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

Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.0499458, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + 3*x^4 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.65705, size = 24, normalized size = 1.04 \[ - \frac{\sqrt{5} \operatorname{atanh}{\left (\sqrt{5} \left (\frac{2 x^{4}}{5} + \frac{3}{5}\right ) \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(5)*atanh(sqrt(5)*(2*x**4/5 + 3/5))/10

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Mathematica [A]  time = 0.0155355, size = 38, normalized size = 1.65 \[ \frac{\log \left (-2 x^4+\sqrt{5}-3\right )-\log \left (2 x^4+\sqrt{5}+3\right )}{4 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 19, normalized size = 0.8 \[ -{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8+3*x^4+1),x)

[Out]

-1/10*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)

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Maxima [A]  time = 0.824023, size = 42, normalized size = 1.83 \[ \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.257924, size = 59, normalized size = 2.57 \[ \frac{1}{20} \, \sqrt{5} \log \left (-\frac{10 \, x^{4} - \sqrt{5}{\left (2 \, x^{8} + 6 \, x^{4} + 7\right )} + 15}{x^{8} + 3 \, x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*sqrt(5)*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 = 0.260736, size = 42, normalized size = 1.83 \[ \frac{\sqrt{5} \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} - \frac{\sqrt{5} \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279537, size = 42, normalized size = 1.83 \[ \frac{1}{20} \, \sqrt{5}{\rm ln}\left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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