Optimal. Leaf size=25 \[ -16-x \left (x+\frac {\log ^2\left (\frac {x}{2}\right )}{x}\right )^2-\log (x) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
time = 0.08, antiderivative size = 52, normalized size of antiderivative = 2.08, number of steps
used = 14, number of rules used = 5, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14, 2333,
2332, 2342, 2341} \begin {gather*} -x^3-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-2 x \log ^2\left (\frac {x}{2}\right )+4 x \log \left (\frac {x}{2}\right )-4 x \log (x)-\log (x)+x \log (16) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-1-3 x^3}{x}+\log (16)-2 \log ^2\left (\frac {x}{2}\right )-\frac {4 \log ^3\left (\frac {x}{2}\right )}{x^2}+\frac {\log ^4\left (\frac {x}{2}\right )}{x^2}-4 \log (x)\right ) \, dx\\ &=x \log (16)-2 \int \log ^2\left (\frac {x}{2}\right ) \, dx-4 \int \frac {\log ^3\left (\frac {x}{2}\right )}{x^2} \, dx-4 \int \log (x) \, dx+\int \frac {-1-3 x^3}{x} \, dx+\int \frac {\log ^4\left (\frac {x}{2}\right )}{x^2} \, dx\\ &=4 x+x \log (16)-2 x \log ^2\left (\frac {x}{2}\right )+\frac {4 \log ^3\left (\frac {x}{2}\right )}{x}-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-4 x \log (x)+4 \int \log \left (\frac {x}{2}\right ) \, dx+4 \int \frac {\log ^3\left (\frac {x}{2}\right )}{x^2} \, dx-12 \int \frac {\log ^2\left (\frac {x}{2}\right )}{x^2} \, dx+\int \left (-\frac {1}{x}-3 x^2\right ) \, dx\\ &=-x^3+x \log (16)+4 x \log \left (\frac {x}{2}\right )+\frac {12 \log ^2\left (\frac {x}{2}\right )}{x}-2 x \log ^2\left (\frac {x}{2}\right )-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-\log (x)-4 x \log (x)+12 \int \frac {\log ^2\left (\frac {x}{2}\right )}{x^2} \, dx-24 \int \frac {\log \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=\frac {24}{x}-x^3+x \log (16)+\frac {24 \log \left (\frac {x}{2}\right )}{x}+4 x \log \left (\frac {x}{2}\right )-2 x \log ^2\left (\frac {x}{2}\right )-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-\log (x)-4 x \log (x)+24 \int \frac {\log \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-x^3+x \log (16)+4 x \log \left (\frac {x}{2}\right )-2 x \log ^2\left (\frac {x}{2}\right )-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-\log (x)-4 x \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.36 \begin {gather*} -x^3-2 x \log ^2\left (\frac {x}{2}\right )-\frac {\log ^4\left (\frac {x}{2}\right )}{x}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 33, normalized size = 1.32
method | result | size |
risch | \(-\frac {\ln \left (\frac {x}{2}\right )^{4}}{x}-2 x \ln \left (\frac {x}{2}\right )^{2}-x^{3}-\ln \left (x \right )\) | \(31\) |
derivativedivides | \(-\frac {\ln \left (\frac {x}{2}\right )^{4}}{x}-2 x \ln \left (\frac {x}{2}\right )^{2}-x^{3}-\ln \left (\frac {x}{2}\right )\) | \(33\) |
default | \(-\frac {\ln \left (\frac {x}{2}\right )^{4}}{x}-2 x \ln \left (\frac {x}{2}\right )^{2}-x^{3}-\ln \left (\frac {x}{2}\right )\) | \(33\) |
norman | \(\frac {-x \ln \left (\frac {x}{2}\right )-x^{4}-\ln \left (\frac {x}{2}\right )^{4}-2 x^{2} \ln \left (\frac {x}{2}\right )^{2}}{x}\) | \(37\) |
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. 99 vs.
\(2 (23) = 46\).
time = 0.26, size = 99, normalized size = 3.96 \begin {gather*} -x^{3} - 2 \, {\left (\log \left (\frac {1}{2} \, x\right )^{2} - 2 \, \log \left (\frac {1}{2} \, x\right ) + 2\right )} x - 4 \, x \log \left (\frac {1}{2} \, x\right ) + 4 \, x - \frac {\log \left (\frac {1}{2} \, x\right )^{4} + 4 \, \log \left (\frac {1}{2} \, x\right )^{3} + 12 \, \log \left (\frac {1}{2} \, x\right )^{2} + 24 \, \log \left (\frac {1}{2} \, x\right ) + 24}{x} + \frac {4 \, {\left (\log \left (\frac {1}{2} \, x\right )^{3} + 3 \, \log \left (\frac {1}{2} \, x\right )^{2} + 6 \, \log \left (\frac {1}{2} \, x\right ) + 6\right )}}{x} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 32, normalized size = 1.28 \begin {gather*} -\frac {x^{4} + 2 \, x^{2} \log \left (\frac {1}{2} \, x\right )^{2} + \log \left (\frac {1}{2} \, x\right )^{4} + x \log \left (\frac {1}{2} \, x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 26, normalized size = 1.04 \begin {gather*} - x^{3} - 2 x \log {\left (\frac {x}{2} \right )}^{2} - \log {\left (x \right )} - \frac {\log {\left (\frac {x}{2} \right )}^{4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 32, normalized size = 1.28 \begin {gather*} -x^{3} - 2 \, x \log \left (\frac {1}{2} \, x\right )^{2} - \frac {\log \left (\frac {1}{2} \, x\right )^{4}}{x} - \log \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 30, normalized size = 1.20 \begin {gather*} -\ln \left (x\right )-2\,x\,{\ln \left (\frac {x}{2}\right )}^2-x^3-\frac {{\ln \left (\frac {x}{2}\right )}^4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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