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

Optimal. Leaf size=75 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]

[Out]

-(ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]) + (Sqrt[(3 + Sqrt[5
])/10]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi [A]  time = 0.0846217, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 - 3*x^4 + x^8),x]

[Out]

-(ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]) + (Sqrt[(3 + Sqrt[5
])/10]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi in Sympy [A]  time = 8.71116, size = 73, normalized size = 0.97 \[ \frac{\sqrt{10} \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{10 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{10} \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{10 \sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0536746, size = 91, normalized size = 1.21 \[ \frac{1}{40} \left (-\left (5+\sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}-1\right )-\left (\sqrt{5}-5\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+\left (5+\sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )+\left (\sqrt{5}-5\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2]) - (-5 + Sqrt[5])*Log[1 + Sqrt[5] - 2
*x^2] + (5 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + (-5 + Sqrt[5])*Log[1 + Sqrt[5]
+ 2*x^2])/40

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Maple [A]  time = 0.005, size = 62, normalized size = 0.8 \[{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}+{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{8}}+{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^8-3*x^4+1),x)

[Out]

1/8*ln(x^4-x^2-1)+1/20*5^(1/2)*arctanh(1/5*(2*x^2-1)*5^(1/2))-1/8*ln(x^4+x^2-1)+
1/20*5^(1/2)*arctanh(1/5*(2*x^2+1)*5^(1/2))

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Maxima [A]  time = 0.822112, size = 117, normalized size = 1.56 \[ -\frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/40*sqrt(5)*lo
g((2*x^2 - sqrt(5) - 1)/(2*x^2 + sqrt(5) - 1)) - 1/8*log(x^4 + x^2 - 1) + 1/8*lo
g(x^4 - x^2 - 1)

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Fricas [A]  time = 0.292682, size = 150, normalized size = 2. \[ -\frac{1}{40} \, \sqrt{5}{\left (\sqrt{5} \log \left (x^{4} + x^{2} - 1\right ) - \sqrt{5} \log \left (x^{4} - x^{2} - 1\right ) - \log \left (\frac{10 \, x^{2} + \sqrt{5}{\left (2 \, x^{4} + 2 \, x^{2} + 3\right )} + 5}{x^{4} + x^{2} - 1}\right ) - \log \left (\frac{10 \, x^{2} + \sqrt{5}{\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} - 5}{x^{4} - x^{2} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.90747, size = 165, normalized size = 2.2 \[ \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \log{\left (x^{2} - \frac{7}{2} - \frac{7 \sqrt{5}}{10} + 960 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right )^{3} \right )} + \left (- \frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \log{\left (x^{2} - \frac{7}{2} + 960 \left (- \frac{\sqrt{5}}{40} + \frac{1}{8}\right )^{3} + \frac{7 \sqrt{5}}{10} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{7 \sqrt{5}}{10} + 960 \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right )^{3} + \frac{7}{2} \right )} + \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} + 960 \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right )^{3} + \frac{7 \sqrt{5}}{10} + \frac{7}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**8-3*x**4+1),x)

[Out]

(sqrt(5)/40 + 1/8)*log(x**2 - 7/2 - 7*sqrt(5)/10 + 960*(sqrt(5)/40 + 1/8)**3) +
(-sqrt(5)/40 + 1/8)*log(x**2 - 7/2 + 960*(-sqrt(5)/40 + 1/8)**3 + 7*sqrt(5)/10)
+ (-1/8 + sqrt(5)/40)*log(x**2 - 7*sqrt(5)/10 + 960*(-1/8 + sqrt(5)/40)**3 + 7/2
) + (-1/8 - sqrt(5)/40)*log(x**2 + 960*(-1/8 - sqrt(5)/40)**3 + 7*sqrt(5)/10 + 7
/2)

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GIAC/XCAS [A]  time = 0.293638, size = 124, normalized size = 1.65 \[ -\frac{1}{40} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{40} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*sqrt(5)*ln(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/40*sqrt(5)*
ln(abs(2*x^2 - sqrt(5) - 1)/abs(2*x^2 + sqrt(5) - 1)) - 1/8*ln(abs(x^4 + x^2 - 1
)) + 1/8*ln(abs(x^4 - x^2 - 1))