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

Optimal. Leaf size=170 \[ x-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]

[Out]

x - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5])
+ ((984 - 440*Sqrt[5])^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - (
((123 + 55*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (
(984 - 440*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

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Rubi [A]  time = 0.325785, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ x-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (123-55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (123-55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(1 - 3*x^4 + x^8),x]

[Out]

x - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5])
+ (((123 - 55*Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5])
- (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5])
+ (((123 - 55*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 = 26.3513, size = 238, normalized size = 1.4 \[ x - \frac{\sqrt [4]{2} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{7 \sqrt{5}}{10} + \frac{3}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \left (\frac{3}{2} + \frac{7 \sqrt{5}}{10}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{7 \sqrt{5}}{10} + \frac{3}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \left (\frac{3}{2} + \frac{7 \sqrt{5}}{10}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2**(1/4)*sqrt(-2*sqrt(5) + 6)*(-7*sqrt(5)/10 + 3/2)*atan(2**(1/4)*x/(-sqrt(5
) + 3)**(1/4))/(2*(-sqrt(5) + 3)**(5/4)) - 2**(1/4)*(3/2 + 7*sqrt(5)/10)*sqrt(2*
sqrt(5) + 6)*atan(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(2*(sqrt(5) + 3)**(5/4)) - 2*
*(1/4)*sqrt(-2*sqrt(5) + 6)*(-7*sqrt(5)/10 + 3/2)*atanh(2**(1/4)*x/(-sqrt(5) + 3
)**(1/4))/(2*(-sqrt(5) + 3)**(5/4)) - 2**(1/4)*(3/2 + 7*sqrt(5)/10)*sqrt(2*sqrt(
5) + 6)*atanh(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(2*(sqrt(5) + 3)**(5/4))

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Mathematica [A]  time = 0.514682, size = 160, normalized size = 0.94 \[ x+\frac{\left (\sqrt{5}-2\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (2+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\left (\sqrt{5}-2\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (2+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

x + ((-2 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] -
((2 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] + ((-2 +
Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((2 + Sqrt
[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])]

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Maple [A]  time = 0.082, size = 205, normalized size = 1.2 \[ x-{\frac{2\,\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^8-3*x^4+1),x)

[Out]

x-2/5*5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-1/(2*5^(1/2)+2
)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-2/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh
(2*x/(-2+2*5^(1/2))^(1/2))+1/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/
2))-2/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+1/(-2+2*5^
(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-2/5*5^(1/2)/(2*5^(1/2)+2)^(1/2)*ar
ctanh(2*x/(2*5^(1/2)+2)^(1/2))-1/(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. \[ x + \frac{1}{2} \, \int \frac{2 \, x^{2} + 1}{x^{4} - x^{2} - 1}\,{d x} - \frac{1}{2} \, \int \frac{2 \, x^{2} - 1}{x^{4} + x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.314602, size = 383, normalized size = 2.25 \[ \frac{1}{40} \, \sqrt{10}{\left (4 \, \sqrt{10} x + 4 \, \sqrt{5 \, \sqrt{5} + 11} \arctan \left (\frac{{\left (3 \, \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{5 \, \sqrt{5} + 11}}{2 \,{\left (\sqrt{10} \sqrt{2} x + \sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}\right )}}\right ) - 4 \, \sqrt{5 \, \sqrt{5} - 11} \arctan \left (\frac{{\left (3 \, \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{5 \, \sqrt{5} - 11}}{2 \,{\left (\sqrt{10} \sqrt{2} x + \sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}\right )}}\right ) + \sqrt{5 \, \sqrt{5} - 11} \log \left (2 \, \sqrt{10} x + \sqrt{5 \, \sqrt{5} - 11}{\left (3 \, \sqrt{5} + 5\right )}\right ) - \sqrt{5 \, \sqrt{5} - 11} \log \left (2 \, \sqrt{10} x - \sqrt{5 \, \sqrt{5} - 11}{\left (3 \, \sqrt{5} + 5\right )}\right ) - \sqrt{5 \, \sqrt{5} + 11} \log \left (2 \, \sqrt{10} x + \sqrt{5 \, \sqrt{5} + 11}{\left (3 \, \sqrt{5} - 5\right )}\right ) + \sqrt{5 \, \sqrt{5} + 11} \log \left (2 \, \sqrt{10} x - \sqrt{5 \, \sqrt{5} + 11}{\left (3 \, \sqrt{5} - 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*sqrt(10)*(4*sqrt(10)*x + 4*sqrt(5*sqrt(5) + 11)*arctan(1/2*(3*sqrt(5)*sqrt(
2) - 5*sqrt(2))*sqrt(5*sqrt(5) + 11)/(sqrt(10)*sqrt(2)*x + sqrt(10)*sqrt(2*x^2 +
sqrt(5) + 1))) - 4*sqrt(5*sqrt(5) - 11)*arctan(1/2*(3*sqrt(5)*sqrt(2) + 5*sqrt(
2))*sqrt(5*sqrt(5) - 11)/(sqrt(10)*sqrt(2)*x + sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1
))) + sqrt(5*sqrt(5) - 11)*log(2*sqrt(10)*x + sqrt(5*sqrt(5) - 11)*(3*sqrt(5) +
5)) - sqrt(5*sqrt(5) - 11)*log(2*sqrt(10)*x - sqrt(5*sqrt(5) - 11)*(3*sqrt(5) +
5)) - sqrt(5*sqrt(5) + 11)*log(2*sqrt(10)*x + sqrt(5*sqrt(5) + 11)*(3*sqrt(5) -
5)) + sqrt(5*sqrt(5) + 11)*log(2*sqrt(10)*x - sqrt(5*sqrt(5) + 11)*(3*sqrt(5) -
5)))

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Sympy [A]  time = 3.24894, size = 58, normalized size = 0.34 \[ x + \operatorname{RootSum}{\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log{\left (- \frac{15360 t^{5}}{11} + \frac{1288 t}{55} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log{\left (- \frac{15360 t^{5}}{11} + \frac{1288 t}{55} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + RootSum(6400*_t**4 - 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t**5/11 + 1288
*_t/55 + x))) + RootSum(6400*_t**4 + 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t*
*5/11 + 1288*_t/55 + x)))

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GIAC/XCAS [A]  time = 0.343088, size = 200, normalized size = 1.18 \[ -\frac{1}{20} \, \sqrt{50 \, \sqrt{5} + 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{50 \, \sqrt{5} - 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(50*sqrt(5) + 110)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(50*sq
rt(5) - 110)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(50*sqrt(5) + 110)*ln(
abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) + 110)*ln(abs(x - sqrt(
1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) - 110)*ln(abs(x + sqrt(1/2*sqrt(5) -
1/2))) - 1/40*sqrt(50*sqrt(5) - 110)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2))) + x