Optimal. Leaf size=67 \[ \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (-1+x)}{\sqrt [3]{(-1+x)^2 (1+x)}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)-\frac {3}{2} \log \left (1-\frac {-1+x}{\sqrt [3]{(-1+x)^2 (1+x)}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(188\) vs. \(2(67)=134\).
time = 0.08, antiderivative size = 188, normalized size of antiderivative = 2.81, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2092, 2089, 62}
\begin {gather*} -\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt [6]{3} \sqrt [3]{3-3 x}}\right )}{\sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1}}-\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (-\frac {8}{3} (x-1)\right )}{2\ 3^{2/3} \sqrt [3]{x^3-x^2-x+1}}-\frac {\sqrt [3]{3} (3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{3} \sqrt [3]{x+1}}{\sqrt [3]{3-3 x}}+1\right )}{2 \sqrt [3]{x^3-x^2-x+1}} \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]{(-1+x)^2 (1+x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {16}{27}-\frac {4 x}{3}+x^3}} \, dx,x,-\frac {1}{3}+x\right )\\ &=\frac {\left (4\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{3 \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {\sqrt {3} (1-x)^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{\sqrt [3]{1-x-x^2+x^3}}-\frac {(1-x)^{2/3} \sqrt [3]{1+x} \log (1-x)}{2 \sqrt [3]{1-x-x^2+x^3}}-\frac {3 (1-x)^{2/3} \sqrt [3]{1+x} \log \left (\frac {3 \left (\sqrt [3]{1-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{1-x}}\right )}{2 \sqrt [3]{1-x-x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 110, normalized size = 1.64 \begin {gather*} \frac {(-1+x)^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {-1+x}{1+x}}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{\frac {-1+x}{1+x}}\right )+\log \left (1+\sqrt [3]{\frac {-1+x}{1+x}}+\left (\frac {-1+x}{1+x}\right )^{2/3}\right )\right )}{2 \sqrt [3]{(-1+x)^2 (1+x)}} \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 = 0.25, size = 370, normalized size = 5.52
method | result | size |
trager | \(-\ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{-1+x}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-4 x +2}{-1+x}\right )\) | \(370\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (56) = 112\).
time = 0.38, size = 128, normalized size = 1.91 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\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]{\left (x - 1\right )^{2} \left (x + 1\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{{\left ({\left (x-1\right )}^2\,\left (x+1\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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