3.2379 \(\int \frac{1}{\left (1+x^{2/3}\right ) x^{2/3}} \, dx\)

Optimal. Leaf size=8 \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0149051, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.82581, size = 7, normalized size = 0.88 \[ 3 \operatorname{atan}{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00567746, size = 8, normalized size = 1. \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \[ 3\,\arctan \left ( \sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(2/3))/x^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.56723, size = 8, normalized size = 1. \[ 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224865, size = 8, normalized size = 1. \[ 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.918791, size = 7, normalized size = 0.88 \[ 3 \operatorname{atan}{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215738, size = 8, normalized size = 1. \[ 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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