Optimal. Leaf size=53 \[ \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )+\log (1+x)-\frac {3}{2} \log \left (2+x-\sqrt [3]{2+x^3}\right ) \]
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Rubi [A]
time = 0.04, 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 = {2176}
\begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
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 [A]
time = 0.61, size = 92, normalized size = 1.74 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.30, size = 544, normalized size = 10.26
method | result | size |
trager | \(-\ln \left (-\frac {787 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x -9678 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-1574 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-904 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+18016 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}}+1340 x \left (x^{3}+2\right )^{\frac {2}{3}}-38712 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +18016 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-3148 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +23844 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-16208 x^{3}+2680 \left (x^{3}+2\right )^{\frac {2}{3}}-38712 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+72064 \left (x^{3}+2\right )^{\frac {1}{3}} x +47688 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -81040 x^{2}+72064 \left (x^{3}+2\right )^{\frac {1}{3}}+22036 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-162080 x -113456}{\left (1+x \right )^{2}}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {1013 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+4504 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +335 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-2026 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-6865 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-9678 x \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +9008 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-4052 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -14634 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+4722 x^{3}-19356 \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+36032 \left (x^{3}+2\right )^{\frac {1}{3}} x -29268 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +12592 x^{2}+36032 \left (x^{3}+2\right )^{\frac {1}{3}}-28364 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+25184 x +22036}{\left (1+x \right )^{2}}\right )}{2}\) | \(544\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x - 1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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