3.2952 \(\int \frac{1}{1+\left (x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=17 \[ \frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \]

[Out]

(x*ArcTan[(x^3)^(1/3)])/(x^3)^(1/3)

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Rubi [A]  time = 0.0124582, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (x^3)^(2/3))^(-1),x]

[Out]

(x*ArcTan[(x^3)^(1/3)])/(x^3)^(1/3)

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Rubi in Sympy [A]  time = 1.13289, size = 15, normalized size = 0.88 \[ \frac{x \operatorname{atan}{\left (\sqrt [3]{x^{3}} \right )}}{\sqrt [3]{x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+(x**3)**(2/3)),x)

[Out]

x*atan((x**3)**(1/3))/(x**3)**(1/3)

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Mathematica [A]  time = 0.0295114, size = 0, normalized size = 0. \[ \int \frac{1}{1+\left (x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(1 + (x^3)^(2/3))^(-1),x]

[Out]

Integrate[(1 + (x^3)^(2/3))^(-1), x]

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Maple [A]  time = 0.053, size = 14, normalized size = 0.8 \[{x\arctan \left ( \sqrt [3]{{x}^{3}} \right ){\frac{1}{\sqrt [3]{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(x^3)^(2/3)),x)

[Out]

x*arctan((x^3)^(1/3))/(x^3)^(1/3)

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Maxima [A]  time = 1.51243, size = 3, normalized size = 0.18 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3)^(2/3) + 1),x, algorithm="maxima")

[Out]

arctan(x)

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Fricas [A]  time = 0.20932, size = 8, normalized size = 0.47 \[ \arctan \left ({\left (x^{3}\right )}^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3)^(2/3) + 1),x, algorithm="fricas")

[Out]

arctan((x^3)^(1/3))

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Sympy [A]  time = 0.159552, size = 2, normalized size = 0.12 \[ \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+(x**3)**(2/3)),x)

[Out]

atan(x)

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GIAC/XCAS [A]  time = 0.215818, size = 3, normalized size = 0.18 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3)^(2/3) + 1),x, algorithm="giac")

[Out]

arctan(x)