Integrand size = 18, antiderivative size = 92 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2176} \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\log (x+1) \]
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Rule 2176
Rubi steps \begin{align*} \text {integral}& = \sqrt {3} \arctan \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*}
Time = 0.86 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.85 (sec) , antiderivative size = 543, normalized size of antiderivative = 5.90
method | result | size |
trager | \(-\ln \left (-\frac {787 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x -4839 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-1574 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-452 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}+335 x \left (x^{3}+2\right )^{\frac {2}{3}}-19356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +4504 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-3148 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +11922 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-4052 x^{3}+670 \left (x^{3}+2\right )^{\frac {2}{3}}-19356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+18016 x \left (x^{3}+2\right )^{\frac {1}{3}}+23844 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -20260 x^{2}+18016 \left (x^{3}+2\right )^{\frac {1}{3}}+11018 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-40520 x -28364}{\left (1+x \right )^{2}}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2026 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +335 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-4052 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-6865 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-4839 x \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +4504 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-8104 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -14634 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2361 x^{3}-9678 \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+18016 x \left (x^{3}+2\right )^{\frac {1}{3}}-29268 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +6296 x^{2}+18016 \left (x^{3}+2\right )^{\frac {1}{3}}-28364 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+12592 x +11018}{\left (1+x \right )^{2}}\right )\) | \(543\) |
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Exception generated. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {x - 1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]
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\[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]
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