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

Optimal. Leaf size=89 \[ -\frac{1}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

[Out]

-1/(2*x^2) - (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + ((
3 + Sqrt[5])^(3/2)*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

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Rubi [A]  time = 0.156714, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x^2) - (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + ((
3 + Sqrt[5])^(3/2)*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

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Rubi in Sympy [A]  time = 20.0375, size = 104, normalized size = 1.17 \[ \frac{\sqrt{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} + \frac{\sqrt{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105769, size = 103, normalized size = 1.16 \[ \frac{1}{20} \left (-\frac{10}{x^2}-\left (5+2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5-2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+\left (5+2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )+\left (2 \sqrt{5}-5\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-10/x^2 - (5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 - 2*Sqrt[5])*Log[1 + S
qrt[5] - 2*x^2] + (5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + (-5 + 2*Sqrt[5])*L
og[1 + Sqrt[5] + 2*x^2])/20

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Maple [A]  time = 0.013, size = 67, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}}-{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{4}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) }+{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{4}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8-3*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.825652, size = 124, normalized size = 1.39 \[ -\frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{4} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{4} \, \log \left (x^{4} - x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/10*sqrt(5)*lo
g((2*x^2 - sqrt(5) - 1)/(2*x^2 + sqrt(5) - 1)) - 1/2/x^2 - 1/4*log(x^4 + x^2 - 1
) + 1/4*log(x^4 - x^2 - 1)

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Fricas [A]  time = 0.311054, size = 177, normalized size = 1.99 \[ -\frac{\sqrt{5}{\left (\sqrt{5} x^{2} \log \left (x^{4} + x^{2} - 1\right ) - \sqrt{5} x^{2} \log \left (x^{4} - x^{2} - 1\right ) - 2 \, x^{2} \log \left (\frac{10 \, x^{2} + \sqrt{5}{\left (2 \, x^{4} + 2 \, x^{2} + 3\right )} + 5}{x^{4} + x^{2} - 1}\right ) - 2 \, x^{2} \log \left (\frac{10 \, x^{2} + \sqrt{5}{\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} - 5}{x^{4} - x^{2} - 1}\right ) + 2 \, \sqrt{5}\right )}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(5)*(sqrt(5)*x^2*log(x^4 + x^2 - 1) - sqrt(5)*x^2*log(x^4 - x^2 - 1) -
2*x^2*log((10*x^2 + sqrt(5)*(2*x^4 + 2*x^2 + 3) + 5)/(x^4 + x^2 - 1)) - 2*x^2*l
og((10*x^2 + sqrt(5)*(2*x^4 - 2*x^2 + 3) - 5)/(x^4 - x^2 - 1)) + 2*sqrt(5))/x^2

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Sympy [A]  time = 1.85077, size = 172, normalized size = 1.93 \[ \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - \frac{123}{8} - \frac{123 \sqrt{5}}{20} + 280 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} \right )} + \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - \frac{123}{8} + 280 \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} + \frac{123 \sqrt{5}}{20} \right )} + \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{123 \sqrt{5}}{20} + 280 \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right )^{3} + \frac{123}{8} \right )} + \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} + 280 \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} + \frac{123 \sqrt{5}}{20} + \frac{123}{8} \right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(5)/10 + 1/4)*log(x**2 - 123/8 - 123*sqrt(5)/20 + 280*(sqrt(5)/10 + 1/4)**3
) + (-sqrt(5)/10 + 1/4)*log(x**2 - 123/8 + 280*(-sqrt(5)/10 + 1/4)**3 + 123*sqrt
(5)/20) + (-1/4 + sqrt(5)/10)*log(x**2 - 123*sqrt(5)/20 + 280*(-1/4 + sqrt(5)/10
)**3 + 123/8) + (-1/4 - sqrt(5)/10)*log(x**2 + 280*(-1/4 - sqrt(5)/10)**3 + 123*
sqrt(5)/20 + 123/8) - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.296997, size = 131, normalized size = 1.47 \[ -\frac{1}{10} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{4} \,{\rm ln}\left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*sqrt(5)*ln(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/10*sqrt(5)*
ln(abs(2*x^2 - sqrt(5) - 1)/abs(2*x^2 + sqrt(5) - 1)) - 1/2/x^2 - 1/4*ln(abs(x^4
+ x^2 - 1)) + 1/4*ln(abs(x^4 - x^2 - 1))