3.179 \(\int \frac{1+x^{2/3}}{-1+x^{2/3}} \, dx\)

Optimal. Leaf size=17 \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]

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Rubi [A]  time = 0.0500649, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]

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Rubi in Sympy [A]  time = 10.3831, size = 15, normalized size = 0.88 \[ 6 \sqrt [3]{x} + x - 6 \operatorname{atanh}{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x**(1/3) + x - 6*atanh(x**(1/3))

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Mathematica [A]  time = 0.0102708, size = 31, normalized size = 1.82 \[ x+6 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right )-3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 24, normalized size = 1.4 \[ x+6\,\sqrt [3]{x}+3\,\ln \left ( -1+\sqrt [3]{x} \right ) -3\,\ln \left ( 1+\sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(2/3))/(-1+x^(2/3)),x)

[Out]

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

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Maxima [A]  time = 1.39064, size = 31, normalized size = 1.82 \[ x + 6 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226716, size = 31, normalized size = 1.82 \[ x + 6 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.649011, size = 27, normalized size = 1.59 \[ 6 \sqrt [3]{x} + x + 3 \log{\left (\sqrt [3]{x} - 1 \right )} - 3 \log{\left (\sqrt [3]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216685, size = 32, normalized size = 1.88 \[ x + 6 \, x^{\frac{1}{3}} - 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) + 3 \,{\rm ln}\left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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