Optimal. Leaf size=32 \[ \frac {-3-\frac {3}{x \log \left (\frac {3}{2}\right )}}{x^2 \left (-3+\frac {6}{x}+\frac {x}{\log (x)}\right )} \]
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Rubi [F]
time = 1.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-x^2 \left (1+x \log \left (\frac {3}{2}\right )\right )+x^2 \left (4+x \log \left (\frac {27}{8}\right )\right ) \log (x)+\left (12-6 x^2 \log \left (\frac {3}{2}\right )+x \left (-9+6 \log \left (\frac {3}{2}\right )\right )\right ) \log ^2(x)\right )}{x^3 \log \left (\frac {3}{2}\right ) \left (x^2-3 (-2+x) \log (x)\right )^2} \, dx\\ &=\frac {3 \int \frac {-x^2 \left (1+x \log \left (\frac {3}{2}\right )\right )+x^2 \left (4+x \log \left (\frac {27}{8}\right )\right ) \log (x)+\left (12-6 x^2 \log \left (\frac {3}{2}\right )+x \left (-9+6 \log \left (\frac {3}{2}\right )\right )\right ) \log ^2(x)}{x^3 \left (x^2-3 (-2+x) \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {3 \int \left (\frac {4-x \left (3-\log \left (\frac {9}{4}\right )\right )-x^2 \log \left (\frac {9}{4}\right )}{3 (2-x)^2 x^3}+\frac {-12-x^2 \left (7-12 \log \left (\frac {3}{2}\right )\right )+12 x \left (1-\log \left (\frac {3}{2}\right )\right )+x^4 \log \left (\frac {3}{2}\right )+x^3 \left (1-\log \left (\frac {2187}{128}\right )\right )}{3 (2-x)^2 x \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {2+x \log \left (\frac {3}{2}\right )+\log \left (\frac {9}{4}\right )}{3 (-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {\int \frac {4-x \left (3-\log \left (\frac {9}{4}\right )\right )-x^2 \log \left (\frac {9}{4}\right )}{(2-x)^2 x^3} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \frac {-12-x^2 \left (7-12 \log \left (\frac {3}{2}\right )\right )+12 x \left (1-\log \left (\frac {3}{2}\right )\right )+x^4 \log \left (\frac {3}{2}\right )+x^3 \left (1-\log \left (\frac {2187}{128}\right )\right )}{(2-x)^2 x \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \frac {2+x \log \left (\frac {3}{2}\right )+\log \left (\frac {9}{4}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {\int \left (\frac {1}{x^3}+\frac {-1-\log \left (\frac {9}{4}\right )}{4 (-2+x)^2}+\frac {1+\log \left (\frac {9}{4}\right )}{4 x^2}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \left (-\frac {3}{x \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {x \log \left (\frac {3}{2}\right )}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {1-\log \left (\frac {27}{8}\right )}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2}-\frac {\log \left (\frac {81}{16}\right )}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {-4-\log \left (\frac {6561}{256}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )^2}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \left (\frac {\log \left (\frac {3}{2}\right )}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )}+\frac {2+\log \left (\frac {81}{16}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=-\frac {1}{2 x^2 \log \left (\frac {3}{2}\right )}-\frac {1+\log \left (\frac {9}{4}\right )}{4 (2-x) \log \left (\frac {3}{2}\right )}-\frac {1+\log \left (\frac {9}{4}\right )}{4 x \log \left (\frac {3}{2}\right )}-\frac {3 \int \frac {1}{x \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\left (1-\log \left (\frac {27}{8}\right )\right ) \int \frac {1}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {\log \left (\frac {81}{16}\right ) \int \frac {1}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\left (2+\log \left (\frac {81}{16}\right )\right ) \int \frac {1}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {\left (4+\log \left (\frac {6561}{256}\right )\right ) \int \frac {1}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\int \frac {x}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx+\int \frac {1}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 34, normalized size = 1.06 \begin {gather*} -\frac {3 \left (1+x \log \left (\frac {3}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \left (x^4-3 (-2+x) x^2 \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs.
