Optimal. Leaf size=23 \[ 4-\frac {4}{5} \left (1+e^x\right )-x-\log (\log (4+2 x)) \]
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Rubi [A]
time = 0.16, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps
used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6820, 2225,
2437, 12, 2339, 29} \begin {gather*} -x-\frac {4 e^x}{5}-\log (\log (2 (x+2))) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2225
Rule 2339
Rule 2437
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {4 e^x}{5}-\frac {1}{(2+x) \log (4+2 x)}\right ) \, dx\\ &=-x-\frac {4 \int e^x \, dx}{5}-\int \frac {1}{(2+x) \log (4+2 x)} \, dx\\ &=-\frac {4 e^x}{5}-x-\frac {1}{2} \text {Subst}\left (\int \frac {2}{x \log (x)} \, dx,x,4+2 x\right )\\ &=-\frac {4 e^x}{5}-x-\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+2 x\right )\\ &=-\frac {4 e^x}{5}-x-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (2 (2+x))\right )\\ &=-\frac {4 e^x}{5}-x-\log (\log (2 (2+x)))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 20, normalized size = 0.87 \begin {gather*} -\frac {4 e^x}{5}-x-\log (\log (2 (2+x))) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 18, normalized size = 0.78
method | result | size |
default | \(-x -\ln \left (\ln \left (2 x +4\right )\right )-\frac {4 \,{\mathrm e}^{x}}{5}\) | \(18\) |
norman | \(-x -\ln \left (\ln \left (2 x +4\right )\right )-\frac {4 \,{\mathrm e}^{x}}{5}\) | \(18\) |
risch | \(-x -\ln \left (\ln \left (2 x +4\right )\right )-\frac {4 \,{\mathrm e}^{x}}{5}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 17, normalized size = 0.74 \begin {gather*} -x - \frac {4}{5} \, e^{x} - \log \left (\log \left (2 \, x + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 17, normalized size = 0.74 \begin {gather*} - x - \frac {4 e^{x}}{5} - \log {\left (\log {\left (2 x + 4 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.74 \begin {gather*} -x - \frac {4}{5} \, e^{x} - \log \left (\log \left (2 \, x + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 17, normalized size = 0.74 \begin {gather*} -x-\ln \left (\ln \left (2\,x+4\right )\right )-\frac {4\,{\mathrm {e}}^x}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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