3.376 \(\int \frac{1}{x^5 \left (1+3 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=66 \[ -\frac{1}{4 x^4}+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-3 \log (x) \]

[Out]

-1/(4*x^4) - 3*Log[x] + ((15 + 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 + ((15 -
7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Rubi [A]  time = 0.144261, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{1}{4 x^4}+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(4*x^4) - 3*Log[x] + ((15 + 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 + ((15 -
7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Rubi in Sympy [A]  time = 16.5915, size = 78, normalized size = 1.18 \[ - \frac{3 \log{\left (x^{4} \right )}}{4} + \frac{\sqrt{5} \left (\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right ) \log{\left (2 x^{4} - \sqrt{5} + 3 \right )}}{20} - \frac{\sqrt{5} \left (- \frac{3 \sqrt{5}}{2} + \frac{7}{2}\right ) \log{\left (2 x^{4} + \sqrt{5} + 3 \right )}}{20} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8+3*x**4+1),x)

[Out]

-3*log(x**4)/4 + sqrt(5)*(3*sqrt(5)/2 + 7/2)*log(2*x**4 - sqrt(5) + 3)/20 - sqrt
(5)*(-3*sqrt(5)/2 + 7/2)*log(2*x**4 + sqrt(5) + 3)/20 - 1/(4*x**4)

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Mathematica [A]  time = 0.059073, size = 60, normalized size = 0.91 \[ \frac{1}{40} \left (-\frac{10}{x^4}+\left (15+7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}-3\right )+\left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-120 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 + 3*x^4 + x^8)),x]

[Out]

(-10/x^4 - 120*Log[x] + (15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] + (15 - 7*Sqr
t[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Maple [A]  time = 0.011, size = 42, normalized size = 0.6 \[ -{\frac{1}{4\,{x}^{4}}}-3\,\ln \left ( x \right ) +{\frac{3\,\ln \left ({x}^{8}+3\,{x}^{4}+1 \right ) }{8}}-{\frac{7\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8+3*x^4+1),x)

[Out]

-1/4/x^4-3*ln(x)+3/8*ln(x^8+3*x^4+1)-7/20*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)

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Maxima [A]  time = 0.823955, size = 76, normalized size = 1.15 \[ \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{1}{4 \, x^{4}} + \frac{3}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) - \frac{3}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^5),x, algorithm="maxima")

[Out]

7/40*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) - 1/4/x^4 + 3/8*lo
g(x^8 + 3*x^4 + 1) - 3/4*log(x^4)

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Fricas [A]  time = 0.251386, size = 117, normalized size = 1.77 \[ \frac{\sqrt{5}{\left (3 \, \sqrt{5} x^{4} \log \left (x^{8} + 3 \, x^{4} + 1\right ) - 24 \, \sqrt{5} x^{4} \log \left (x\right ) + 7 \, x^{4} \log \left (-\frac{10 \, x^{4} - \sqrt{5}{\left (2 \, x^{8} + 6 \, x^{4} + 7\right )} + 15}{x^{8} + 3 \, x^{4} + 1}\right ) - 2 \, \sqrt{5}\right )}}{40 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^5),x, algorithm="fricas")

[Out]

1/40*sqrt(5)*(3*sqrt(5)*x^4*log(x^8 + 3*x^4 + 1) - 24*sqrt(5)*x^4*log(x) + 7*x^4
*log(-(10*x^4 - sqrt(5)*(2*x^8 + 6*x^4 + 7) + 15)/(x^8 + 3*x^4 + 1)) - 2*sqrt(5)
)/x^4

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Sympy [A]  time = 0.529782, size = 65, normalized size = 0.98 \[ - 3 \log{\left (x \right )} + \left (\frac{3}{8} + \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (- \frac{7 \sqrt{5}}{40} + \frac{3}{8}\right ) \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8+3*x**4+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.282319, size = 85, normalized size = 1.29 \[ \frac{7}{40} \, \sqrt{5}{\rm ln}\left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) + \frac{3 \, x^{4} - 1}{4 \, x^{4}} + \frac{3}{8} \,{\rm ln}\left (x^{8} + 3 \, x^{4} + 1\right ) - \frac{3}{4} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^5),x, algorithm="giac")

[Out]

7/40*sqrt(5)*ln((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) + 1/4*(3*x^4 - 1)/x
^4 + 3/8*ln(x^8 + 3*x^4 + 1) - 3/4*ln(x^4)