Optimal. Leaf size=24 \[ x+\frac {\log (2)}{2}-\frac {(-2+2 x) \log (2) \log ^2(x)}{x} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 2.17, number of steps
used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {14, 45, 2372,
2338, 2342, 2341} \begin {gather*} x+\frac {\log (4) \log ^2(x)}{x}-2 \log (2) \log ^2(x)+\frac {2 \log (4) \log (x)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 45
Rule 2338
Rule 2341
Rule 2342
Rule 2372
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {4 (-1+x) \log (2) \log (x)}{x^2}-\frac {\log (4) \log ^2(x)}{x^2}\right ) \, dx\\ &=x-(4 \log (2)) \int \frac {(-1+x) \log (x)}{x^2} \, dx-\log (4) \int \frac {\log ^2(x)}{x^2} \, dx\\ &=x+\frac {\log (4) \log ^2(x)}{x}-4 \log (2) \log (x) \left (\frac {1}{x}+\log (x)\right )+(4 \log (2)) \int \frac {1+x \log (x)}{x^2} \, dx-(2 \log (4)) \int \frac {\log (x)}{x^2} \, dx\\ &=x+\frac {2 \log (4)}{x}+\frac {2 \log (4) \log (x)}{x}+\frac {\log (4) \log ^2(x)}{x}-4 \log (2) \log (x) \left (\frac {1}{x}+\log (x)\right )+(4 \log (2)) \int \left (\frac {1}{x^2}+\frac {\log (x)}{x}\right ) \, dx\\ &=x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}+\frac {2 \log (4) \log (x)}{x}+\frac {\log (4) \log ^2(x)}{x}-4 \log (2) \log (x) \left (\frac {1}{x}+\log (x)\right )+(4 \log (2)) \int \frac {\log (x)}{x} \, dx\\ &=x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}+\frac {2 \log (4) \log (x)}{x}+2 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x}-4 \log (2) \log (x) \left (\frac {1}{x}+\log (x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
time = 0.01, size = 52, normalized size = 2.17 \begin {gather*} x-\frac {4 \log (2)}{x}+\frac {2 \log (4)}{x}-\frac {4 \log (2) \log (x)}{x}+\frac {2 \log (4) \log (x)}{x}-2 \log (2) \log ^2(x)+\frac {\log (4) \log ^2(x)}{x} \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. \(53\) vs.
\(2(22)=44\).
time = 0.07, size = 54, normalized size = 2.25
method | result | size |
risch | \(-\frac {2 \left (x -1\right ) \ln \left (2\right ) \ln \left (x \right )^{2}}{x}+x\) | \(17\) |
norman | \(\frac {x^{2}+2 \ln \left (2\right ) \ln \left (x \right )^{2}-2 x \ln \left (2\right ) \ln \left (x \right )^{2}}{x}\) | \(26\) |
default | \(-2 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )^{2}}{x}-\frac {2 \ln \left (x \right )}{x}-\frac {2}{x}\right )-2 \ln \left (2\right ) \ln \left (x \right )^{2}+4 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+x\) | \(54\) |
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. 41 vs.
\(2 (20) = 40\).
time = 0.28, size = 41, normalized size = 1.71 \begin {gather*} -2 \, \log \left (2\right ) \log \left (x\right )^{2} - 4 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (2\right ) + x + \frac {2 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )} \log \left (2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 22, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (x - 1\right )} \log \left (2\right ) \log \left (x\right )^{2} - x^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 19, normalized size = 0.79 \begin {gather*} x + \frac {\left (- 2 x \log {\left (2 \right )} + 2 \log {\left (2 \right )}\right ) \log {\left (x \right )}^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 0.79 \begin {gather*} 2 \, {\left (\frac {\log \left (2\right )}{x} - \log \left (2\right )\right )} \log \left (x\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 21, normalized size = 0.88 \begin {gather*} x-2\,\ln \left (2\right )\,{\ln \left (x\right )}^2+\frac {2\,\ln \left (2\right )\,{\ln \left (x\right )}^2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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