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

Optimal. Leaf size=189 \[ -\frac{1}{7 x^7}-\frac{1}{x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

[Out]

-1/(7*x^7) - x^(-3) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5])
)^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5
])/2)^(1/4)*x])/(2*Sqrt[5]) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 +
Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTanh[((
3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5])

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Rubi [A]  time = 0.403161, antiderivative size = 189, 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 \[ -\frac{1}{7 x^7}-\frac{1}{x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(7*x^7) - x^(-3) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5])
)^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5
])/2)^(1/4)*x])/(2*Sqrt[5]) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 +
Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTanh[((
3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5])

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Rubi in Sympy [A]  time = 31.4174, size = 248, normalized size = 1.31 \[ \frac{\sqrt [4]{2} \left (\frac{63}{2} + \frac{147 \sqrt{5}}{10}\right ) \sqrt{- 2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{42 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{\sqrt [4]{2} \left (- \frac{147 \sqrt{5}}{10} + \frac{63}{2}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{42 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{\sqrt [4]{2} \left (\frac{63}{2} + \frac{147 \sqrt{5}}{10}\right ) \sqrt{- 2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{42 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{\sqrt [4]{2} \left (- \frac{147 \sqrt{5}}{10} + \frac{63}{2}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{42 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{1}{x^{3}} - \frac{1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*(63/2 + 147*sqrt(5)/10)*sqrt(-2*sqrt(5) + 6)*atan(2**(1/4)*x/(-sqrt(5)
+ 3)**(1/4))/(42*(-sqrt(5) + 3)**(5/4)) + 2**(1/4)*(-147*sqrt(5)/10 + 63/2)*sqrt
(2*sqrt(5) + 6)*atan(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(42*(sqrt(5) + 3)**(5/4))
+ 2**(1/4)*(63/2 + 147*sqrt(5)/10)*sqrt(-2*sqrt(5) + 6)*atanh(2**(1/4)*x/(-sqrt(
5) + 3)**(1/4))/(42*(-sqrt(5) + 3)**(5/4)) + 2**(1/4)*(-147*sqrt(5)/10 + 63/2)*s
qrt(2*sqrt(5) + 6)*atanh(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(42*(sqrt(5) + 3)**(5/
4)) - 1/x**3 - 1/(7*x**7)

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Mathematica [A]  time = 0.528083, size = 189, normalized size = 1. \[ -\frac{1}{7 x^7}-\frac{1}{x^3}+\frac{\left (11+5 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\left (11-5 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (-11-5 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (5 \sqrt{5}-11\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(7*x^7) - x^(-3) + ((11 + 5*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqr
t[10*(-1 + Sqrt[5])]) + ((11 - 5*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sq
rt[10*(1 + Sqrt[5])]) - ((-11 - 5*Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2
*Sqrt[10*(-1 + Sqrt[5])]) - ((-11 + 5*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])
/(2*Sqrt[10*(1 + Sqrt[5])])

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Maple [A]  time = 0.057, size = 216, normalized size = 1.1 \[ -{\frac{1}{7\,{x}^{7}}}-{x}^{-3}+{\frac{11\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{5}{2\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{11\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{5}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{11\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{5}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{11\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{5}{2\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^8-3*x^4+1),x)

[Out]

-1/7/x^7-1/x^3+11/10*5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))
-5/2/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+11/10*5^(1/2)/(-2+2*5^(
1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+5/2/(-2+2*5^(1/2))^(1/2)*arctanh(2
*x/(-2+2*5^(1/2))^(1/2))+11/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(
1/2))^(1/2))+5/2/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+11/10*5^(
1/2)/(2*5^(1/2)+2)^(1/2)*arctanh(2*x/(2*5^(1/2)+2)^(1/2))-5/2/(2*5^(1/2)+2)^(1/2
)*arctanh(2*x/(2*5^(1/2)+2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{7 \, x^{4} + 1}{7 \, x^{7}} - \frac{1}{2} \, \int \frac{5 \, x^{2} + 8}{x^{4} + x^{2} - 1}\,{d x} + \frac{1}{2} \, \int \frac{5 \, x^{2} - 8}{x^{4} - x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*(7*x^4 + 1)/x^7 - 1/2*integrate((5*x^2 + 8)/(x^4 + x^2 - 1), x) + 1/2*integ
rate((5*x^2 - 8)/(x^4 - x^2 - 1), x)

