Optimal. Leaf size=75 \[ \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (-3+x)}{\sqrt [3]{9+3 x-5 x^2+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)-\frac {3}{2} \log \left (1-\frac {-3+x}{\sqrt [3]{9+3 x-5 x^2+x^3}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(188\) vs. \(2(75)=150\).
time = 0.08, antiderivative size = 188, normalized size of antiderivative = 2.51, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2092, 2089, 62}
\begin {gather*} -\frac {(9-3 x)^{2/3} \sqrt [3]{x+1} \log \left (-\frac {32}{3} (x-3)\right )}{2\ 3^{2/3} \sqrt [3]{x^3-5 x^2+3 x+9}}-\frac {\sqrt [3]{3} (9-3 x)^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{3} \sqrt [3]{x+1}}{\sqrt [3]{9-3 x}}+1\right )}{2 \sqrt [3]{x^3-5 x^2+3 x+9}}-\frac {(9-3 x)^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt [6]{3} \sqrt [3]{9-3 x}}\right )}{\sqrt [6]{3} \sqrt [3]{x^3-5 x^2+3 x+9}} \end {gather*}
Antiderivative was successfully verified.
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Rule 62
Rule 2089
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {128}{27}-\frac {16 x}{3}+x^3}} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {\left (16\ 2^{2/3} (3-x)^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right )^{2/3} \sqrt [3]{\frac {128}{9}+\frac {16 x}{3}}} \, dx,x,-\frac {5}{3}+x\right )}{3 \sqrt [3]{9+3 x-5 x^2+x^3}}\\ &=-\frac {\sqrt {3} (3-x)^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{3-x}}\right )}{\sqrt [3]{9+3 x-5 x^2+x^3}}-\frac {(3-x)^{2/3} \sqrt [3]{1+x} \log (3-x)}{2 \sqrt [3]{9+3 x-5 x^2+x^3}}-\frac {3 (3-x)^{2/3} \sqrt [3]{1+x} \log \left (\frac {3 \left (\sqrt [3]{3-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{3-x}}\right )}{2 \sqrt [3]{9+3 x-5 x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 110, normalized size = 1.47 \begin {gather*} \frac {(-3+x)^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {-3+x}{1+x}}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{\frac {-3+x}{1+x}}\right )+\log \left (1+\sqrt [3]{\frac {-3+x}{1+x}}+\left (\frac {-3+x}{1+x}\right )^{2/3}\right )\right )}{2 \sqrt [3]{(-3+x)^2 (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 670, normalized size = 8.93
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \ln \left (-\frac {20 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -60 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -33 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-216 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}-216 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -6 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -36 x^{2}+648 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+360 x -756}{-3+x}\right )}{3}-\frac {\ln \left (\frac {-20 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x +60 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x +87 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -366 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -45 x^{2}-405 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+198 x -189}{-3+x}\right ) \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )}{3}+\ln \left (\frac {-20 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x +60 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x +87 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -366 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -45 x^{2}-405 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+198 x -189}{-3+x}\right )\) | \(670\) |
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 [A]
time = 0.32, size = 128, normalized size = 1.71 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x - 3\right )} + 2 \, \sqrt {3} {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}}{3 \, {\left (x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}} {\left (x - 3\right )} - 6 \, x + {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) - \log \left (-\frac {x - {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}} - 3}{x - 3}\right ) \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 {1}{\sqrt [3]{x^{3} - 5 x^{2} + 3 x + 9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = -0.01 \begin {gather*} \int \frac {1}{{\left (x^3-5\,x^2+3\,x+9\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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