3.403 \(\int \frac{1}{x^6 \left (1-3 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=173 \[ -\frac{1}{5 x^5}-\frac{3}{x}+\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

[Out]

-1/(5*x^5) - 3/x + ((2889 - 1292*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x
])/(2*Sqrt[5]) - ((2889 + 1292*Sqrt[5])^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])
/(2*Sqrt[5]) - ((2889 - 1292*Sqrt[5])^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/
(2*Sqrt[5]) + ((2889 + 1292*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(
2*Sqrt[5])

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Rubi [A]  time = 0.371805, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{1}{5 x^5}-\frac{3}{x}+\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(5*x^5) - 3/x + ((2889 - 1292*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x
])/(2*Sqrt[5]) - ((2889 + 1292*Sqrt[5])^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])
/(2*Sqrt[5]) - ((2889 - 1292*Sqrt[5])^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/
(2*Sqrt[5]) + ((2889 + 1292*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(
2*Sqrt[5])

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Rubi in Sympy [A]  time = 37.4362, size = 199, normalized size = 1.15 \[ - \frac{\sqrt [4]{2} \left (\frac{15}{2} + \frac{7 \sqrt{5}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \sqrt [4]{- \sqrt{5} + 3}} - \frac{\sqrt [4]{2} \left (- \frac{7 \sqrt{5}}{2} + \frac{15}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \sqrt [4]{\sqrt{5} + 3}} + \frac{\sqrt [4]{2} \left (\frac{15}{2} + \frac{7 \sqrt{5}}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \sqrt [4]{- \sqrt{5} + 3}} + \frac{\sqrt [4]{2} \left (- \frac{7 \sqrt{5}}{2} + \frac{15}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \sqrt [4]{\sqrt{5} + 3}} - \frac{3}{x} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**8-3*x**4+1),x)

[Out]

-2**(1/4)*(15/2 + 7*sqrt(5)/2)*atan(2**(1/4)*x/(-sqrt(5) + 3)**(1/4))/(10*(-sqrt
(5) + 3)**(1/4)) - 2**(1/4)*(-7*sqrt(5)/2 + 15/2)*atan(2**(1/4)*x/(sqrt(5) + 3)*
*(1/4))/(10*(sqrt(5) + 3)**(1/4)) + 2**(1/4)*(15/2 + 7*sqrt(5)/2)*atanh(2**(1/4)
*x/(-sqrt(5) + 3)**(1/4))/(10*(-sqrt(5) + 3)**(1/4)) + 2**(1/4)*(-7*sqrt(5)/2 +
15/2)*atanh(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(10*(sqrt(5) + 3)**(1/4)) - 3/x - 1
/(5*x**5)

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Mathematica [A]  time = 0.518811, size = 189, normalized size = 1.09 \[ -\frac{1}{5 x^5}-\frac{3}{x}+\frac{\left (-7-3 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\left (7-3 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (-7-3 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (7-3 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(5*x^5) - 3/x + ((-7 - 3*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[1
0*(-1 + Sqrt[5])]) + ((7 - 3*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[1
0*(1 + Sqrt[5])]) - ((-7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt
[10*(-1 + Sqrt[5])]) - ((7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqr
t[10*(1 + Sqrt[5])])

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Maple [A]  time = 0.051, size = 216, normalized size = 1.3 \[ -{\frac{1}{5\,{x}^{5}}}-3\,{x}^{-1}+{\frac{7\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{3}{2\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{7\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{3}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{7\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{7\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{3}{2\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8-3*x^4+1),x)

[Out]

-1/5/x^5-3/x+7/10*5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-3/
2/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+7/10*5^(1/2)/(-2+2*5^(1/2)
)^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+3/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(
-2+2*5^(1/2))^(1/2))-7/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))
^(1/2))-3/2/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-7/10*5^(1/2)/(
2*5^(1/2)+2)^(1/2)*arctanh(2*x/(2*5^(1/2)+2)^(1/2))+3/2/(2*5^(1/2)+2)^(1/2)*arct
anh(2*x/(2*5^(1/2)+2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{15 \, x^{4} + 1}{5 \, x^{5}} - \frac{1}{2} \, \int \frac{3 \, x^{2} + 5}{x^{4} + x^{2} - 1}\,{d x} - \frac{1}{2} \, \int \frac{3 \, x^{2} - 5}{x^{4} - x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 3*x^4 + 1)*x^6),x, algorithm="maxima")

[Out]

-1/5*(15*x^4 + 1)/x^5 - 1/2*integrate((3*x^2 + 5)/(x^4 + x^2 - 1), x) - 1/2*inte
grate((3*x^2 - 5)/(x^4 - x^2 - 1), x)

