Optimal. Leaf size=21 \[ e^4 \left (4-(2+x) \log \left (\frac {x}{4+2 x}\right )\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.62, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {14, 2486, 31} \begin {gather*} -2 e^4 \log (x)-e^4 x \log \left (\frac {x}{2 (x+2)}\right )+2 e^4 \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 31
Rule 2486
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 e^4}{x}-e^4 \log \left (\frac {x}{4+2 x}\right )\right ) \, dx\\ &=-2 e^4 \log (x)-e^4 \int \log \left (\frac {x}{4+2 x}\right ) \, dx\\ &=-2 e^4 \log (x)-e^4 x \log \left (\frac {x}{2 (2+x)}\right )+\left (4 e^4\right ) \int \frac {1}{4+2 x} \, dx\\ &=-2 e^4 \log (x)-e^4 x \log \left (\frac {x}{2 (2+x)}\right )+2 e^4 \log (2+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 33, normalized size = 1.57 \begin {gather*} -2 e^4 \log (x)+2 e^4 \log (2+x)-e^4 x \log \left (\frac {x}{4+2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 16, normalized size = 0.76 \begin {gather*} -{\left (x + 2\right )} e^{4} \log \left (\frac {x}{2 \, {\left (x + 2\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 1.14 \begin {gather*} \frac {2 \, e^{4} \log \left (\frac {x}{2 \, {\left (x + 2\right )}}\right )}{\frac {x}{x + 2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 31, normalized size = 1.48
method | result | size |
risch | \(-x \,{\mathrm e}^{4} \ln \left (\frac {x}{2 x +4}\right )-2 \,{\mathrm e}^{4} \ln \relax (x )+2 \,{\mathrm e}^{4} \ln \left (2+x \right )\) | \(31\) |
norman | \(-2 \,{\mathrm e}^{4} \ln \left (\frac {x}{2 x +4}\right )-x \,{\mathrm e}^{4} \ln \left (\frac {x}{2 x +4}\right )\) | \(35\) |
derivativedivides | \(-2 \,{\mathrm e}^{4} \ln \left (\frac {1}{2}-\frac {1}{2+x}\right ) \left (\frac {1}{2}-\frac {1}{2+x}\right ) \left (2+x \right )-2 \,{\mathrm e}^{4} \ln \left (\frac {1}{2}-\frac {1}{2+x}\right )\) | \(46\) |
default | \(-2 \,{\mathrm e}^{4} \ln \left (\frac {1}{2}-\frac {1}{2+x}\right ) \left (\frac {1}{2}-\frac {1}{2+x}\right ) \left (2+x \right )-2 \,{\mathrm e}^{4} \ln \left (\frac {1}{2}-\frac {1}{2+x}\right )\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 29, normalized size = 1.38 \begin {gather*} -{\left (x \log \left (\frac {x}{2 \, {\left (x + 2\right )}}\right ) - 2 \, \log \left (x + 2\right )\right )} e^{4} - 2 \, e^{4} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 17, normalized size = 0.81 \begin {gather*} -{\mathrm {e}}^4\,\ln \left (\frac {x}{2\,x+4}\right )\,\left (x+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 31, normalized size = 1.48 \begin {gather*} - x e^{4} \log {\left (\frac {x}{2 x + 4} \right )} - 4 \left (\frac {\log {\relax (x )}}{2} - \frac {\log {\left (x + 2 \right )}}{2}\right ) e^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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