Optimal. Leaf size=110 \[ \frac {3 \log \left (-2^{2/3} \sqrt [3]{x^2-3 x+2}-x+2\right )}{4 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{2} (2-x)}{\sqrt {3} \sqrt [3]{x^2-3 x+2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log (2-x)}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}} \]
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Rubi [A] time = 0.02, antiderivative size = 176, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {755, 123} \[ \frac {3 \sqrt [3]{x-2} \sqrt [3]{x-1} \log \left (-\frac {(x-2)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{x-1}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac {\sqrt [3]{x-2} \sqrt [3]{x-1} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac {\sqrt {3} \sqrt [3]{x-2} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} (x-2)^{2/3}}{\sqrt {3} \sqrt [3]{x-1}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}} \]
Antiderivative was successfully verified.
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Rule 123
Rule 755
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx &=\frac {\left (\sqrt [3]{-4+2 x} \sqrt [3]{-2+2 x}\right ) \int \frac {1}{x \sqrt [3]{-4+2 x} \sqrt [3]{-2+2 x}} \, dx}{\sqrt [3]{2-3 x+x^2}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-2+x} \sqrt [3]{-1+x} \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} \sqrt [3]{2-3 x+x^2}}+\frac {3 \sqrt [3]{-2+x} \sqrt [3]{-1+x} \log \left (-\frac {(-2+x)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{4 \sqrt [3]{2} \sqrt [3]{2-3 x+x^2}}-\frac {\sqrt [3]{-2+x} \sqrt [3]{-1+x} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{2-3 x+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.54 \[ -\frac {3 \sqrt [3]{1-\frac {2}{x}} \sqrt [3]{1-\frac {1}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {1}{x},\frac {2}{x}\right )}{2 \sqrt [3]{x^2-3 x+2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 4.94, size = 277, normalized size = 2.52 \[ -\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 36 \, x^{5} - 612 \, x^{4} + 2880 \, x^{3} - 5760 \, x^{2} + 5184 \, x - 1728\right )} + 12 \, \sqrt {2} {\left (x^{5} - 38 \, x^{4} + 252 \, x^{3} - 648 \, x^{2} + 720 \, x - 288\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} + 48 \cdot 2^{\frac {1}{6}} {\left (x^{4} - 6 \, x^{3} + 6 \, x^{2}\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{6} - 108 \, x^{5} + 972 \, x^{4} - 3456 \, x^{3} + 6048 \, x^{2} - 5184 \, x + 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 6 \cdot 2^{\frac {1}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} {\left (x - 2\right )} + 12 \, {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6 \, x + 6\right )} + 2^{\frac {1}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} - 6 \, {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}}}{x^{4}}\right ) \]
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 \[ \int \frac {1}{{\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.32, size = 1069, normalized size = 9.72 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (x^2-3\,x+2\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{x \sqrt [3]{\left (x - 2\right ) \left (x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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