3.1486 \(\int \frac{1}{x^6 \left (1-x^8\right )} \, dx\)

Optimal. Leaf size=104 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]

[Out]

-1/(5*x^5) - ArcTan[x]/4 - ArcTan[1 - Sqrt[2]*x]/(4*Sqrt[2]) + ArcTan[1 + Sqrt[2
]*x]/(4*Sqrt[2]) + ArcTanh[x]/4 + Log[1 - Sqrt[2]*x + x^2]/(8*Sqrt[2]) - Log[1 +
Sqrt[2]*x + x^2]/(8*Sqrt[2])

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Rubi [A]  time = 0.140047, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 - x^8)),x]

[Out]

-1/(5*x^5) - ArcTan[x]/4 - ArcTan[1 - Sqrt[2]*x]/(4*Sqrt[2]) + ArcTan[1 + Sqrt[2
]*x]/(4*Sqrt[2]) + ArcTanh[x]/4 + Log[1 - Sqrt[2]*x + x^2]/(8*Sqrt[2]) - Log[1 +
Sqrt[2]*x + x^2]/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 20.9346, size = 90, normalized size = 0.87 \[ \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{16} - \frac{\operatorname{atan}{\left (x \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{8} + \frac{\operatorname{atanh}{\left (x \right )}}{4} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-x**8+1),x)

[Out]

sqrt(2)*log(x**2 - sqrt(2)*x + 1)/16 - sqrt(2)*log(x**2 + sqrt(2)*x + 1)/16 - at
an(x)/4 + sqrt(2)*atan(sqrt(2)*x - 1)/8 + sqrt(2)*atan(sqrt(2)*x + 1)/8 + atanh(
x)/4 - 1/(5*x**5)

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Mathematica [A]  time = 0.0783238, size = 104, normalized size = 1. \[ \frac{1}{80} \left (-\frac{16}{x^5}+5 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-5 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-10 \log (1-x)+10 \log (x+1)-20 \tan ^{-1}(x)-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 - x^8)),x]

[Out]

(-16/x^5 - 20*ArcTan[x] - 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 10*Sqrt[2]*ArcTan[1
+ Sqrt[2]*x] - 10*Log[1 - x] + 10*Log[1 + x] + 5*Sqrt[2]*Log[1 - Sqrt[2]*x + x^
2] - 5*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/80

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Maple [A]  time = 0.016, size = 79, normalized size = 0.8 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}+{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{8}}+{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{8}}+{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{1}{5\,{x}^{5}}}-{\frac{\arctan \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-x^8+1),x)

[Out]

-1/8*ln(-1+x)+1/8*arctan(1+x*2^(1/2))*2^(1/2)+1/8*arctan(x*2^(1/2)-1)*2^(1/2)+1/
16*2^(1/2)*ln((1+x^2-x*2^(1/2))/(1+x^2+x*2^(1/2)))+1/8*ln(1+x)-1/5/x^5-1/4*arcta
n(x)

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Maxima [A]  time = 1.59259, size = 126, normalized size = 1.21 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{5 \, x^{5}} - \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^6),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*sqrt(2)
*(2*x - sqrt(2))) - 1/16*sqrt(2)*log(x^2 + sqrt(2)*x + 1) + 1/16*sqrt(2)*log(x^2
- sqrt(2)*x + 1) - 1/5/x^5 - 1/4*arctan(x) + 1/8*log(x + 1) - 1/8*log(x - 1)

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Fricas [A]  time = 0.235309, size = 194, normalized size = 1.87 \[ -\frac{\sqrt{2}{\left (10 \, \sqrt{2} x^{5} \arctan \left (x\right ) - 5 \, \sqrt{2} x^{5} \log \left (x + 1\right ) + 5 \, \sqrt{2} x^{5} \log \left (x - 1\right ) + 20 \, x^{5} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 20 \, x^{5} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + 5 \, x^{5} \log \left (x^{2} + \sqrt{2} x + 1\right ) - 5 \, x^{5} \log \left (x^{2} - \sqrt{2} x + 1\right ) + 8 \, \sqrt{2}\right )}}{80 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^6),x, algorithm="fricas")

[Out]

-1/80*sqrt(2)*(10*sqrt(2)*x^5*arctan(x) - 5*sqrt(2)*x^5*log(x + 1) + 5*sqrt(2)*x
^5*log(x - 1) + 20*x^5*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x + 1) +
1)) + 20*x^5*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) + 5*
x^5*log(x^2 + sqrt(2)*x + 1) - 5*x^5*log(x^2 - sqrt(2)*x + 1) + 8*sqrt(2))/x^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(-x**8+1),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236809, size = 128, normalized size = 1.23 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{5 \, x^{5}} - \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^6),x, algorithm="giac")

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*sqrt(2)
*(2*x - sqrt(2))) - 1/16*sqrt(2)*ln(x^2 + sqrt(2)*x + 1) + 1/16*sqrt(2)*ln(x^2 -
sqrt(2)*x + 1) - 1/5/x^5 - 1/4*arctan(x) + 1/8*ln(abs(x + 1)) - 1/8*ln(abs(x -
1))