3.298 \(\int \frac{1}{x^5 \left (1-2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0394494, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.1395, size = 29, normalized size = 0.78 \[ \frac{\log{\left (x^{4} \right )}}{2} - \frac{\log{\left (- x^{4} + 1 \right )}}{2} + \frac{1}{4 \left (- x^{4} + 1\right )} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8-2*x**4+1),x)

[Out]

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

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Mathematica [A]  time = 0.0207327, size = 35, normalized size = 0.95 \[ -\frac{1}{4 \left (x^4-1\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 54, normalized size = 1.5 \[ -{\frac{1}{-16+16\,x}}-{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{1}{16+16\,x}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{1}{4\,{x}^{4}}}+2\,\ln \left ( x \right ) +{\frac{1}{8\,{x}^{2}+8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8-2*x^4+1),x)

[Out]

-1/16/(-1+x)-1/2*ln(-1+x)+1/16/(1+x)-1/2*ln(1+x)-1/4/x^4+2*ln(x)+1/8/(x^2+1)-1/2
*ln(x^2+1)

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Maxima [A]  time = 0.765369, size = 47, normalized size = 1.27 \[ -\frac{2 \, x^{4} - 1}{4 \,{\left (x^{8} - x^{4}\right )}} - \frac{1}{2} \, \log \left (x^{4} - 1\right ) + \frac{1}{2} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250569, size = 68, normalized size = 1.84 \[ -\frac{2 \, x^{4} + 2 \,{\left (x^{8} - x^{4}\right )} \log \left (x^{4} - 1\right ) - 8 \,{\left (x^{8} - x^{4}\right )} \log \left (x\right ) - 1}{4 \,{\left (x^{8} - x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.426497, size = 29, normalized size = 0.78 \[ - \frac{2 x^{4} - 1}{4 x^{8} - 4 x^{4}} + 2 \log{\left (x \right )} - \frac{\log{\left (x^{4} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8-2*x**4+1),x)

[Out]

-(2*x**4 - 1)/(4*x**8 - 4*x**4) + 2*log(x) - log(x**4 - 1)/2

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GIAC/XCAS [A]  time = 0.268617, size = 49, normalized size = 1.32 \[ -\frac{2 \, x^{4} - 1}{4 \,{\left (x^{8} - x^{4}\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{4}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x^{4} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^5),x, algorithm="giac")

[Out]

-1/4*(2*x^4 - 1)/(x^8 - x^4) + 1/2*ln(x^4) - 1/2*ln(abs(x^4 - 1))