Optimal. Leaf size=99 \[ \frac {3 \log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A] time = 0.01, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {123} \[ \frac {3 \log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Rule 123
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 32, normalized size = 0.32 \[ -\frac {3}{2} (1-x)^{2/3} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x-1,1-x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 3.61, size = 296, normalized size = 2.99 \[ \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \cdot 2^{\frac {1}{6}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {2} {\left (x^{5} - 14 \, x^{4} + 36 \, x^{3} - 24 \, x^{2}\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}} {\left (x^{6} - 72 \, x^{5} + 792 \, x^{4} - 3168 \, x^{3} + 5904 \, x^{2} - 5184 \, x + 1728\right )}\right )}}{6 \, {\left (x^{6} - 432 \, x^{4} + 2592 \, x^{3} - 5616 \, x^{2} + 5184 \, x - 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {1}{3}} x^{2} + 6 \cdot 2^{\frac {2}{3}} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} + 6 \, {\left (x - 2\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {24 \, {\left (x^{2} - 6 \, x + 6\right )} {\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 2^{\frac {2}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )}}{x^{4}}\right ) \]
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 \[ \int \frac {1}{x {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x +1\right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}} x}\, dx \]
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 \[ \int \frac {1}{x {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{x \sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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