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Fricas [A]  time = 0.297299, size = 478, normalized size = 2.76 \[ -\frac{4 \, \sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}} x^{5} \arctan \left (\frac{\sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}}{\left (5 \, \sqrt{5} + 11\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) - 4 \, \sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}} x^{5} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}}{\left (5 \, \sqrt{5} - 11\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) + \sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}} x^{5} \log \left (\sqrt{5} x + \frac{1}{2} \, \sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}}{\left (5 \, \sqrt{5} + 11\right )}\right ) - \sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}} x^{5} \log \left (\sqrt{5} x - \frac{1}{2} \, \sqrt{-\sqrt{5}{\left (38 \, \sqrt{5} - 85\right )}}{\left (5 \, \sqrt{5} + 11\right )}\right ) - \sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}} x^{5} \log \left (\sqrt{5} x + \frac{1}{2} \, \sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}}{\left (5 \, \sqrt{5} - 11\right )}\right ) + \sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}} x^{5} \log \left (\sqrt{5} x - \frac{1}{2} \, \sqrt{\sqrt{5}{\left (38 \, \sqrt{5} + 85\right )}}{\left (5 \, \sqrt{5} - 11\right )}\right ) + 60 \, x^{4} + 4}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 3*x^4 + 1)*x^6),x, algorithm="fricas")

[Out]

-1/20*(4*sqrt(-sqrt(5)*(38*sqrt(5) - 85))*x^5*arctan(1/2*sqrt(-sqrt(5)*(38*sqrt(
5) - 85))*(5*sqrt(5) + 11)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 1)
+ 5)) + sqrt(5)*x)) - 4*sqrt(sqrt(5)*(38*sqrt(5) + 85))*x^5*arctan(1/2*sqrt(sqr
t(5)*(38*sqrt(5) + 85))*(5*sqrt(5) - 11)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(
5)*(2*x^2 - 1) + 5)) + sqrt(5)*x)) + sqrt(-sqrt(5)*(38*sqrt(5) - 85))*x^5*log(sq
rt(5)*x + 1/2*sqrt(-sqrt(5)*(38*sqrt(5) - 85))*(5*sqrt(5) + 11)) - sqrt(-sqrt(5)
*(38*sqrt(5) - 85))*x^5*log(sqrt(5)*x - 1/2*sqrt(-sqrt(5)*(38*sqrt(5) - 85))*(5*
sqrt(5) + 11)) - sqrt(sqrt(5)*(38*sqrt(5) + 85))*x^5*log(sqrt(5)*x + 1/2*sqrt(sq
rt(5)*(38*sqrt(5) + 85))*(5*sqrt(5) - 11)) + sqrt(sqrt(5)*(38*sqrt(5) + 85))*x^5
*log(sqrt(5)*x - 1/2*sqrt(sqrt(5)*(38*sqrt(5) + 85))*(5*sqrt(5) - 11)) + 60*x^4
+ 4)/x^5

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Sympy [A]  time = 3.48181, size = 71, normalized size = 0.41 \[ \operatorname{RootSum}{\left (6400 t^{4} - 6080 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{215808000 t^{7}}{323} - \frac{194833880 t^{3}}{323} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 6080 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{215808000 t^{7}}{323} - \frac{194833880 t^{3}}{323} + x \right )} \right )\right )} - \frac{15 x^{4} + 1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 6080*_t**2 - 1, Lambda(_t, _t*log(215808000*_t**7/323 - 194
833880*_t**3/323 + x))) + RootSum(6400*_t**4 + 6080*_t**2 - 1, Lambda(_t, _t*log
(215808000*_t**7/323 - 194833880*_t**3/323 + x))) - (15*x**4 + 1)/(5*x**5)

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GIAC/XCAS [A]  time = 0.344612, size = 215, normalized size = 1.24 \[ \frac{1}{10} \, \sqrt{85 \, \sqrt{5} - 190} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{10} \, \sqrt{85 \, \sqrt{5} + 190} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{85 \, \sqrt{5} - 190}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{85 \, \sqrt{5} - 190}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{85 \, \sqrt{5} + 190}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{20} \, \sqrt{85 \, \sqrt{5} + 190}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{15 \, x^{4} + 1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 3*x^4 + 1)*x^6),x, algorithm="giac")

[Out]

1/10*sqrt(85*sqrt(5) - 190)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt(85*sqr
t(5) + 190)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(85*sqrt(5) - 190)*ln(a
bs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) - 190)*ln(abs(x - sqrt(1
/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) + 190)*ln(abs(x + sqrt(1/2*sqrt(5) -
1/2))) - 1/20*sqrt(85*sqrt(5) + 190)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/5*
(15*x^4 + 1)/x^5