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

Optimal. Leaf size=268 \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]

[Out]

-ArcTan[(3 - (1 + 4/x)^2)/(6*Sqrt[7])]/(12*Sqrt[7]) - (Sqrt[(109 + 67*Sqrt[29])/
1218]*ArcTan[(2 - Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (
Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 + Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*
(-1 + Sqrt[29])]])/12 - (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] - Sqrt[6
*(1 + Sqrt[29])]*(1 + 4/x) + (1 + 4/x)^2])/24 + (Sqrt[(-109 + 67*Sqrt[29])/1218]
*Log[3*Sqrt[29] + Sqrt[6*(1 + Sqrt[29])]*(1 + 4/x) + (1 + 4/x)^2])/24

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Rubi [A]  time = 0.868381, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(3 - (1 + 4/x)^2)/(6*Sqrt[7])]/(12*Sqrt[7]) - (Sqrt[(109 + 67*Sqrt[29])/
1218]*ArcTan[(2 - Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (
Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 + Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*
(-1 + Sqrt[29])]])/12 - (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] - Sqrt[6
*(1 + Sqrt[29])]*(1 + 4/x) + (1 + 4/x)^2])/24 + (Sqrt[(-109 + 67*Sqrt[29])/1218]
*Log[3*Sqrt[29] + Sqrt[6*(1 + Sqrt[29])]*(1 + 4/x) + (1 + 4/x)^2])/24

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Rubi in Sympy [A]  time = 65.2582, size = 382, normalized size = 1.43 \[ \frac{\sqrt{174} \left (- 96 \sqrt{29} + 32\right ) \log{\left (\sqrt{6} \left (- \frac{1}{16} - \frac{1}{4 x}\right ) \sqrt{1 + \sqrt{29}} + \left (\frac{1}{4} + \frac{1}{x}\right )^{2} + \frac{3 \sqrt{29}}{16} \right )}}{133632 \sqrt{1 + \sqrt{29}}} - \frac{\sqrt{174} \left (- 96 \sqrt{29} + 32\right ) \log{\left (\sqrt{6} \left (\frac{1}{16} + \frac{1}{4 x}\right ) \sqrt{1 + \sqrt{29}} + \left (\frac{1}{4} + \frac{1}{x}\right )^{2} + \frac{3 \sqrt{29}}{16} \right )}}{133632 \sqrt{1 + \sqrt{29}}} + \frac{\sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{8 \left (\frac{1}{4} + \frac{1}{x}\right )^{2}}{21} - \frac{1}{14}\right ) \right )}}{84} - \frac{\sqrt{29} \left (16 \sqrt{6} \sqrt{1 + \sqrt{29}} - \frac{\sqrt{6} \sqrt{1 + \sqrt{29}} \left (- 192 \sqrt{29} + 64\right )}{8}\right ) \operatorname{atan}{\left (\frac{\sqrt{6} \left (\frac{1}{3} + \frac{\sqrt{6 + 6 \sqrt{29}}}{6} + \frac{4}{3 x}\right )}{\sqrt{-1 + \sqrt{29}}} \right )}}{16704 \sqrt{-1 + \sqrt{29}} \sqrt{1 + \sqrt{29}}} - \frac{\sqrt{29} \left (16 \sqrt{6} \sqrt{1 + \sqrt{29}} - \frac{\sqrt{6} \sqrt{1 + \sqrt{29}} \left (- 192 \sqrt{29} + 64\right )}{8}\right ) \operatorname{atan}{\left (\frac{\sqrt{6} \left (- \frac{\sqrt{6 + 6 \sqrt{29}}}{6} + \frac{1}{3} + \frac{4}{3 x}\right )}{\sqrt{-1 + \sqrt{29}}} \right )}}{16704 \sqrt{-1 + \sqrt{29}} \sqrt{1 + \sqrt{29}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(174)*(-96*sqrt(29) + 32)*log(sqrt(6)*(-1/16 - 1/(4*x))*sqrt(1 + sqrt(29)) +
(1/4 + 1/x)**2 + 3*sqrt(29)/16)/(133632*sqrt(1 + sqrt(29))) - sqrt(174)*(-96*sq
rt(29) + 32)*log(sqrt(6)*(1/16 + 1/(4*x))*sqrt(1 + sqrt(29)) + (1/4 + 1/x)**2 +
3*sqrt(29)/16)/(133632*sqrt(1 + sqrt(29))) + sqrt(7)*atan(sqrt(7)*(8*(1/4 + 1/x)
**2/21 - 1/14))/84 - sqrt(29)*(16*sqrt(6)*sqrt(1 + sqrt(29)) - sqrt(6)*sqrt(1 +
sqrt(29))*(-192*sqrt(29) + 64)/8)*atan(sqrt(6)*(1/3 + sqrt(6 + 6*sqrt(29))/6 + 4
/(3*x))/sqrt(-1 + sqrt(29)))/(16704*sqrt(-1 + sqrt(29))*sqrt(1 + sqrt(29))) - sq
rt(29)*(16*sqrt(6)*sqrt(1 + sqrt(29)) - sqrt(6)*sqrt(1 + sqrt(29))*(-192*sqrt(29
) + 64)/8)*atan(sqrt(6)*(-sqrt(6 + 6*sqrt(29))/6 + 1/3 + 4/(3*x))/sqrt(-1 + sqrt
(29)))/(16704*sqrt(-1 + sqrt(29))*sqrt(1 + sqrt(29)))

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Mathematica [C]  time = 0.0137583, size = 45, normalized size = 0.17 \[ \text{RootSum}\left [8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,\frac{\log (x-\text{$\#$1})}{32 \text{$\#$1}^3-3 \text{$\#$1}^2+8}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

RootSum[8 + 8*#1 - #1^3 + 8*#1^4 & , Log[x - #1]/(8 - 3*#1^2 + 32*#1^3) & ]

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Maple [C]  time = 0.007, size = 41, normalized size = 0.2 \[ \sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+8\,{\it \_Z}+8 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(8*x^4-x^3+8*x+8),x)

[Out]

sum(1/(32*_R^3-3*_R^2+8)*ln(x-_R),_R=RootOf(8*_Z^4-_Z^3+8*_Z+8))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.04107, size = 41, normalized size = 0.15 \[ \operatorname{RootSum}{\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log{\left (\frac{35914274424 t^{3}}{2109763} - \frac{1504863360 t^{2}}{2109763} + \frac{102851343 t}{2109763} + x + \frac{6055613}{16878104} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(66298176*_t**4 + 74088*_t**2 + 4095*_t + 64, Lambda(_t, _t*log(359142744
24*_t**3/2109763 - 1504863360*_t**2/2109763 + 102851343*_t/2109763 + x + 6055613
/16878104)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{8 \, x^{4} - x^{3} + 8 \, x + 8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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