Optimal. Leaf size=77 \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac {1}{3} \log (x-1)+\log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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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 = {50, 59} \begin {gather*} \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 )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 59
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 [C] time = 0.02, size = 48, normalized size = 0.62 \begin {gather*} \frac {3 \left (\frac {x-1}{x+1}\right )^{4/3} (x+1)^{4/3} \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};\frac {1-x}{2}\right )}{4 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 113, normalized size = 1.47 \begin {gather*} \sqrt [3]{x-1} (x+1)^{2/3}+\frac {2}{3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{3} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x - 1\right )}^{\frac {1}{3}}}{{\left (x + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 573, normalized size = 7.44 \begin {gather*} \left (x -1\right )^{\frac {1}{3}} \left (x +1\right )^{\frac {2}{3}}+\frac {\left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-2 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -2 x +3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+1}{x -1}\right )}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {2 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+5 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+2 x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-4 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x -1}\right )}{3}-\frac {2 \ln \left (\frac {-2 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -2 x +3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+1}{x -1}\right )}{3}\right ) \left (\left (x -1\right )^{2} \left (x +1\right )\right )^{\frac {1}{3}}}{\left (x -1\right )^{\frac {2}{3}} \left (x +1\right )^{\frac {1}{3}}} \end {gather*}
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*} \int \frac {{\left (x - 1\right )}^{\frac {1}{3}}}{{\left (x + 1\right )}^{\frac {1}{3}}}\,{d x} \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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sympy [C] time = 2.55, 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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