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

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

[Out]

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

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Rubi [A]  time = 0.230677, antiderivative size = 173, 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 \[ -\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 - 3*x^4 + x^8),x]

[Out]

-(((3 + Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((3
- Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - (((3 + Sqrt
[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((3 - 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 = 20.263, size = 238, normalized size = 1.38 \[ - \frac{\sqrt [4]{2} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \sqrt{2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/4)*sqrt(-2*sqrt(5) + 6)*(-3*sqrt(5)/10 + 1/2)*atan(2**(1/4)*x/(-sqrt(5) +
3)**(1/4))/(2*(-sqrt(5) + 3)**(5/4)) - 2**(1/4)*(1/2 + 3*sqrt(5)/10)*sqrt(2*sqr
t(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)*(-3*sqrt(5)/10 + 1/2)*atanh(2**(1/4)*x/(-sqrt(5) + 3)**
(1/4))/(2*(-sqrt(5) + 3)**(5/4)) - 2**(1/4)*(1/2 + 3*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 = 0.361066, size = 132, normalized size = 0.76 \[ \frac{\sqrt{\sqrt{5}-1} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )-\sqrt{1+\sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\sqrt{5}-1} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )-\sqrt{1+\sqrt{5}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.046, size = 206, normalized size = 1.2 \[ -{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^8-3*x^4+1),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(x^8 - 3*x^4 + 1), x)

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Fricas [A]  time = 0.287486, size = 374, normalized size = 2.16 \[ \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{5} \sqrt{2} \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}}{5 \,{\left (\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} + 1}\right )}}\right ) - \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{5} \sqrt{2} \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}}{5 \,{\left (\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} - 1}\right )}}\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (\frac{1}{5} \, \sqrt{5} \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} + x\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (-\frac{1}{5} \, \sqrt{5} \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} + x\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (\frac{1}{5} \, \sqrt{5} \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} + x\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (-\frac{1}{5} \, \sqrt{5} \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*arctan(1/5*sqrt(5)*sqrt(2)*sqrt(1/2)*s
qrt(sqrt(5)*(sqrt(5) + 5))/(sqrt(2)*x + sqrt(2*x^2 + sqrt(5) + 1))) - 1/5*sqrt(1
/2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*arctan(1/5*sqrt(5)*sqrt(2)*sqrt(1/2)*sqrt(-sqrt
(5)*(sqrt(5) - 5))/(sqrt(2)*x + sqrt(2*x^2 + sqrt(5) - 1))) - 1/20*sqrt(1/2)*sqr
t(sqrt(5)*(sqrt(5) + 5))*log(1/5*sqrt(5)*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5)) +
x) + 1/20*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*log(-1/5*sqrt(5)*sqrt(1/2)*sqrt
(sqrt(5)*(sqrt(5) + 5)) + x) + 1/20*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*log(1
/5*sqrt(5)*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5)) + x) - 1/20*sqrt(1/2)*sqrt(-sq
rt(5)*(sqrt(5) - 5))*log(-1/5*sqrt(5)*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5)) + x
)

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Sympy [A]  time = 3.1531, size = 49, normalized size = 0.28 \[ \operatorname{RootSum}{\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log{\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log{\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x)))
+ RootSum(6400*_t**4 + 80*_t**2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x
)))

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GIAC/XCAS [A]  time = 0.350041, size = 198, normalized size = 1.14 \[ -\frac{1}{20} \, \sqrt{10 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{10 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(10*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(10*sqr
t(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(10*sqrt(5) + 10)*ln(abs
(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)*ln(abs(x - sqrt(1/2*
sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2)
)) - 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))