Optimal. Leaf size=111 \[ -\frac {\log \left (3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \log (6 x)}{\sqrt [6]{3}}\right )}{2^{2/3} 3^{5/6}} \]
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Rubi [A] time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \log (6 x)}{\sqrt [6]{3}}\right )}{2^{2/3} 3^{5/6}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x \left (2+3 \log ^3(6 x)\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{2+3 x^3} \, dx,x,\log (6 x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}+\sqrt [3]{3} x} \, dx,x,\log (6 x)\right )}{3\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{2}-\sqrt [3]{3} x}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{3\ 2^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{2 \sqrt [3]{2}}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ &=\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (2^{2/3}-\sqrt [3]{6} \log (6 x)+3^{2/3} \log ^2(6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{3} \log (6 x)\right )}{2^{2/3} \sqrt [3]{3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1-2^{2/3} \sqrt [3]{3} \log (6 x)}{\sqrt {3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (2^{2/3}-\sqrt [3]{6} \log (6 x)+3^{2/3} \log ^2(6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 106, normalized size = 0.95 \[ \frac {\sqrt {3} \left (2 \log \left (2^{2/3} \sqrt [3]{3} \log (6 x)+2\right )-\log \left (\sqrt [3]{2} 3^{2/3} \log ^2(6 x)-2^{2/3} \sqrt [3]{3} \log (6 x)+2\right )\right )+6 \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{3} \log (6 x)-1}{\sqrt {3}}\right )}{6\ 2^{2/3} 3^{5/6}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 71, normalized size = 0.64 \[ -\frac {1}{72} \cdot 12^{\frac {2}{3}} \log \left (6 \, \log \left (6 \, x\right )^{2} - 12^{\frac {2}{3}} \log \left (6 \, x\right ) + 2 \cdot 12^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 12^{\frac {2}{3}} \log \left (12^{\frac {2}{3}} + 6 \, \log \left (6 \, x\right )\right ) + \frac {1}{6} \cdot 12^{\frac {1}{6}} \arctan \left (\frac {1}{6} \cdot 12^{\frac {1}{6}} {\left (12^{\frac {2}{3}} \log \left (6 \, x\right ) - 12^{\frac {1}{3}}\right )}\right ) \]
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 \[ \int \frac {1}{{\left (3 \, \log \left (6 \, x\right )^{3} + 2\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 87, normalized size = 0.78 \[ \frac {2^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (6 x \right )-1\right )}{3}\right )}{6}+\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )+\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{18}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\ln \left (6 x \right )^{2}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (6 x \right )}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 97, normalized size = 0.87 \[ -\frac {1}{36} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{3}} \log \left (3^{\frac {2}{3}} \log \left (6 \, x\right )^{2} - 3^{\frac {1}{3}} 2^{\frac {1}{3}} \log \left (6 \, x\right ) + 2^{\frac {2}{3}}\right ) + \frac {1}{18} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{3}} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} \log \left (6 \, x\right ) + 2^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {1}{6}} 2^{\frac {2}{3}} {\left (2 \cdot 3^{\frac {2}{3}} \log \left (6 \, x\right ) - 3^{\frac {1}{3}} 2^{\frac {1}{3}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 120, normalized size = 1.08 \[ \frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}+\frac {2^{1/3}\,3^{2/3}}{3\,x^2}\right )}{18}+\frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}+\frac {2^{1/3}\,3^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,x^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18}-\frac {2^{1/3}\,3^{2/3}\,\ln \left (\frac {\ln \left (6\,x\right )}{x^2}-\frac {2^{1/3}\,3^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,x^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 17, normalized size = 0.15 \[ \operatorname {RootSum} {\left (324 z^{3} - 1, \left (i \mapsto i \log {\left (6 i + \log {\left (6 x \right )} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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