3.293 \(\int \frac{1}{\frac{1}{\sqrt [3]{x}}+x} \, dx\)

Optimal. Leaf size=12 \[ \frac{3}{4} \log \left (x^{4/3}+1\right ) \]

[Out]

(3*Log[1 + x^(4/3)])/4

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Rubi [A]  time = 0.00931854, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3}{4} \log \left (x^{4/3}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1/3) + x)^(-1),x]

[Out]

(3*Log[1 + x^(4/3)])/4

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Rubi in Sympy [A]  time = 0.895945, size = 10, normalized size = 0.83 \[ \frac{3 \log{\left (x^{\frac{4}{3}} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x**(4/3) + 1)/4

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Mathematica [A]  time = 0.00507557, size = 12, normalized size = 1. \[ \frac{3}{4} \log \left (x^{4/3}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[1 + x^(4/3)])/4

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Maple [A]  time = 0.003, size = 9, normalized size = 0.8 \[{\frac{3}{4}\ln \left ( 1+{x}^{{\frac{4}{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/x^(1/3)+x),x)

[Out]

3/4*ln(1+x^(4/3))

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Maxima [A]  time = 1.49971, size = 11, normalized size = 0.92 \[ \frac{3}{4} \, \log \left (x^{\frac{4}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*log(x^(4/3) + 1)

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Fricas [A]  time = 0.208122, size = 11, normalized size = 0.92 \[ \frac{3}{4} \, \log \left (x^{\frac{4}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*log(x^(4/3) + 1)

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Sympy [A]  time = 0.606153, size = 10, normalized size = 0.83 \[ \frac{3 \log{\left (x^{\frac{4}{3}} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1/x**(1/3)+x),x)

[Out]

3*log(x**(4/3) + 1)/4

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GIAC/XCAS [A]  time = 0.227831, size = 43, normalized size = 3.58 \[ \frac{3}{4} \,{\rm ln}\left (\sqrt{2} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) + \frac{3}{4} \,{\rm ln}\left (-\sqrt{2} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*ln(sqrt(2)*x^(1/3) + x^(2/3) + 1) + 3/4*ln(-sqrt(2)*x^(1/3) + x^(2/3) + 1)