3.1375 \(\int \frac{1}{x^8 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=90 \[ -\frac{1}{7 x^7}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-1/(7*x^7) + x^(-1) - ArcTan[Sqrt[3] - 2*x]/6 + ArcTan[x]/3 + ArcTan[Sqrt[3] + 2
*x]/6 + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[
3])

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Rubi [A]  time = 0.510526, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{7 x^7}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 + x^6)),x]

[Out]

-1/(7*x^7) + x^(-1) - ArcTan[Sqrt[3] - 2*x]/6 + ArcTan[x]/3 + ArcTan[Sqrt[3] + 2
*x]/6 + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[
3])

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Rubi in Sympy [A]  time = 92.5218, size = 78, normalized size = 0.87 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} + \frac{1}{x} - \frac{1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + at
an(x)/3 + atan(2*x - sqrt(3))/6 + atan(2*x + sqrt(3))/6 + 1/x - 1/(7*x**7)

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Mathematica [A]  time = 0.0510462, size = 84, normalized size = 0.93 \[ \frac{1}{84} \left (-\frac{12}{x^7}+7 \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-7 \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+\frac{84}{x}-14 \tan ^{-1}\left (\sqrt{3}-2 x\right )+28 \tan ^{-1}(x)+14 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(1 + x^6)),x]

[Out]

(-12/x^7 + 84/x - 14*ArcTan[Sqrt[3] - 2*x] + 28*ArcTan[x] + 14*ArcTan[Sqrt[3] +
2*x] + 7*Sqrt[3]*Log[1 - Sqrt[3]*x + x^2] - 7*Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/
84

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Maple [A]  time = 0.025, size = 69, normalized size = 0.8 \[ -{\frac{1}{7\,{x}^{7}}}+{x}^{-1}+{\frac{\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^6+1),x)

[Out]

-1/7/x^7+1/x+1/3*arctan(x)+1/6*arctan(2*x-3^(1/2))+1/6*arctan(2*x+3^(1/2))+1/12*
ln(1+x^2-x*3^(1/2))*3^(1/2)-1/12*ln(1+x^2+x*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.58837, size = 97, normalized size = 1.08 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{7 \, x^{6} - 1}{7 \, x^{7}} + \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) +
1/7*(7*x^6 - 1)/x^7 + 1/6*arctan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1
/3*arctan(x)

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Fricas [A]  time = 0.235124, size = 161, normalized size = 1.79 \[ -\frac{7 \, \sqrt{3} x^{7} \log \left (x^{2} + \sqrt{3} x + 1\right ) - 7 \, \sqrt{3} x^{7} \log \left (x^{2} - \sqrt{3} x + 1\right ) - 28 \, x^{7} \arctan \left (x\right ) + 28 \, x^{7} \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + 28 \, x^{7} \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - 84 \, x^{6} + 12}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84*(7*sqrt(3)*x^7*log(x^2 + sqrt(3)*x + 1) - 7*sqrt(3)*x^7*log(x^2 - sqrt(3)*
x + 1) - 28*x^7*arctan(x) + 28*x^7*arctan(1/(2*x + sqrt(3) + 2*sqrt(x^2 + sqrt(3
)*x + 1))) + 28*x^7*arctan(1/(2*x - sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1))) - 84
*x^6 + 12)/x^7

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Sympy [A]  time = 0.878678, size = 80, normalized size = 0.89 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} + \frac{7 x^{6} - 1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(x**6+1),x)

[Out]

sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + at
an(x)/3 + atan(2*x - sqrt(3))/6 + atan(2*x + sqrt(3))/6 + (7*x**6 - 1)/(7*x**7)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 1\right )} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^6 + 1)*x^8), x)