Optimal. Leaf size=77 \[ \sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log (-1+x)+\log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {52, 61}
\begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [3]{x-1} (x+1)^{2/3}+\frac {1}{3} \log (x-1)+\log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx &=\sqrt [3]{-1+x} (1+x)^{2/3}-\frac {2}{3} \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{1+x}} \, dx\\ &=\sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log (-1+x)+\log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(77)=154\).
time = 0.22, size = 158, normalized size = 2.05 \begin {gather*} \frac {\sqrt [3]{\frac {-1+x}{1+x}} \left (3 \sqrt [3]{-1+x}+3 \sqrt [3]{-1+x} x+2 \sqrt {3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )+2 \sqrt [3]{1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\sqrt [3]{1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{3 \sqrt [3]{-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 = 0.44, size = 577, normalized size = 7.49
method | result | size |
risch | \(\left (-1+x \right )^{\frac {1}{3}} \left (1+x \right )^{\frac {2}{3}}+\frac {\left (\frac {2 \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}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +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 +5 \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}}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{-1+x}\right )}{3}-\frac {2 \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -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 -\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 {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 x -1}{-1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {2 \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -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 -\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 {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 x -1}{-1+x}\right )}{3}\right ) \left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{\left (-1+x \right )^{\frac {2}{3}} \left (1+x \right )^{\frac {1}{3}}}\) | \(577\) |
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 = 1.11, size = 107, normalized size = 1.39 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) + {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} + {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {2}{3}} + x + 1}{x + 1}\right ) + \frac {2}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - x - 1}{x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.59, size = 39, normalized size = 0.51 \begin {gather*} \frac {2^{\frac {2}{3}} \left (x - 1\right )^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{2 \Gamma \left (\frac {7}{3}\right )} \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 {{\left (x-1\right )}^{1/3}}{{\left (x+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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