3.3029 \(\int \frac{1}{1+4 \sqrt [3]{x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0140665, 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 \[ \frac{x \tan ^{-1}\left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 4*(x^6)^(1/3))^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.25537, size = 19, normalized size = 0.86 \[ \frac{x \operatorname{atan}{\left (2 \sqrt [6]{x^{6}} \right )}}{2 \sqrt [6]{x^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.174093, size = 142, normalized size = 6.45 \[ \frac{2 x \left (x^6\right )^{2/3} B_{-64 x^6}\left (\frac{5}{6},0\right )-2 x \sqrt [3]{-x^{12}} B_{-64 x^6}\left (\frac{1}{2},0\right )+\left (-x^6\right )^{5/6} \left (-\sqrt{3} \log \left (4 x^2-2 \sqrt{3} x+1\right )+\sqrt{3} \log \left (4 x^2+2 \sqrt{3} x+1\right )-2 \tan ^{-1}\left (\sqrt{3}-4 x\right )+4 \tan ^{-1}(2 x)+2 \tan ^{-1}\left (4 x+\sqrt{3}\right )\right )}{24 \left (-x^6\right )^{5/6}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*x*(-x^12)^(1/3)*Beta[-64*x^6, 1/2, 0] + 2*x*(x^6)^(2/3)*Beta[-64*x^6, 5/6, 0
] + (-x^6)^(5/6)*(-2*ArcTan[Sqrt[3] - 4*x] + 4*ArcTan[2*x] + 2*ArcTan[Sqrt[3] +
4*x] - Sqrt[3]*Log[1 - 2*Sqrt[3]*x + 4*x^2] + Sqrt[3]*Log[1 + 2*Sqrt[3]*x + 4*x^
2]))/(24*(-x^6)^(5/6))

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Maple [A]  time = 0.091, size = 17, normalized size = 0.8 \[{\frac{x}{2}\arctan \left ( 2\,\sqrt [6]{{x}^{6}} \right ){\frac{1}{\sqrt [6]{{x}^{6}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+4*(x^6)^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.53686, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)

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Fricas [A]  time = 0.23834, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)

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Sympy [A]  time = 0.176389, size = 5, normalized size = 0.23 \[ \frac{\operatorname{atan}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(2*x)/2

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GIAC/XCAS [A]  time = 0.213025, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)