3.3030 \(\int \frac{1}{1-4 \sqrt [3]{x^6}} \, dx\)

Optimal. Leaf size=22 \[ \frac{x \tanh ^{-1}\left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

[Out]

(x*ArcTanh[2*(x^6)^(1/6)])/(2*(x^6)^(1/6))

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Rubi [A]  time = 0.0113293, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x \tanh ^{-1}\left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 4*(x^6)^(1/3))^(-1),x]

[Out]

(x*ArcTanh[2*(x^6)^(1/6)])/(2*(x^6)^(1/6))

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Rubi in Sympy [A]  time = 1.25759, size = 19, normalized size = 0.86 \[ \frac{x \operatorname{atanh}{\left (2 \sqrt [6]{x^{6}} \right )}}{2 \sqrt [6]{x^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-4*(x**6)**(1/3)),x)

[Out]

x*atanh(2*(x**6)**(1/6))/(2*(x**6)**(1/6))

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Mathematica [C]  time = 0.17318, size = 123, normalized size = 5.59 \[ \frac{1}{24} \left (\frac{2 x B_{64 x^6}\left (\frac{1}{2},0\right )}{\sqrt [6]{x^6}}+\frac{2 x B_{64 x^6}\left (\frac{5}{6},0\right )}{\sqrt [6]{x^6}}-\log \left (4 x^2-2 x+1\right )+\log \left (4 x^2+2 x+1\right )-2 \log (1-2 x)+2 \log (2 x+1)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 4*(x^6)^(1/3))^(-1),x]

[Out]

(2*Sqrt[3]*ArcTan[(-1 + 4*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 4*x)/Sqrt[3]] + (2
*x*Beta[64*x^6, 1/2, 0])/(x^6)^(1/6) + (2*x*Beta[64*x^6, 5/6, 0])/(x^6)^(1/6) -
2*Log[1 - 2*x] + 2*Log[1 + 2*x] - Log[1 - 2*x + 4*x^2] + Log[1 + 2*x + 4*x^2])/2
4

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Maple [A]  time = 0.079, size = 17, normalized size = 0.8 \[{\frac{x}{2}{\it Artanh} \left ( 2\,\sqrt [6]{{x}^{6}} \right ){\frac{1}{\sqrt [6]{{x}^{6}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-4*(x^6)^(1/3)),x)

[Out]

1/2*x*arctanh(2*(x^6)^(1/6))/(x^6)^(1/6)

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Maxima [A]  time = 1.33761, size = 23, normalized size = 1.05 \[ \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^6)^(1/3) - 1),x, algorithm="maxima")

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Fricas [A]  time = 0.241611, size = 23, normalized size = 1.05 \[ \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^6)^(1/3) - 1),x, algorithm="fricas")

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Sympy [A]  time = 0.174726, size = 15, normalized size = 0.68 \[ - \frac{\log{\left (x - \frac{1}{2} \right )}}{4} + \frac{\log{\left (x + \frac{1}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-4*(x**6)**(1/3)),x)

[Out]

-log(x - 1/2)/4 + log(x + 1/2)/4

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GIAC/XCAS [A]  time = 0.213384, size = 20, normalized size = 0.91 \[ \frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{2} \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - \frac{1}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^6)^(1/3) - 1),x, algorithm="giac")

[Out]

1/4*ln(abs(x + 1/2)) - 1/4*ln(abs(x - 1/2))