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

Optimal. Leaf size=33 \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0267663, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + (1 + x)^(1/3))^(-1),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0116135, size = 33, normalized size = 1. \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 47, normalized size = 1.4 \[ \ln \left ( 2+x \right ) +{\frac{3}{2} \left ( 1+x \right ) ^{{\frac{2}{3}}}}+2\,\ln \left ( 1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{1+x}+1 \right ) -3\,\sqrt [3]{1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(1+x)^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.32782, size = 34, normalized size = 1.03 \[ \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212089, size = 34, normalized size = 1.03 \[ \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.131618, size = 29, normalized size = 0.88 \[ \frac{3 \left (x + 1\right )^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log{\left (\sqrt [3]{x + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+(1+x)**(1/3)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210932, size = 34, normalized size = 1.03 \[ \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \,{\rm ln}\left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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