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

Optimal. Leaf size=89 \[ -\frac{1}{2 x^2}+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )-\frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

[Out]

-1/(2*x^2) + (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 - ((3
+ Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

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Rubi [A]  time = 0.167353, antiderivative size = 89, 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 \[ -\frac{1}{2 x^2}+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )-\frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x^2) + (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 - ((3
+ Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

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Rubi in Sympy [A]  time = 18.0556, size = 105, normalized size = 1.18 \[ - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(1/2 + 3*sqrt(5)/10)*atan(sqrt(2)*x**2/sqrt(-sqrt(5) + 3))/(2*sqrt(-sqr
t(5) + 3)) - sqrt(2)*(-3*sqrt(5)/10 + 1/2)*atan(sqrt(2)*x**2/sqrt(sqrt(5) + 3))/
(2*sqrt(sqrt(5) + 3)) - 1/(2*x**2)

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Mathematica [C]  time = 0.0270219, size = 65, normalized size = 0.73 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^6+3 \text{$\#$1}^2}\&\right ]-\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x^2) - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3
*#1^2 + 2*#1^6) & ]/4

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Maple [B]  time = 0.006, size = 117, normalized size = 1.3 \[ -{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{{x}^{2}}{2\,\sqrt{5}+2}} \right ) }+{\frac{3\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2\,\sqrt{5}+2}} \right ) }-{\frac{3\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{1}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8+3*x^4+1),x)

[Out]

-1/(2*5^(1/2)+2)*arctan(4*x^2/(2*5^(1/2)+2))+3/5*5^(1/2)/(2*5^(1/2)+2)*arctan(4*
x^2/(2*5^(1/2)+2))-3/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-1/(-2
+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-1/2/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \, x^{2}} - \int \frac{{\left (x^{4} + 3\right )} x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/x^2 - integrate((x^4 + 3)*x/(x^8 + 3*x^4 + 1), x)

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Fricas [A]  time = 0.288127, size = 217, normalized size = 2.44 \[ -\frac{2 \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}} x^{2} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}}{\left (3 \, \sqrt{5} + 7\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} + 5\right )}}\right )}}\right ) + 2 \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}} x^{2} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}}{\left (3 \, \sqrt{5} - 7\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} - 5\right )}}\right )}}\right ) + 5}{10 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(2*sqrt(sqrt(5)*(9*sqrt(5) - 20))*x^2*arctan(1/2*sqrt(sqrt(5)*(9*sqrt(5) -
20))*(3*sqrt(5) + 7)/(sqrt(5)*x^2 + sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2
*x^4 + 3) + 5)))) + 2*sqrt(sqrt(5)*(9*sqrt(5) + 20))*x^2*arctan(1/2*sqrt(sqrt(5)
*(9*sqrt(5) + 20))*(3*sqrt(5) - 7)/(sqrt(5)*x^2 + sqrt(5)*sqrt(1/10)*sqrt(sqrt(5
)*(sqrt(5)*(2*x^4 + 3) - 5)))) + 5)/x^2

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Sympy [A]  time = 0.723562, size = 56, normalized size = 0.63 \[ - 2 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} + 2 \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(sqrt(5)/10 + 1/4)*atan(2*x**2/(-1 + sqrt(5))) + 2*(-sqrt(5)/10 + 1/4)*atan(2
*x**2/(1 + sqrt(5))) - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.280451, size = 92, normalized size = 1.03 \[ -\frac{1}{20} \,{\left (x^{4}{\left (\sqrt{5} - 5\right )} + 3 \, \sqrt{5} - 15\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) - \frac{1}{20} \,{\left (x^{4}{\left (\sqrt{5} + 5\right )} + 3 \, \sqrt{5} + 15\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) - \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(x^4*(sqrt(5) - 5) + 3*sqrt(5) - 15)*arctan(2*x^2/(sqrt(5) + 1)) - 1/20*(x
^4*(sqrt(5) + 5) + 3*sqrt(5) + 15)*arctan(2*x^2/(sqrt(5) - 1)) - 1/2/x^2