3.50 \(\int \frac{1}{-1+x^8} \, dx\)

Optimal. Leaf size=97 \[ \frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(x) \]

[Out]

-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 = 0.0964736, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.429 \[ \frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^8)^(-1),x]

[Out]

-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 = 6.33561, size = 83, normalized size = 0.86 \[ \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{16} - \frac{\operatorname{atan}{\left (x \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{8} - \frac{\operatorname{atanh}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(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

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Mathematica [A]  time = 0.0362864, size = 98, normalized size = 1.01 \[ \frac{1}{16} \left (\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+2 \log (1-x)-2 \log (x+1)-4 \tan ^{-1}(x)+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^8)^(-1),x]

[Out]

(-4*ArcTan[x] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*x
] + 2*Log[1 - x] - 2*Log[1 + x] + Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - Sqrt[2]*Log
[1 + Sqrt[2]*x + x^2])/16

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Maple [A]  time = 0.001, size = 66, normalized size = 0.7 \[ -{\frac{{\it Artanh} \left ( x \right ) }{4}}-{\frac{\arctan \left ( x \right ) }{4}}-{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{8}}-{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }-{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8-1),x)

[Out]

-1/4*arctanh(x)-1/4*arctan(x)-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*arctan(1+x*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.56852, size = 119, normalized size = 1.23 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{4} \, \arctan \left (x\right ) - \frac{1}{8} \, \log \left (x + 1\right ) + \frac{1}{8} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.219247, size = 151, normalized size = 1.56 \[ -\frac{1}{16} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left (x\right ) + \sqrt{2} \log \left (x + 1\right ) - \sqrt{2} \log \left (x - 1\right ) - 4 \, \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - 4 \, \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + \log \left (x^{2} + \sqrt{2} x + 1\right ) - \log \left (x^{2} - \sqrt{2} x + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*sqrt(2)*(2*sqrt(2)*arctan(x) + sqrt(2)*log(x + 1) - sqrt(2)*log(x - 1) - 4
*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x + 1) + 1)) - 4*arctan(1/(sqr
t(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) + log(x^2 + sqrt(2)*x + 1) - lo
g(x^2 - sqrt(2)*x + 1))

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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**8-1),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.199278, size = 122, normalized size = 1.26 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{4} \, \arctan \left (x\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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