Optimal. Leaf size=15 \[ \frac {3 x (1+\log (3))}{x+\log (4+x)} \]
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Rubi [A] time = 0.15, antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6, 6688, 12, 6711, 32} \begin {gather*} -\frac {3 (1+\log (3))}{\frac {x}{\log (x+4)}+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 32
Rule 6688
Rule 6711
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x (-3-3 \log (3))+(12+3 x+(12+3 x) \log (3)) \log (4+x)}{4 x^2+x^3+\left (8 x+2 x^2\right ) \log (4+x)+(4+x) \log ^2(4+x)} \, dx\\ &=\int \frac {3 (1+\log (3)) (-x+(4+x) \log (4+x))}{(4+x) (x+\log (4+x))^2} \, dx\\ &=(3 (1+\log (3))) \int \frac {-x+(4+x) \log (4+x)}{(4+x) (x+\log (4+x))^2} \, dx\\ &=(3 (1+\log (3))) \operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{\log (4+x)}\right )\\ &=-\frac {3 (1+\log (3))}{1+\frac {x}{\log (4+x)}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.20, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 x (1+\log (3))}{x+\log (4+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 16, normalized size = 1.07 \begin {gather*} \frac {3 \, {\left (x \log \relax (3) + x\right )}}{x + \log \left (x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 16, normalized size = 1.07 \begin {gather*} \frac {3 \, {\left (x \log \relax (3) + x\right )}}{x + \log \left (x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 16, normalized size = 1.07
method | result | size |
risch | \(\frac {3 x \left (\ln \relax (3)+1\right )}{\ln \left (4+x \right )+x}\) | \(16\) |
norman | \(\frac {\left (-3 \ln \relax (3)-3\right ) \ln \left (4+x \right )}{\ln \left (4+x \right )+x}\) | \(20\) |
derivativedivides | \(\frac {3 x}{\ln \left (4+x \right )+x}-\frac {3 \ln \relax (3) \ln \left (4+x \right )}{\ln \left (4+x \right )+x}\) | \(29\) |
default | \(\frac {3 x}{\ln \left (4+x \right )+x}-\frac {3 \ln \relax (3) \ln \left (4+x \right )}{\ln \left (4+x \right )+x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 \, x {\left (\log \relax (3) + 1\right )}}{x + \log \left (x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 63, normalized size = 4.20 \begin {gather*} \frac {\left (\ln \left (27\right )+3\right )\,x^3+\left (24\,\ln \relax (3)+24\right )\,x^2+\left (48\,\ln \relax (3)+48\right )\,x}{16\,x+16\,\ln \left (x+4\right )+8\,x\,\ln \left (x+4\right )+x^2\,\ln \left (x+4\right )+8\,x^2+x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 15, normalized size = 1.00 \begin {gather*} \frac {3 x + 3 x \log {\relax (3 )}}{x + \log {\left (x + 4 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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