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

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

[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

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Rubi [A]  time = 0.0312722, antiderivative size = 53, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0197673, size = 51, normalized size = 0.96 \[ -\frac{1}{x^2}-\frac{1}{3 x^3}+\frac{1}{4} \log \left (x^2+1\right )-\frac{1}{2 (x-1)}-\frac{2}{x}-\frac{5}{2} \log (1-x)+2 \log (x) \]

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 + 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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Maple [A]  time = 0.01, size = 42, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}-{\frac{1}{3\,{x}^{3}}}-{x}^{-2}-2\,{x}^{-1}+2\,\ln \left ( x \right ) -{\frac{1}{2\,x-2}}-{\frac{5\,\ln \left ( -1+x \right ) }{2}} \end{align*}

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]

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

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Maxima [A]  time = 1.41625, size = 63, normalized size = 1.19 \begin{align*} -\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 \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.79574, size = 173, normalized size = 3.26 \begin{align*} -\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 \left (x\right ) - 8 \, x - 4}{12 \,{\left (x^{4} - x^{3}\right )}} \end{align*}

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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Sympy [A]  time = 0.160552, size = 46, normalized size = 0.87 \begin{align*} 2 \log{\left (x \right )} - \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}} \end{align*}

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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Giac [A]  time = 1.08327, size = 62, normalized size = 1.17 \begin{align*} -\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 ) \end{align*}

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))