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

Optimal. Leaf size=167 \[ \frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{144-64 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{144-64 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]

[Out]

((3 + Sqrt[5])^(3/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ((
144 - 64*Sqrt[5])^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - ((3 + S
qrt[5])^(3/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) + ((144 -
64*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

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Rubi [A]  time = 0.224657, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(1 - 3*x^4 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.368, size = 189, normalized size = 1.13 \[ \frac{\sqrt [4]{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \sqrt [4]{- \sqrt{5} + 3}} + \frac{\sqrt [4]{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \sqrt [4]{\sqrt{5} + 3}} - \frac{\sqrt [4]{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \sqrt [4]{- \sqrt{5} + 3}} - \frac{\sqrt [4]{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \sqrt [4]{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.25764, size = 160, normalized size = 0.96 \[ \frac{\frac{\left (\sqrt{5}-3\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}+\frac{\left (3+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}-\frac{\left (\sqrt{5}-3\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (3+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(((-3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] + ((3 + Sq
rt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] - ((-3 + Sqrt[5])*ArcT
anh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((3 + Sqrt[5])*ArcTanh[Sqrt[
2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])

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Maple [A]  time = 0.036, size = 206, normalized size = 1.2 \[{\frac{3\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(x^8-3*x^4+1),x)

[Out]

3/10*5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+1/2/(2*5^(1/2)+
2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+3/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arcta
nh(2*x/(-2+2*5^(1/2))^(1/2))-1/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))
^(1/2))-3/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+1/2/(
-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-3/10*5^(1/2)/(2*5^(1/2)+2)^
(1/2)*arctanh(2*x/(2*5^(1/2)+2)^(1/2))-1/2/(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. \[ \int \frac{x^{6}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^6/(x^8 - 3*x^4 + 1), x)

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Fricas [A]  time = 0.303137, size = 425, normalized size = 2.54 \[ \frac{1}{5} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) - \frac{1}{5} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) + \frac{1}{20} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \log \left (\sqrt{5} x + \frac{1}{2} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )}\right ) - \frac{1}{20} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \log \left (\sqrt{5} x - \frac{1}{2} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )}\right ) - \frac{1}{20} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \log \left (\sqrt{5} x + \frac{1}{2} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )}\right ) + \frac{1}{20} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \log \left (\sqrt{5} x - \frac{1}{2} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*arctan(1/2*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*(sq
rt(5) + 1)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 - 1) + 5)) + sqrt(5)
*x)) - 1/5*sqrt(sqrt(5)*(2*sqrt(5) + 5))*arctan(1/2*sqrt(sqrt(5)*(2*sqrt(5) + 5)
)*(sqrt(5) - 1)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 1) + 5)) + sq
rt(5)*x)) + 1/20*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*log(sqrt(5)*x + 1/2*sqrt(-sqrt(5
)*(2*sqrt(5) - 5))*(sqrt(5) + 1)) - 1/20*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*log(sqrt
(5)*x - 1/2*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*(sqrt(5) + 1)) - 1/20*sqrt(sqrt(5)*(2
*sqrt(5) + 5))*log(sqrt(5)*x + 1/2*sqrt(sqrt(5)*(2*sqrt(5) + 5))*(sqrt(5) - 1))
+ 1/20*sqrt(sqrt(5)*(2*sqrt(5) + 5))*log(sqrt(5)*x - 1/2*sqrt(sqrt(5)*(2*sqrt(5)
+ 5))*(sqrt(5) - 1))

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Sympy [A]  time = 3.20836, size = 53, normalized size = 0.32 \[ \operatorname{RootSum}{\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(-1792000*_t**7 + 4920*_t**
3 + x))) + RootSum(6400*_t**4 + 320*_t**2 - 1, Lambda(_t, _t*log(-1792000*_t**7
+ 4920*_t**3 + x)))

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GIAC/XCAS [A]  time = 0.350257, size = 198, normalized size = 1.19 \[ \frac{1}{10} \, \sqrt{5 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{10} \, \sqrt{5 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*sqrt(5*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt(5*sqrt(5
) - 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) + 10)*ln(abs(x +
sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*ln(abs(x - sqrt(1/2*sqrt(
5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1
/20*sqrt(5*sqrt(5) - 10)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))