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

Optimal. Leaf size=32 \[ -\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac{1}{\sqrt [3]{x}}+1\right )-\log (x) \]

[Out]

3*x^(1/3) - (3*x^(2/3))/2 + x - 3*Log[1 + x^(-1/3)] - Log[x]

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Rubi [A]  time = 0.0275681, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac{1}{\sqrt [3]{x}}+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(-1/3))^(-1),x]

[Out]

3*x^(1/3) - (3*x^(2/3))/2 + x - 3*Log[1 + x^(-1/3)] - Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 \sqrt [3]{x} + x - 3 \log{\left (\sqrt [3]{x} + 1 \right )} - 3 \int ^{\sqrt [3]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**(1/3) + x - 3*log(x**(1/3) + 1) - 3*Integral(x, (x, x**(1/3)))

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Mathematica [A]  time = 0.00853203, size = 28, normalized size = 0.88 \[ -\frac{3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

3*x^(1/3) - (3*x^(2/3))/2 + x - 3*Log[1 + x^(1/3)]

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Maple [A]  time = 0.004, size = 21, normalized size = 0.7 \[ x-{\frac{3}{2}{x}^{{\frac{2}{3}}}}+3\,\sqrt [3]{x}-3\,\ln \left ( \sqrt [3]{x}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+1/x^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.34434, size = 38, normalized size = 1.19 \[ -\frac{1}{2} \, x{\left (\frac{3}{x^{\frac{1}{3}}} - \frac{6}{x^{\frac{2}{3}}} - 2\right )} - \log \left (x\right ) - 3 \, \log \left (\frac{1}{x^{\frac{1}{3}}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x*(3/x^(1/3) - 6/x^(2/3) - 2) - log(x) - 3*log(1/x^(1/3) + 1)

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Fricas [A]  time = 0.20167, size = 27, normalized size = 0.84 \[ x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.145261, size = 26, normalized size = 0.81 \[ - \frac{3 x^{\frac{2}{3}}}{2} + 3 \sqrt [3]{x} + x - 3 \log{\left (\sqrt [3]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+1/x**(1/3)),x)

[Out]

-3*x**(2/3)/2 + 3*x**(1/3) + x - 3*log(x**(1/3) + 1)

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GIAC/XCAS [A]  time = 0.208532, size = 27, normalized size = 0.84 \[ x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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