Optimal. Leaf size=21 \[ 24-\frac {20 x \log \left (\frac {1}{2 x}\right ) \log (x)}{4+\log (x)} \]
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Rubi [C] time = 0.24, antiderivative size = 79, normalized size of antiderivative = 3.76, number of steps used = 15, number of rules used = 9, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6688, 6742, 2295, 2297, 2299, 2178, 2361, 6482, 12} \begin {gather*} -\frac {80 \left (1-\log \left (\frac {1}{2 x}\right )\right ) \text {Ei}(\log (x)+4)}{e^4}-\frac {80 \log \left (\frac {1}{2 x}\right ) \text {Ei}(\log (x)+4)}{e^4}+\frac {80 \text {Ei}(\log (x)+4)}{e^4}-20 x \log \left (\frac {1}{2 x}\right )+\frac {80 x \log \left (\frac {1}{2 x}\right )}{\log (x)+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2295
Rule 2297
Rule 2299
Rule 2361
Rule 6482
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-20 \log \left (\frac {1}{2 x}\right ) (2+\log (x))^2+20 \log (x) (4+\log (x))}{(4+\log (x))^2} \, dx\\ &=\int \left (-20 \left (-1+\log \left (\frac {1}{2 x}\right )\right )-\frac {80 \log \left (\frac {1}{2 x}\right )}{(4+\log (x))^2}+\frac {80 \left (-1+\log \left (\frac {1}{2 x}\right )\right )}{4+\log (x)}\right ) \, dx\\ &=-\left (20 \int \left (-1+\log \left (\frac {1}{2 x}\right )\right ) \, dx\right )-80 \int \frac {\log \left (\frac {1}{2 x}\right )}{(4+\log (x))^2} \, dx+80 \int \frac {-1+\log \left (\frac {1}{2 x}\right )}{4+\log (x)} \, dx\\ &=20 x-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}-20 \int \log \left (\frac {1}{2 x}\right ) \, dx+80 \int \frac {\text {Ei}(4+\log (x))}{e^4 x} \, dx-80 \int \left (\frac {\text {Ei}(4+\log (x))}{e^4 x}-\frac {1}{4+\log (x)}\right ) \, dx\\ &=-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}+80 \int \frac {1}{4+\log (x)} \, dx\\ &=-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}+80 \operatorname {Subst}\left (\int \frac {e^x}{4+x} \, dx,x,\log (x)\right )\\ &=\frac {80 \text {Ei}(4+\log (x))}{e^4}-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 19, normalized size = 0.90 \begin {gather*} -\frac {20 x \log \left (\frac {1}{2 x}\right ) \log (x)}{4+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 35, normalized size = 1.67 \begin {gather*} -\frac {20 \, {\left (x \log \relax (2) \log \left (\frac {1}{2 \, x}\right ) + x \log \left (\frac {1}{2 \, x}\right )^{2}\right )}}{\log \relax (2) + \log \left (\frac {1}{2 \, x}\right ) - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 27, normalized size = 1.29 \begin {gather*} \frac {20 \, x \log \relax (2) \log \relax (x)}{\log \relax (x) + 4} + \frac {20 \, x \log \relax (x)^{2}}{\log \relax (x) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 18, normalized size = 0.86
method | result | size |
norman | \(-\frac {20 \ln \relax (x ) \ln \left (\frac {1}{2 x}\right ) x}{\ln \relax (x )+4}\) | \(18\) |
risch | \(20 x \ln \relax (x )+20 x \ln \relax (2)-80 x -\frac {40 x \left (2 \ln \relax (2)-8\right )}{\ln \relax (x )+4}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 21, normalized size = 1.00 \begin {gather*} \frac {20 \, {\left (x \log \relax (2) \log \relax (x) + x \log \relax (x)^{2}\right )}}{\log \relax (x) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 17, normalized size = 0.81 \begin {gather*} -\frac {20\,x\,\ln \left (\frac {1}{2\,x}\right )\,\ln \relax (x)}{\ln \relax (x)+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 29, normalized size = 1.38 \begin {gather*} 20 x \log {\relax (x )} + x \left (-80 + 20 \log {\relax (2 )}\right ) + \frac {- 80 x \log {\relax (2 )} + 320 x}{\log {\relax (x )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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