Optimal. Leaf size=15 \[ \frac {8}{9} \log \left (2 e^{e^x}+\log (3)\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 2282, 2246, 31} \begin {gather*} \frac {8}{9} \log \left (2 e^{e^x}+\log (3)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 2246
Rule 2282
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=16 \int \frac {e^{e^x+x}}{18 e^{e^x}+9 \log (3)} \, dx\\ &=16 \operatorname {Subst}\left (\int \frac {e^x}{9 \left (2 e^x+\log (3)\right )} \, dx,x,e^x\right )\\ &=\frac {16}{9} \operatorname {Subst}\left (\int \frac {e^x}{2 e^x+\log (3)} \, dx,x,e^x\right )\\ &=\frac {16}{9} \operatorname {Subst}\left (\int \frac {1}{2 x+\log (3)} \, dx,x,e^{e^x}\right )\\ &=\frac {8}{9} \log \left (2 e^{e^x}+\log (3)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {8}{9} \log \left (2 e^{e^x}+\log (3)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 20, normalized size = 1.33 \begin {gather*} -\frac {8}{9} \, x + \frac {8}{9} \, \log \left (e^{x} \log \relax (3) + 2 \, e^{\left (x + e^{x}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 11, normalized size = 0.73 \begin {gather*} \frac {8}{9} \, \log \left (2 \, e^{\left (e^{x}\right )} + \log \relax (3)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {8 \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{x}}+\ln \relax (3)\right )}{9}\) | \(12\) |
default | \(\frac {8 \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{x}}+\ln \relax (3)\right )}{9}\) | \(12\) |
risch | \(\frac {8 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {\ln \relax (3)}{2}\right )}{9}\) | \(12\) |
norman | \(\frac {8 \ln \left (18 \,{\mathrm e}^{{\mathrm e}^{x}}+9 \ln \relax (3)\right )}{9}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 11, normalized size = 0.73 \begin {gather*} \frac {8}{9} \, \log \left (2 \, e^{\left (e^{x}\right )} + \log \relax (3)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 11, normalized size = 0.73 \begin {gather*} \frac {8\,\ln \left (2\,{\mathrm {e}}^{{\mathrm {e}}^x}+\ln \relax (3)\right )}{9} \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 {8 \log {\left (e^{e^{x}} + \frac {\log {\relax (3 )}}{2} \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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