3.1373 \(\int \frac{1}{x^6 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=87 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-1/(5*x^5) + ArcTan[Sqrt[3] - 2*x]/6 - ArcTan[x]/3 - ArcTan[Sqrt[3] + 2*x]/6 + L
og[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 = 0.339371, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^6)),x]

[Out]

-1/(5*x^5) + ArcTan[Sqrt[3] - 2*x]/6 - ArcTan[x]/3 - ArcTan[Sqrt[3] + 2*x]/6 + L
og[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 = 58.2856, size = 75, normalized size = 0.86 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(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/(5*x**5)

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Mathematica [A]  time = 0.0493651, size = 79, normalized size = 0.91 \[ \frac{1}{60} \left (-\frac{12}{x^5}+5 \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-5 \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+10 \tan ^{-1}\left (\sqrt{3}-2 x\right )-20 \tan ^{-1}(x)-10 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + x^6)),x]

[Out]

(-12/x^5 + 10*ArcTan[Sqrt[3] - 2*x] - 20*ArcTan[x] - 10*ArcTan[Sqrt[3] + 2*x] +
5*Sqrt[3]*Log[1 - Sqrt[3]*x + x^2] - 5*Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/60

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Maple [A]  time = 0.01, size = 66, normalized size = 0.8 \[ -{\frac{1}{5\,{x}^{5}}}-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^6+1),x)

[Out]

-1/5/x^5-1/3*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 = 1.58665, size = 88, normalized size = 1.01 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) - \frac{1}{5 \, x^{5}} - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^6),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) -
1/5/x^5 - 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 = 0.233486, size = 154, normalized size = 1.77 \[ -\frac{5 \, \sqrt{3} x^{5} \log \left (x^{2} + \sqrt{3} x + 1\right ) - 5 \, \sqrt{3} x^{5} \log \left (x^{2} - \sqrt{3} x + 1\right ) + 20 \, x^{5} \arctan \left (x\right ) - 20 \, x^{5} \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) - 20 \, x^{5} \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) + 12}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(5*sqrt(3)*x^5*log(x^2 + sqrt(3)*x + 1) - 5*sqrt(3)*x^5*log(x^2 - sqrt(3)*
x + 1) + 20*x^5*arctan(x) - 20*x^5*arctan(1/(2*x + sqrt(3) + 2*sqrt(x^2 + sqrt(3
)*x + 1))) - 20*x^5*arctan(1/(2*x - sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1))) + 12
)/x^5

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Sympy [A]  time = 0.813958, size = 75, normalized size = 0.86 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(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/(5*x**5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^6 + 1)*x^6), x)