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Fricas [A]  time = 0.28179, size = 489, normalized size = 2.59 \[ \frac{28 \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}} x^{7} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}}{\left (9 \, \sqrt{5} + 20\right )}}{5 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + x\right )}}\right ) - 28 \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}} x^{7} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}}{\left (9 \, \sqrt{5} - 20\right )}}{5 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + x\right )}}\right ) - 7 \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}} x^{7} \log \left (\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}}{\left (9 \, \sqrt{5} + 20\right )} + x\right ) + 7 \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}} x^{7} \log \left (-\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (199 \, \sqrt{5} - 445\right )}}{\left (9 \, \sqrt{5} + 20\right )} + x\right ) + 7 \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}} x^{7} \log \left (\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}}{\left (9 \, \sqrt{5} - 20\right )} + x\right ) - 7 \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}} x^{7} \log \left (-\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (199 \, \sqrt{5} + 445\right )}}{\left (9 \, \sqrt{5} - 20\right )} + x\right ) - 140 \, x^{4} - 20}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/140*(28*sqrt(1/2)*sqrt(-sqrt(5)*(199*sqrt(5) - 445))*x^7*arctan(1/5*sqrt(1/2)*
sqrt(-sqrt(5)*(199*sqrt(5) - 445))*(9*sqrt(5) + 20)/(sqrt(1/10)*sqrt(sqrt(5)*(sq
rt(5)*(2*x^2 + 1) + 5)) + x)) - 28*sqrt(1/2)*sqrt(sqrt(5)*(199*sqrt(5) + 445))*x
^7*arctan(1/5*sqrt(1/2)*sqrt(sqrt(5)*(199*sqrt(5) + 445))*(9*sqrt(5) - 20)/(sqrt
(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 - 1) + 5)) + x)) - 7*sqrt(1/2)*sqrt(-sqrt(5)
*(199*sqrt(5) - 445))*x^7*log(1/5*sqrt(1/2)*sqrt(-sqrt(5)*(199*sqrt(5) - 445))*(
9*sqrt(5) + 20) + x) + 7*sqrt(1/2)*sqrt(-sqrt(5)*(199*sqrt(5) - 445))*x^7*log(-1
/5*sqrt(1/2)*sqrt(-sqrt(5)*(199*sqrt(5) - 445))*(9*sqrt(5) + 20) + x) + 7*sqrt(1
/2)*sqrt(sqrt(5)*(199*sqrt(5) + 445))*x^7*log(1/5*sqrt(1/2)*sqrt(sqrt(5)*(199*sq
rt(5) + 445))*(9*sqrt(5) - 20) + x) - 7*sqrt(1/2)*sqrt(sqrt(5)*(199*sqrt(5) + 44
5))*x^7*log(-1/5*sqrt(1/2)*sqrt(sqrt(5)*(199*sqrt(5) + 445))*(9*sqrt(5) - 20) +
x) - 140*x^4 - 20)/x^7

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Sympy [A]  time = 3.56258, size = 68, normalized size = 0.36 \[ \operatorname{RootSum}{\left (6400 t^{4} - 15920 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{460800 t^{5}}{17711} - \frac{2842588 t}{17711} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 15920 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{460800 t^{5}}{17711} - \frac{2842588 t}{17711} + x \right )} \right )\right )} - \frac{7 x^{4} + 1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(6400*_t**4 - 15920*_t**2 - 1, Lambda(_t, _t*log(460800*_t**5/17711 - 284
2588*_t/17711 + x))) + RootSum(6400*_t**4 + 15920*_t**2 - 1, Lambda(_t, _t*log(4
60800*_t**5/17711 - 2842588*_t/17711 + x))) - (7*x**4 + 1)/(7*x**7)

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GIAC/XCAS [A]  time = 0.353901, size = 215, normalized size = 1.14 \[ -\frac{1}{20} \, \sqrt{890 \, \sqrt{5} - 1990} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{890 \, \sqrt{5} + 1990} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{890 \, \sqrt{5} - 1990}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{890 \, \sqrt{5} - 1990}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{890 \, \sqrt{5} + 1990}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{890 \, \sqrt{5} + 1990}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{7 \, x^{4} + 1}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(890*sqrt(5) - 1990)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(890
*sqrt(5) + 1990)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(890*sqrt(5) - 199
0)*ln(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqrt(5) - 1990)*ln(abs(x
- sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqrt(5) + 1990)*ln(abs(x + sqrt(1/2
*sqrt(5) - 1/2))) - 1/40*sqrt(890*sqrt(5) + 1990)*ln(abs(x - sqrt(1/2*sqrt(5) -
1/2))) - 1/7*(7*x^4 + 1)/x^7