Optimal. Leaf size=17 \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {376, 459, 321, 207} \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 376
Rule 459
Rubi steps
\begin {align*} \int \frac {1+x^{2/3}}{-1+x^{2/3}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2 \left (1+x^2\right )}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=x+6 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=6 \sqrt [3]{x}+x+6 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=6 \sqrt [3]{x}+x-6 \tanh ^{-1}\left (\sqrt [3]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ x+6 \sqrt [3]{x}-6 \tanh ^{-1}\left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 23, normalized size = 1.35 \[ x + 6 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 24, normalized size = 1.41 \[ x + 6 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) + 3 \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.41 \[ x -3 \ln \left (x^{\frac {1}{3}}+1\right )+3 \ln \left (x^{\frac {1}{3}}-1\right )+6 x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 23, normalized size = 1.35 \[ x + 6 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) + 3 \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 13, normalized size = 0.76 \[ x-6\,\mathrm {atanh}\left (x^{1/3}\right )+6\,x^{1/3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, 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.
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