3.1369 \(\int \frac{1}{x^2 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=85 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-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 = 0.507893, antiderivative size = 85, 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 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 + x^6)),x]

[Out]

-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 = 106.733, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(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 - a
tan(x)/3 - atan(2*x - sqrt(3))/6 - atan(2*x + sqrt(3))/6 - 1/x

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Mathematica [A]  time = 0.0415165, size = 82, normalized size = 0.96 \[ -\frac{\sqrt{3} x \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} x \log \left (x^2+\sqrt{3} x+1\right )-2 x \tan ^{-1}\left (\sqrt{3}-2 x\right )+4 x \tan ^{-1}(x)+2 x \tan ^{-1}\left (2 x+\sqrt{3}\right )+12}{12 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 + x^6)),x]

[Out]

-(12 - 2*x*ArcTan[Sqrt[3] - 2*x] + 4*x*ArcTan[x] + 2*x*ArcTan[Sqrt[3] + 2*x] + S
qrt[3]*x*Log[1 - Sqrt[3]*x + x^2] - Sqrt[3]*x*Log[1 + Sqrt[3]*x + x^2])/(12*x)

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Maple [A]  time = 0.02, size = 66, normalized size = 0.8 \[ -{x}^{-1}-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^6+1),x)

[Out]

-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 = 1.59405, size = 88, normalized size = 1.04 \[ \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) - \frac{1}{x} - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^2),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/x - 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 = 0.234086, size = 139, normalized size = 1.64 \[ \frac{\sqrt{3} x \log \left (x^{2} + \sqrt{3} x + 1\right ) - \sqrt{3} x \log \left (x^{2} - \sqrt{3} x + 1\right ) - 4 \, x \arctan \left (x\right ) + 4 \, x \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + 4 \, x \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - 12}{12 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^2),x, algorithm="fricas")

[Out]

1/12*(sqrt(3)*x*log(x^2 + sqrt(3)*x + 1) - sqrt(3)*x*log(x^2 - sqrt(3)*x + 1) -
4*x*arctan(x) + 4*x*arctan(1/(2*x + sqrt(3) + 2*sqrt(x^2 + sqrt(3)*x + 1))) + 4*
x*arctan(1/(2*x - sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1))) - 12)/x

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Sympy [A]  time = 0.7333, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(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 - a
tan(x)/3 - atan(2*x - sqrt(3))/6 - atan(2*x + sqrt(3))/6 - 1/x

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 1\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^2),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 1)*x^2), x)