Optimal. Leaf size=15 \[ \frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \]
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Rubi [B] time = 0.55, antiderivative size = 75, normalized size of antiderivative = 5.00, number of steps used = 22, number of rules used = 11, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {6741, 6742, 2411, 12, 2353, 2297, 2298, 2302, 30, 29, 2390} \begin {gather*} \frac {x+4+\log (4)}{\log \left (\frac {x+4+\log (4)}{\log (2)}\right )}-(4+\log (4)) \log \left (\log \left (\frac {x+4+\log (4)}{\log (2)}\right )\right )+2 (2+\log (2)) \log \left (\log \left (\frac {x+4+\log (4)}{\log (2)}\right )\right )-\frac {4+\log (4)}{\log \left (\frac {x+4+\log (4)}{\log (2)}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 30
Rule 2297
Rule 2298
Rule 2302
Rule 2353
Rule 2390
Rule 2411
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )} \, dx\\ &=\int \left (\frac {x}{(-4-x-\log (4)) \log ^2\left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )}+\frac {x}{(4+x+\log (4)) \log \left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )}+\frac {4 \left (1+\frac {\log (2)}{2}\right )}{(4+x+\log (4)) \log \left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )}\right ) \, dx\\ &=\left (4 \left (1+\frac {\log (2)}{2}\right )\right ) \int \frac {1}{(4+x+\log (4)) \log \left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )} \, dx+\int \frac {x}{(-4-x-\log (4)) \log ^2\left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )} \, dx+\int \frac {x}{(4+x+\log (4)) \log \left (\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )} \, dx\\ &=-\left (\log (2) \operatorname {Subst}\left (\int \frac {-4+x \log (2)-\log (4)}{x \log (2) \log ^2(x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )\right )+\log (2) \operatorname {Subst}\left (\int \frac {-4+x \log (2)-\log (4)}{x \log (2) \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+(2 \log (2) (2+\log (2))) \operatorname {Subst}\left (\int \frac {1}{x \log (2) \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )\\ &=(2 (2+\log (2))) \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-\operatorname {Subst}\left (\int \frac {-4+x \log (2)-\log (4)}{x \log ^2(x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+\operatorname {Subst}\left (\int \frac {-4+x \log (2)-\log (4)}{x \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )\\ &=(2 (2+\log (2))) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )-\operatorname {Subst}\left (\int \left (\frac {\log (2)}{\log ^2(x)}+\frac {-4-\log (4)}{x \log ^2(x)}\right ) \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+\operatorname {Subst}\left (\int \left (\frac {\log (2)}{\log (x)}+\frac {-4-\log (4)}{x \log (x)}\right ) \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )\\ &=2 (2+\log (2)) \log \left (\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )-\log (2) \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+\log (2) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-(-4-\log (4)) \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+(-4-\log (4)) \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )\\ &=\frac {4+x+\log (4)}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )}+2 (2+\log (2)) \log \left (\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )+\log (2) \text {li}\left (\frac {4+x+\log (4)}{\log (2)}\right )-\log (2) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-(-4-\log (4)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )+(-4-\log (4)) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )\\ &=-\frac {4+\log (4)}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )}+\frac {4+x+\log (4)}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )}+2 (2+\log (2)) \log \left (\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )-(4+\log (4)) \log \left (\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 15, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 17, normalized size = 1.13 \begin {gather*} \frac {x}{\log \left (\frac {x + 2 \, \log \relax (2) + 4}{\log \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 18, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (x + 2 \, \log \relax (2) + 4\right ) - \log \left (\log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 18, normalized size = 1.20
method | result | size |
norman | \(\frac {x}{\ln \left (\frac {2 \ln \relax (2)+4+x}{\ln \relax (2)}\right )}\) | \(18\) |
risch | \(\frac {x}{\ln \left (\frac {2 \ln \relax (2)+4+x}{\ln \relax (2)}\right )}\) | \(18\) |
derivativedivides | \(\ln \relax (2) \left (\frac {\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {4}{\ln \relax (2) \ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}\right )\) | \(91\) |
default | \(\ln \relax (2) \left (\frac {\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {4}{\ln \relax (2) \ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}\right )\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 18, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (x + 2 \, \log \relax (2) + 4\right ) - \log \left (\log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 15, normalized size = 1.00 \begin {gather*} \frac {x}{\ln \left (\frac {x+\ln \relax (4)+4}{\ln \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.93 \begin {gather*} \frac {x}{\log {\left (\frac {x + 2 \log {\relax (2 )} + 4}{\log {\relax (2 )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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