Optimal. Leaf size=97 \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{3 x^2-6 x+4}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.0133008, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {750} \[ \frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{3 x^2-6 x+4}-3 x+6\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{3 x^2-6 x+4}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 750
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (2-x)}{\sqrt{3} \sqrt [3]{4-6 x+3 x^2}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2\ 2^{2/3}}+\frac{\log \left (6-3 x-3 \sqrt [3]{2} \sqrt [3]{4-6 x+3 x^2}\right )}{2\ 2^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0602103, size = 111, normalized size = 1.14 \[ -\frac{\sqrt [3]{\frac{3 x+i \sqrt{3}-3}{x}} \sqrt [3]{\frac{9 x-3 i \sqrt{3}-9}{x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3-i \sqrt{3}}{3 x},\frac{3+i \sqrt{3}}{3 x}\right )}{2 \sqrt [3]{3 x^2-6 x+4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.166, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{3\,{x}^{2}-6\,x+4}}}}\, 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{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 20.7382, size = 495, normalized size = 5.1 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (4^{\frac{1}{3}} x^{3} + 2 \cdot 4^{\frac{2}{3}}{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{2}{3}}{\left (x - 2\right )} + 4 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}{\left (x^{2} - 4 \, x + 4\right )}\right )}}{6 \,{\left (x^{3} - 12 \, x^{2} + 24 \, x - 16\right )}}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{1}{3}}{\left (x - 2\right )} + 2 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}}{x}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{2}{3}}{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{2} - 4 \, x + 4\right )} - 2 \,{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}}{\left (x - 2\right )}}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt [3]{3 x^{2} - 6 x + 4}}\, dx \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{1}{{\left (3 \, x^{2} - 6 \, x + 4\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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