Optimal. Leaf size=27 \[ 4 \left (1-x+\frac {1}{4} \log \left (\frac {4}{3}\right ) \left (-x+\frac {\log (x)}{x}\right )^2\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(27)=54\).
time = 0.04, antiderivative size = 80, normalized size of antiderivative = 2.96, number of steps
used = 7, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2341, 2342}
\begin {gather*} \frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-4 x-2 \log \left (\frac {4}{3}\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2341
Rule 2342
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x}+\frac {\log \left (\frac {16}{9}\right ) \log (x)}{x^3}-\frac {2 \log \left (\frac {4}{3}\right ) \log ^2(x)}{x^3}\right ) \, dx\\ &=-\left (\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log ^2(x)}{x^3} \, dx\right )+\log \left (\frac {16}{9}\right ) \int \frac {\log (x)}{x^3} \, dx+\int \frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x} \, dx\\ &=-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}-\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log (x)}{x^3} \, dx+\int \left (-4-\frac {2 \log \left (\frac {4}{3}\right )}{x}+x \log \left (\frac {16}{9}\right )\right ) \, dx\\ &=-4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(27)=54\).
time = 0.02, size = 80, normalized size = 2.96 \begin {gather*} -4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2} \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. \(116\) vs.
\(2(30)=60\).
time = 0.34, size = 117, normalized size = 4.33
method | result | size |
risch | \(\frac {\left (-\ln \left (3\right )+2 \ln \left (2\right )\right ) \ln \left (x \right )^{2}}{x^{2}}+2 x^{2} \ln \left (2\right )-x^{2} \ln \left (3\right )-4 \ln \left (2\right ) \ln \left (x \right )+2 \ln \left (3\right ) \ln \left (x \right )-4 x\) | \(48\) |
norman | \(\frac {\left (-\ln \left (3\right )+2 \ln \left (2\right )\right ) x^{4}+\left (-\ln \left (3\right )+2 \ln \left (2\right )\right ) \ln \left (x \right )^{2}+\left (2 \ln \left (3\right )-4 \ln \left (2\right )\right ) x^{2} \ln \left (x \right )-4 x^{3}}{x^{2}}\) | \(53\) |
default | \(-x^{2} \ln \left (3\right )+2 x^{2} \ln \left (2\right )+2 \ln \left (3\right ) \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-4 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+2 \ln \left (3\right ) \ln \left (x \right )-4 \ln \left (2\right ) \ln \left (x \right )-4 x -2 \ln \left (3\right ) \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+4 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (20) = 40\).
time = 0.30, size = 51, normalized size = 1.89 \begin {gather*} -x^{2} \log \left (\frac {3}{4}\right ) + \frac {1}{2} \, {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (\frac {3}{4}\right ) + 2 \, \log \left (\frac {3}{4}\right ) \log \left (x\right ) - 4 \, x - \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} \log \left (\frac {3}{4}\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.22 \begin {gather*} -\frac {x^{4} \log \left (\frac {3}{4}\right ) - 2 \, x^{2} \log \left (\frac {3}{4}\right ) \log \left (x\right ) + 4 \, x^{3} + \log \left (\frac {3}{4}\right ) \log \left (x\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 44, normalized size = 1.63 \begin {gather*} x^{2} \left (- \log {\left (3 \right )} + 2 \log {\left (2 \right )}\right ) - 4 x - 2 \left (- \log {\left (3 \right )} + 2 \log {\left (2 \right )}\right ) \log {\left (x \right )} + \frac {\left (- \log {\left (3 \right )} + 2 \log {\left (2 \right )}\right ) \log {\left (x \right )}^{2}}{x^{2}} \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. 43 vs.
\(2 (20) = 40\).
time = 0.41, size = 43, normalized size = 1.59 \begin {gather*} -x^{2} {\left (\log \left (3\right ) - 2 \, \log \left (2\right )\right )} + 2 \, {\left (\log \left (3\right ) - 2 \, \log \left (2\right )\right )} \log \left (x\right ) - 4 \, x - \frac {{\left (\log \left (3\right ) - 2 \, \log \left (2\right )\right )} \log \left (x\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.15, size = 25, normalized size = 0.93 \begin {gather*} x^2\,\ln \left (\frac {4}{3}\right )-4\,x+\ln \left (\frac {9}{16}\right )\,\ln \left (x\right )+\frac {\ln \left (\frac {4}{3}\right )\,{\ln \left (x\right )}^2}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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