3.336 \(\int \frac{1}{x^5 \left (1+x^4+x^8\right )} \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{4 x^4}-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{8} \log \left (x^8+x^4+1\right )-\log (x) \]

[Out]

-1/(4*x^4) - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[x] + Log[1 + x^4 + x^
8]/8

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Rubi [A]  time = 0.10025, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{4 x^4}-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{8} \log \left (x^8+x^4+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + x^4 + x^8)),x]

[Out]

-1/(4*x^4) - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[x] + Log[1 + x^4 + x^
8]/8

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Rubi in Sympy [A]  time = 14.3619, size = 48, normalized size = 1. \[ - \frac{\log{\left (x^{4} \right )}}{4} + \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} + \frac{1}{3}\right ) \right )}}{12} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8+x**4+1),x)

[Out]

-log(x**4)/4 + log(x**8 + x**4 + 1)/8 - sqrt(3)*atan(sqrt(3)*(2*x**4/3 + 1/3))/1
2 - 1/(4*x**4)

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Mathematica [C]  time = 0.253038, size = 141, normalized size = 2.94 \[ \frac{1}{24} \left (-\frac{6}{x^4}+\sqrt{3} \left (\sqrt{3}+i\right ) \log \left (x^2-\frac{i \sqrt{3}}{2}-\frac{1}{2}\right )+\sqrt{3} \left (\sqrt{3}-i\right ) \log \left (x^2+\frac{1}{2} i \left (\sqrt{3}+i\right )\right )+3 \log \left (x^2-x+1\right )+3 \log \left (x^2+x+1\right )-24 \log (x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 + x^4 + x^8)),x]

[Out]

(-6/x^4 + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt
[3]] - 24*Log[x] + Sqrt[3]*(I + Sqrt[3])*Log[-1/2 - (I/2)*Sqrt[3] + x^2] + Sqrt[
3]*(-I + Sqrt[3])*Log[(I/2)*(I + Sqrt[3]) + x^2] + 3*Log[1 - x + x^2] + 3*Log[1
+ x + x^2])/24

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Maple [B]  time = 0.012, size = 94, normalized size = 2. \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{4}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8+x^4+1),x)

[Out]

1/8*ln(x^2+x+1)-1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/8*ln(x^4-x^2+1)-1/12*
3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))-1/4/x^4-ln(x)+1/8*ln(x^2-x+1)+1/12*3^(1/2)
*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 0.822466, size = 55, normalized size = 1.15 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) - \frac{1}{4 \, x^{4}} + \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) - \frac{1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254229, size = 78, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (\sqrt{3} x^{4} \log \left (x^{8} + x^{4} + 1\right ) - 8 \, \sqrt{3} x^{4} \log \left (x\right ) - 2 \, x^{4} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) - 2 \, \sqrt{3}\right )}}{24 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^5),x, algorithm="fricas")

[Out]

1/24*sqrt(3)*(sqrt(3)*x^4*log(x^8 + x^4 + 1) - 8*sqrt(3)*x^4*log(x) - 2*x^4*arct
an(1/3*sqrt(3)*(2*x^4 + 1)) - 2*sqrt(3))/x^4

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Sympy [A]  time = 0.522131, size = 48, normalized size = 1. \[ - \log{\left (x \right )} + \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8+x**4+1),x)

[Out]

-log(x) + log(x**8 + x**4 + 1)/8 - sqrt(3)*atan(2*sqrt(3)*x**4/3 + sqrt(3)/3)/12
- 1/(4*x**4)

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GIAC/XCAS [A]  time = 0.309429, size = 62, normalized size = 1.29 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{x^{4} - 1}{4 \, x^{4}} + \frac{1}{8} \,{\rm ln}\left (x^{8} + x^{4} + 1\right ) - \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^5),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) + 1/4*(x^4 - 1)/x^4 + 1/8*ln(x^8 +
x^4 + 1) - 1/4*ln(x^4)