3.2378 \(\int \frac{1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=12 \[ \frac{3}{2} \log \left (x^{2/3}+1\right ) \]

[Out]

(3*Log[1 + x^(2/3)])/2

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Rubi [A]  time = 0.0115411, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{3}{2} \log \left (x^{2/3}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[1 + x^(2/3)])/2

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Rubi in Sympy [A]  time = 1.71504, size = 10, normalized size = 0.83 \[ \frac{3 \log{\left (x^{\frac{2}{3}} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x**(2/3) + 1)/2

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Mathematica [A]  time = 0.00341518, size = 12, normalized size = 1. \[ \frac{3}{2} \log \left (x^{2/3}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[1 + x^(2/3)])/2

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Maple [A]  time = 0.004, size = 9, normalized size = 0.8 \[{\frac{3}{2}\ln \left ( 1+{x}^{{\frac{2}{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(2/3))/x^(1/3),x)

[Out]

3/2*ln(1+x^(2/3))

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Maxima [A]  time = 1.48496, size = 11, normalized size = 0.92 \[ \frac{3}{2} \, \log \left (x^{\frac{2}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*log(x^(2/3) + 1)

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Fricas [A]  time = 0.22047, size = 11, normalized size = 0.92 \[ \frac{3}{2} \, \log \left (x^{\frac{2}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*log(x^(2/3) + 1)

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Sympy [A]  time = 0.452449, size = 10, normalized size = 0.83 \[ \frac{3 \log{\left (x^{\frac{2}{3}} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x**(2/3) + 1)/2

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GIAC/XCAS [A]  time = 0.215437, size = 11, normalized size = 0.92 \[ \frac{3}{2} \,{\rm ln}\left (x^{\frac{2}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*ln(x^(2/3) + 1)