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

Optimal. Leaf size=26 \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

_______________________________________________________________________________________

Rubi [A]  time = 0.0462667, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 6 \sqrt [3]{x} + x - 6 \log{\left (\sqrt [3]{x} + 1 \right )} - 6 \int ^{\sqrt [3]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x**(1/3) + x - 6*log(x**(1/3) + 1) - 6*Integral(x, (x, x**(1/3)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0100471, size = 26, normalized size = 1. \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \[ 6\,\sqrt [3]{x}-3\,{x}^{2/3}+x-6\,\ln \left ( 1+\sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x^(1/3))/(1+x^(1/3)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.40036, size = 27, normalized size = 1.04 \[ x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.229914, size = 27, normalized size = 1.04 \[ x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.471564, size = 24, normalized size = 0.92 \[ - 3 x^{\frac{2}{3}} + 6 \sqrt [3]{x} + x - 6 \log{\left (\sqrt [3]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215143, size = 27, normalized size = 1.04 \[ x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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