\(2(30)=60\).
time = 1.82, size = 87, normalized size = 2.72
method | result | size |
norman | \(\frac {-3 x \ln \left (x \right )+\frac {3 \ln \left (x \right )}{-\ln \left (3\right )+\ln \left (2\right )}}{x^{2} \left (x^{2}-3 x \ln \left (x \right )+6 \ln \left (x \right )\right )}\) | \(39\) |
risch | \(\frac {x \ln \left (2\right )-x \ln \left (3\right )-1}{x^{2} \left (x \ln \left (2\right )-x \ln \left (3\right )-2 \ln \left (2\right )+2 \ln \left (3\right )\right )}-\frac {x \ln \left (2\right )-x \ln \left (3\right )-1}{\left (x -2\right ) \left (-\ln \left (3\right )+\ln \left (2\right )\right ) \left (x^{2}-3 x \ln \left (x \right )+6 \ln \left (x \right )\right )}\) | \(79\) |
default | \(-\frac {3 \left (-\frac {\ln \left (2\right ) \ln \left (x \right )}{x \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}+\frac {\ln \left (x \right )}{x^{2} \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}+\frac {\ln \left (3\right ) \ln \left (x \right )}{x \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}\right )}{-\ln \left (3\right )+\ln \left (2\right )}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 57, normalized size = 1.78 \begin {gather*} -\frac {3 \, {\left (x {\left (\log \left (3\right ) - \log \left (2\right )\right )} + 1\right )} \log \left (x\right )}{x^{4} {\left (\log \left (3\right ) - \log \left (2\right )\right )} - 3 \, {\left (x^{3} {\left (\log \left (3\right ) - \log \left (2\right )\right )} - 2 \, x^{2} {\left (\log \left (3\right ) - \log \left (2\right )\right )}\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x \log \left (\frac {2}{3}\right ) - 1\right )} \log \left (x\right )}{x^{4} \log \left (\frac {2}{3}\right ) - 3 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (\frac {2}{3}\right ) \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (22) = 44\).
time = 1.65, size = 119, normalized size = 3.72 \begin {gather*} - \frac {x \left (- \log {\left (2 \right )} + \log {\left (3 \right )}\right ) + 1}{x^{3} \left (- \log {\left (3 \right )} + \log {\left (2 \right )}\right ) + x^{2} \left (- 2 \log {\left (2 \right )} + 2 \log {\left (3 \right )}\right )} + \frac {- x \log {\left (3 \right )} + x \log {\left (2 \right )} - 1}{- x^{3} \log {\left (2 \right )} + x^{3} \log {\left (3 \right )} - 2 x^{2} \log {\left (3 \right )} + 2 x^{2} \log {\left (2 \right )} + \left (- 3 x^{2} \log {\left (3 \right )} + 3 x^{2} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} + 12 x \log {\left (3 \right )} - 12 \log {\left (3 \right )} + 12 \log {\left (2 \right )}\right ) \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (30) = 60\).
time = 0.43, size = 151, normalized size = 4.72 \begin {gather*} -\frac {2 \, x \log \left (3\right ) - 2 \, x \log \left (2\right ) + x + 2}{4 \, {\left (x^{2} \log \left (3\right ) - x^{2} \log \left (2\right )\right )}} - \frac {x \log \left (3\right ) - x \log \left (2\right ) + 1}{x^{3} \log \left (3\right ) - x^{3} \log \left (2\right ) - 3 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 3 \, x^{2} \log \left (2\right ) \log \left (x\right ) - 2 \, x^{2} \log \left (3\right ) + 2 \, x^{2} \log \left (2\right ) + 12 \, x \log \left (3\right ) \log \left (x\right ) - 12 \, x \log \left (2\right ) \log \left (x\right ) - 12 \, \log \left (3\right ) \log \left (x\right ) + 12 \, \log \left (2\right ) \log \left (x\right )} + \frac {2 \, \log \left (3\right ) - 2 \, \log \left (2\right ) + 1}{4 \, {\left (x \log \left (3\right ) - x \log \left (2\right ) - 2 \, \log \left (3\right ) + 2 \, \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\ln \left (x\right )}^2\,\left (27\,x+\ln \left (\frac {2}{3}\right )\,\left (18\,x-18\,x^2\right )-36\right )+\ln \left (x\right )\,\left (9\,x^3\,\ln \left (\frac {2}{3}\right )-12\,x^2\right )-3\,x^3\,\ln \left (\frac {2}{3}\right )+3\,x^2}{x^7\,\ln \left (\frac {2}{3}\right )+\ln \left (\frac {2}{3}\right )\,{\ln \left (x\right )}^2\,\left (9\,x^5-36\,x^4+36\,x^3\right )+\ln \left (\frac {2}{3}\right )\,\ln \left (x\right )\,\left (12\,x^5-6\,x^6\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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