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

Optimal. Leaf size=57 \[ -\frac{1}{2 x^2}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0798921, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{2 x^2}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - x^4 + x^8)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.1231, size = 49, normalized size = 0.86 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} - \frac{1}{2 x^{2}} \]

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

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Mathematica [A]  time = 0.0287988, size = 55, normalized size = 0.96 \[ \frac{1}{12} \left (-\frac{6}{x^2}-\sqrt{3} \log \left (-x^4+\sqrt{3} x^2-1\right )+\sqrt{3} \log \left (x^4+\sqrt{3} x^2+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6/x^2 - Sqrt[3]*Log[-1 + Sqrt[3]*x^2 - x^4] + Sqrt[3]*Log[1 + Sqrt[3]*x^2 + x^
4])/12

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Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8-x^4+1),x)

[Out]

-1/2/x^2-1/12*ln(1+x^4-x^2*3^(1/2))*3^(1/2)+1/12*ln(1+x^4+x^2*3^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \, x^{2}} - \int \frac{{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/x^2 - integrate((x^4 - 1)*x/(x^8 - x^4 + 1), x)

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Fricas [A]  time = 0.275455, size = 77, normalized size = 1.35 \[ \frac{\sqrt{3}{\left (x^{2} \log \left (\frac{6 \, x^{6} + 6 \, x^{2} + \sqrt{3}{\left (x^{8} + 5 \, x^{4} + 1\right )}}{x^{8} - x^{4} + 1}\right ) - 2 \, \sqrt{3}\right )}}{12 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*(x^2*log((6*x^6 + 6*x^2 + sqrt(3)*(x^8 + 5*x^4 + 1))/(x^8 - x^4 + 1
)) - 2*sqrt(3))/x^2

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Sympy [A]  time = 0.40514, size = 49, normalized size = 0.86 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.336939, size = 348, normalized size = 6.11 \[ \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2)))
+ 1/48*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2
))) + 1/48*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqr
t(2))) + 1/48*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) -
sqrt(2))) + 1/96*(sqrt(6) + 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) -
1/96*(sqrt(6) + 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/96*(sqrt
(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/96*(sqrt(6) - 3*sqr
t(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/2/x^2