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

Optimal. Leaf size=57 \[ -\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\log (x) \]

[Out]

Log[x] - ((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] - 2*x^4])/40 - ((5 - 3*Sqrt[5])*Log[3
+ Sqrt[5] - 2*x^4])/40

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Rubi [A]  time = 0.0699153, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - 3*x^4 + x^8)),x]

[Out]

Log[x] - ((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] - 2*x^4])/40 - ((5 - 3*Sqrt[5])*Log[3
+ Sqrt[5] - 2*x^4])/40

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Rubi in Sympy [A]  time = 11.6048, size = 66, normalized size = 1.16 \[ \frac{\log{\left (x^{4} \right )}}{4} - \frac{\sqrt{5} \left (\frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \log{\left (- 2 x^{4} - \sqrt{5} + 3 \right )}}{20} + \frac{\sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \log{\left (- 2 x^{4} + \sqrt{5} + 3 \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4)/4 - sqrt(5)*(sqrt(5)/2 + 3/2)*log(-2*x**4 - sqrt(5) + 3)/20 + sqrt(5)*
(-sqrt(5)/2 + 3/2)*log(-2*x**4 + sqrt(5) + 3)/20

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Mathematica [A]  time = 0.047609, size = 55, normalized size = 0.96 \[ \frac{1}{40} \left (3 \sqrt{5}-5\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\frac{1}{40} \left (-5-3 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}-3\right )+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

Log[x] + ((-5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40 + ((-5 - 3*Sqrt[5])*Log[
-3 + Sqrt[5] + 2*x^4])/40

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Maple [A]  time = 0.015, size = 64, normalized size = 1.1 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{8}}+{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}-{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^8-3*x^4+1),x)

[Out]

ln(x)-1/8*ln(x^4+x^2-1)+3/20*5^(1/2)*arctanh(1/5*(2*x^2+1)*5^(1/2))-1/8*ln(x^4-x
^2-1)-3/20*5^(1/2)*arctanh(1/5*(2*x^2-1)*5^(1/2))

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Maxima [A]  time = 0.820638, size = 69, normalized size = 1.21 \[ \frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} - 3}{2 \, x^{4} + \sqrt{5} - 3}\right ) - \frac{1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40*sqrt(5)*log((2*x^4 - sqrt(5) - 3)/(2*x^4 + sqrt(5) - 3)) - 1/8*log(x^8 - 3*
x^4 + 1) + 1/4*log(x^4)

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Fricas [A]  time = 0.274614, size = 93, normalized size = 1.63 \[ -\frac{1}{40} \, \sqrt{5}{\left (\sqrt{5} \log \left (x^{8} - 3 \, x^{4} + 1\right ) - 8 \, \sqrt{5} \log \left (x\right ) - 3 \, \log \left (-\frac{10 \, x^{4} - \sqrt{5}{\left (2 \, x^{8} - 6 \, x^{4} + 7\right )} - 15}{x^{8} - 3 \, x^{4} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*sqrt(5)*(sqrt(5)*log(x^8 - 3*x^4 + 1) - 8*sqrt(5)*log(x) - 3*log(-(10*x^4
- sqrt(5)*(2*x^8 - 6*x^4 + 7) - 15)/(x^8 - 3*x^4 + 1)))

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Sympy [A]  time = 0.369575, size = 58, normalized size = 1.02 \[ \log{\left (x \right )} + \left (- \frac{1}{8} + \frac{3 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right )} + \left (- \frac{3 \sqrt{5}}{40} - \frac{1}{8}\right ) \log{\left (x^{4} - \frac{3}{2} + \frac{\sqrt{5}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + (-1/8 + 3*sqrt(5)/40)*log(x**4 - 3/2 - sqrt(5)/2) + (-3*sqrt(5)/40 - 1/
8)*log(x**4 - 3/2 + sqrt(5)/2)

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GIAC/XCAS [A]  time = 0.292529, size = 73, normalized size = 1.28 \[ \frac{3}{40} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{4} - \sqrt{5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt{5} - 3 \right |}}\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40*sqrt(5)*ln(abs(2*x^4 - sqrt(5) - 3)/abs(2*x^4 + sqrt(5) - 3)) + 1/4*ln(x^4)
- 1/8*ln(abs(x^8 - 3*x^4 + 1))