

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