3.335 \(\int \frac{1}{x^3 \left (1+x^4+x^8\right )} \, dx\)

Optimal. Leaf size=54 \[ -\frac{1}{2 x^2}+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-1/(2*x^2) + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x^2)/Sqrt[3
]]/(2*Sqrt[3])

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Rubi [A]  time = 0.0995506, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{1}{2 x^2}+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 + x^4 + x^8)),x]

[Out]

-1/(2*x^2) + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x^2)/Sqrt[3
]]/(2*Sqrt[3])

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Rubi in Sympy [A]  time = 15.7505, size = 53, normalized size = 0.98 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(x**8+x**4+1),x)

[Out]

sqrt(3)*atan(sqrt(3)*(-2*x**2/3 + 1/3))/6 - sqrt(3)*atan(sqrt(3)*(2*x**2/3 + 1/3
))/6 - 1/(2*x**2)

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Mathematica [C]  time = 0.0849388, size = 100, normalized size = 1.85 \[ \frac{1}{12} \left (-\frac{6}{x^2}+i \sqrt{3} \log \left (2 x^2-i \sqrt{3}-1\right )-i \sqrt{3} \log \left (2 x^2+i \sqrt{3}-1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 + x^4 + x^8)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 57, normalized size = 1.1 \[{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{2\,{x}^{2}}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8+x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.819949, size = 57, normalized size = 1.06 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.247279, size = 59, normalized size = 1.09 \[ -\frac{\sqrt{3}{\left (x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3} x^{2}\right ) + x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{6} + 2 \, x^{2}\right )}\right ) + \sqrt{3}\right )}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.440587, size = 53, normalized size = 0.98 \[ \frac{\sqrt{3} \left (- 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{2}}{3} \right )} - 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{6}}{3} + \frac{2 \sqrt{3} x^{2}}{3} \right )}\right )}{12} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**8+x**4+1),x)

[Out]

sqrt(3)*(-2*atan(sqrt(3)*x**2/3) - 2*atan(sqrt(3)*x**6/3 + 2*sqrt(3)*x**2/3))/12
- 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.311122, size = 57, normalized size = 1.06 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^3),x, algorithm="giac")

[Out]

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