Optimal. Leaf size=24 \[ 3 e^{e^{\frac {\log ^2(10)}{5}} \left (1+4 e^{-x}\right )} \]
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Rubi [A]
time = 0.19, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps
used = 3, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12, 2320, 2240}
\begin {gather*} 3 e^{4 e^{\frac {\log ^2(10)}{5}-x}+e^{\frac {\log ^2(10)}{5}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2240
Rule 2320
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (12 \int \frac {\exp \left (e^{\frac {1}{5} \left (\log ^2(10)+5 \log \left (e^{-x} \left (4+e^x\right )\right )\right )}+\frac {1}{5} \left (\log ^2(10)+5 \log \left (e^{-x} \left (4+e^x\right )\right )\right )\right )}{4+e^x} \, dx\right )\\ &=-\left (12 \text {Subst}\left (\int \frac {e^{e^{\frac {\log ^2(10)}{5}}+\frac {4 e^{\frac {\log ^2(10)}{5}}}{x}+\frac {\log ^2(10)}{5}}}{x^2} \, dx,x,e^x\right )\right )\\ &=3 e^{e^{\frac {\log ^2(10)}{5}}+4 e^{-x+\frac {\log ^2(10)}{5}}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 24, normalized size = 1.00 \begin {gather*} 3 e^{e^{-x+\frac {\log ^2(10)}{5}} \left (4+e^x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs.
\(2(20)=40\).
time = 0.11, size = 61, normalized size = 2.54
method | result | size |
norman | \(3 \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \left (10\right )^{2}}{5}} \left ({\mathrm e}^{x}+4\right ) {\mathrm e}^{-x}}\) | \(20\) |
derivativedivides | \(3 \,{\mathrm e}^{\frac {\ln \left (5\right )^{2}}{5}+\frac {2 \ln \left (2\right ) \ln \left (5\right )}{5}+\frac {\ln \left (2\right )^{2}}{5}+{\mathrm e}^{\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}}+4 \,{\mathrm e}^{\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}} {\mathrm e}^{-x}} {\mathrm e}^{-\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}}\) | \(61\) |
default | \(3 \,{\mathrm e}^{\frac {\ln \left (5\right )^{2}}{5}+\frac {2 \ln \left (2\right ) \ln \left (5\right )}{5}+\frac {\ln \left (2\right )^{2}}{5}+{\mathrm e}^{\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}}+4 \,{\mathrm e}^{\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}} {\mathrm e}^{-x}} {\mathrm e}^{-\frac {\left (\ln \left (2\right )+\ln \left (5\right )\right )^{2}}{5}}\) | \(61\) |
risch | \(3 \,{\mathrm e}^{2^{\frac {2 \ln \left (5\right )}{5}} \left ({\mathrm e}^{x}+4\right ) {\mathrm e}^{-x -\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+4\right )\right )^{3}}{2}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+4\right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )}{2}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+4\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+4\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+4\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+4\right )\right )}{2}+\frac {\ln \left (5\right )^{2}}{5}+\frac {\ln \left (2\right )^{2}}{5}}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (20) = 40\).
time = 0.50, size = 51, normalized size = 2.12 \begin {gather*} 3 \, e^{\left (2^{\frac {2}{5} \, \log \left (5\right ) + 2} e^{\left (\frac {1}{5} \, \log \left (5\right )^{2} + \frac {1}{5} \, \log \left (2\right )^{2} - x\right )} + 2^{\frac {2}{5} \, \log \left (5\right )} e^{\left (\frac {1}{5} \, \log \left (5\right )^{2} + \frac {1}{5} \, \log \left (2\right )^{2}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (20) = 40\).
time = 0.35, size = 58, normalized size = 2.42 \begin {gather*} \frac {3 \, e^{\left (\frac {1}{5} \, {\left (e^{x} \log \left (10\right )^{2} + 5 \, {\left (e^{x} + 4\right )} e^{\left (\frac {1}{5} \, \log \left (10\right )^{2}\right )} + 5 \, e^{x} \log \left ({\left (e^{x} + 4\right )} e^{\left (-x\right )}\right )\right )} e^{\left (-x\right )} - \frac {1}{5} \, \log \left (10\right )^{2} + x\right )}}{e^{x} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 19, normalized size = 0.79 \begin {gather*} 3 e^{\left (e^{x} + 4\right ) e^{- x} e^{\frac {\log {\left (10 \right )}^{2}}{5}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 24, normalized size = 1.00 \begin {gather*} 3 \, e^{\left (e^{\left (\frac {1}{5} \, \log \left (10\right )^{2}\right )} + 4 \, e^{\left (\frac {1}{5} \, \log \left (10\right )^{2} - x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.46, size = 24, normalized size = 1.00 \begin {gather*} 3\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {{\ln \left (10\right )}^2}{5}}}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\ln \left (10\right )}^2}{5}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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