Optimal. Leaf size=53 \[ -\frac{3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )+\log (x+1) \]
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Rubi [A] time = 0.0552897, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2151} \[ -\frac{3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 2151
Rubi steps
\begin{align*} \int \frac{-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx &=\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )+\log (1+x)-\frac{3}{2} \log \left (2+x-\sqrt [3]{2+x^3}\right )\\ \end{align*}
Mathematica [F] time = 0.242957, size = 0, normalized size = 0. \[ \int \frac{-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{-1+x}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - 1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - 1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - 1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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