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 = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, 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*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.00, size = 818, normalized size = 15.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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