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 = 0.0138222, antiderivative size = 77, normalized size of antiderivative = 1., 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} \[ \sqrt [3]{x-1} (x+1)^{2/3}+\frac{1}{3} \log (x-1)+\log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{x-1}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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*}
Mathematica [C] time = 0.0187196, size = 48, normalized size = 0.62 \[ \frac{3 \left (\frac{x-1}{x+1}\right )^{4/3} (x+1)^{4/3} \, _2F_1\left (\frac{1}{3},\frac{4}{3};\frac{7}{3};\frac{1-x}{2}\right )}{4 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{-1+x}{\frac{1}{\sqrt [3]{1+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x - 1\right )}^{\frac{1}{3}}}{{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0401, size = 359, normalized size = 4.66 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.48307, size = 39, normalized size = 0.51 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x - 1\right )}^{\frac{1}{3}}}{{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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