3.1485 \(\int \frac{1}{x^4 \left (1-x^8\right )} \, dx\)

Optimal. Leaf size=104 \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]

[Out]

-1/(3*x^3) + 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 = 0.128195, 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 \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 - x^8)),x]

[Out]

-1/(3*x^3) + 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 = 18.3225, size = 90, normalized size = 0.87 \[ \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{16} + \frac{\operatorname{atan}{\left (x \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{8} + \frac{\operatorname{atanh}{\left (x \right )}}{4} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-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/(3*x**3)

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Mathematica [A]  time = 0.0762081, size = 104, normalized size = 1. \[ \frac{1}{48} \left (-\frac{16}{x^3}+3 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-3 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-6 \log (1-x)+6 \log (x+1)+12 \tan ^{-1}(x)+6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )-6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 - x^8)),x]

[Out]

(-16/x^3 + 12*ArcTan[x] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 6*Sqrt[2]*ArcTan[1 +
Sqrt[2]*x] - 6*Log[1 - x] + 6*Log[1 + x] + 3*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] -
3*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/48

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Maple [A]  time = 0.016, size = 79, normalized size = 0.8 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{8}}-{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }-{\frac{1}{3\,{x}^{3}}}+{\frac{\ln \left ( 1+x \right ) }{8}}+{\frac{\arctan \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-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)))-1/3/x^3+1/8*ln(1+x)+1/4*arcta
n(x)

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Maxima [A]  time = 1.59396, size = 126, normalized size = 1.21 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{3 \, x^{3}} + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^4),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) - 1/3/x^3 + 1/4*arctan(x) + 1/8*log(x + 1) - 1/8*log(x - 1)

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Fricas [A]  time = 0.238078, size = 194, normalized size = 1.87 \[ \frac{\sqrt{2}{\left (6 \, \sqrt{2} x^{3} \arctan \left (x\right ) + 3 \, \sqrt{2} x^{3} \log \left (x + 1\right ) - 3 \, \sqrt{2} x^{3} \log \left (x - 1\right ) + 12 \, x^{3} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 12 \, x^{3} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - 3 \, x^{3} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 3 \, x^{3} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 8 \, \sqrt{2}\right )}}{48 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(2)*(6*sqrt(2)*x^3*arctan(x) + 3*sqrt(2)*x^3*log(x + 1) - 3*sqrt(2)*x^3
*log(x - 1) + 12*x^3*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x + 1) + 1
)) + 12*x^3*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) - 3*x^
3*log(x^2 + sqrt(2)*x + 1) + 3*x^3*log(x^2 - sqrt(2)*x + 1) - 8*sqrt(2))/x^3

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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**4/(-x**8+1),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24, size = 128, normalized size = 1.23 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{3 \, x^{3}} + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^4),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) - 1/3/x^3 + 1/4*arctan(x) + 1/8*ln(abs(x + 1)) - 1/8*ln(abs(x -
1))