3.368 \(\int \frac{x^{11}}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=62 \[ \frac{x^4}{4}-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right ) \]

[Out]

x^4/4 - ((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 = 0.106361, antiderivative size = 62, 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 \[ \frac{x^4}{4}-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^11/(1 + 3*x^4 + x^8),x]

[Out]

x^4/4 - ((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 = 11.6429, size = 68, normalized size = 1.1 \[ \frac{x^{4}}{4} + \frac{\sqrt{5} \left (- \frac{3 \sqrt{5}}{2} + \frac{7}{2}\right ) \log{\left (2 x^{4} - \sqrt{5} + 3 \right )}}{20} - \frac{\sqrt{5} \left (\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right ) \log{\left (2 x^{4} + \sqrt{5} + 3 \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**8+3*x**4+1),x)

[Out]

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

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Mathematica [A]  time = 0.0554396, size = 57, normalized size = 0.92 \[ \frac{1}{40} \left (10 x^4+\left (7 \sqrt{5}-15\right ) \log \left (-2 x^4+\sqrt{5}-3\right )-\left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(1 + 3*x^4 + x^8),x]

[Out]

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

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Maple [A]  time = 0.005, size = 38, normalized size = 0.6 \[{\frac{{x}^{4}}{4}}-{\frac{3\,\ln \left ({x}^{8}+3\,{x}^{4}+1 \right ) }{8}}-{\frac{7\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^8+3*x^4+1),x)

[Out]

1/4*x^4-3/8*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 = 0.826583, size = 68, normalized size = 1.1 \[ \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{3}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248572, size = 96, normalized size = 1.55 \[ \frac{1}{40} \, \sqrt{5}{\left (2 \, \sqrt{5} x^{4} - 3 \, \sqrt{5} \log \left (x^{8} + 3 \, x^{4} + 1\right ) + 7 \, \log \left (-\frac{10 \, x^{4} - \sqrt{5}{\left (2 \, x^{8} + 6 \, x^{4} + 7\right )} + 15}{x^{8} + 3 \, x^{4} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 1),x, algorithm="fricas")

[Out]

1/40*sqrt(5)*(2*sqrt(5)*x^4 - 3*sqrt(5)*log(x^8 + 3*x^4 + 1) + 7*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 = 0.330164, size = 60, normalized size = 0.97 \[ \frac{x^{4}}{4} + \left (- \frac{3}{8} + \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (- \frac{7 \sqrt{5}}{40} - \frac{3}{8}\right ) \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**8+3*x**4+1),x)

[Out]

x**4/4 + (-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)

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GIAC/XCAS [A]  time = 0.271132, size = 68, normalized size = 1.1 \[ \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5}{\rm ln}\left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{3}{8} \,{\rm ln}\left (x^{8} + 3 \, x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 1),x, algorithm="giac")

[Out]

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