3.2377 \(\int \frac{1}{1+x^{2/3}} \, dx\)

Optimal. Leaf size=16 \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

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Rubi [A]  time = 0.0186947, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(2/3))^(-1),x]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

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Rubi in Sympy [A]  time = 3.57968, size = 14, normalized size = 0.88 \[ 3 \sqrt [3]{x} - 3 \operatorname{atan}{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**(1/3) - 3*atan(x**(1/3))

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Mathematica [A]  time = 0.00737561, size = 16, normalized size = 1. \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^(2/3))^(-1),x]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

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Maple [B]  time = 0.01, size = 41, normalized size = 2.6 \[ \arctan \left ( x \right ) +3\,\sqrt [3]{x}-\arctan \left ( 2\,\sqrt [3]{x}-\sqrt{3} \right ) -\arctan \left ( 2\,\sqrt [3]{x}+\sqrt{3} \right ) -2\,\arctan \left ( \sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(2/3)),x)

[Out]

arctan(x)+3*x^(1/3)-arctan(2*x^(1/3)-3^(1/2))-arctan(2*x^(1/3)+3^(1/2))-2*arctan
(x^(1/3))

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Maxima [A]  time = 1.61226, size = 16, normalized size = 1. \[ 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))

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Fricas [A]  time = 0.224141, size = 16, normalized size = 1. \[ 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))

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Sympy [A]  time = 0.376882, size = 14, normalized size = 0.88 \[ 3 \sqrt [3]{x} - 3 \operatorname{atan}{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x**(2/3)),x)

[Out]

3*x**(1/3) - 3*atan(x**(1/3))

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GIAC/XCAS [A]  time = 0.218314, size = 16, normalized size = 1. \[ 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))