Optimal. Leaf size=22 \[ 2 e^x \left (4+\frac {1}{4} e^4 x \left (6+x^4\right ) \log (x)\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 22, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {12, 2196, 2194, 2176, 2554} \begin {gather*} \frac {1}{2} e^{x+4} x^5 \log (x)+8 e^x+3 e^{x+4} x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rule 2554
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (e^x \left (16+e^4 \left (6+x^4\right )\right )+e^{4+x} \left (6+6 x+5 x^4+x^5\right ) \log (x)\right ) \, dx\\ &=\frac {1}{2} \int e^x \left (16+e^4 \left (6+x^4\right )\right ) \, dx+\frac {1}{2} \int e^{4+x} \left (6+6 x+5 x^4+x^5\right ) \log (x) \, dx\\ &=3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)-\frac {1}{2} \int e^{4+x} \left (6+x^4\right ) \, dx+\frac {1}{2} \int \left (16 e^x+e^{4+x} \left (6+x^4\right )\right ) \, dx\\ &=3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)+\frac {1}{2} \int e^{4+x} \left (6+x^4\right ) \, dx-\frac {1}{2} \int \left (6 e^{4+x}+e^{4+x} x^4\right ) \, dx+8 \int e^x \, dx\\ &=8 e^x+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)-\frac {1}{2} \int e^{4+x} x^4 \, dx+\frac {1}{2} \int \left (6 e^{4+x}+e^{4+x} x^4\right ) \, dx-3 \int e^{4+x} \, dx\\ &=8 e^x-3 e^{4+x}-\frac {1}{2} e^{4+x} x^4+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)+\frac {1}{2} \int e^{4+x} x^4 \, dx+2 \int e^{4+x} x^3 \, dx+3 \int e^{4+x} \, dx\\ &=8 e^x+2 e^{4+x} x^3+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)-2 \int e^{4+x} x^3 \, dx-6 \int e^{4+x} x^2 \, dx\\ &=8 e^x-6 e^{4+x} x^2+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)+6 \int e^{4+x} x^2 \, dx+12 \int e^{4+x} x \, dx\\ &=8 e^x+12 e^{4+x} x+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)-12 \int e^{4+x} \, dx-12 \int e^{4+x} x \, dx\\ &=8 e^x-12 e^{4+x}+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)+12 \int e^{4+x} \, dx\\ &=8 e^x+3 e^{4+x} x \log (x)+\frac {1}{2} e^{4+x} x^5 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 24, normalized size = 1.09 \begin {gather*} \frac {1}{2} \left (16 e^x+e^{4+x} x \left (6+x^4\right ) \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 25, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, {\left ({\left (x^{5} + 6 \, x\right )} e^{\left (x + 8\right )} \log \relax (x) + 16 \, e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 74, normalized size = 3.36 \begin {gather*} \frac {1}{2} \, {\left (x^{5} + 6 \, x\right )} e^{\left (x + 4\right )} \log \relax (x) + \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 30\right )} e^{\left (x + 4\right )} - \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{\left (x + 4\right )} - 3 \, e^{\left (x + 4\right )} + 8 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 26, normalized size = 1.18
method | result | size |
risch | \(\frac {\ln \relax (x ) {\mathrm e}^{4+x} x^{5}}{2}+3 \ln \relax (x ) {\mathrm e}^{4+x} x +8 \,{\mathrm e}^{x}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 97, normalized size = 4.41 \begin {gather*} \frac {1}{2} \, {\left (x^{5} e^{4} + 6 \, x e^{4}\right )} e^{x} \log \relax (x) - \frac {1}{2} \, {\left (x^{4} e^{4} - 4 \, x^{3} e^{4} + 12 \, x^{2} e^{4} - 24 \, x e^{4} + 30 \, e^{4}\right )} e^{x} + \frac {1}{2} \, {\left (x^{4} e^{4} - 4 \, x^{3} e^{4} + 12 \, x^{2} e^{4} - 24 \, x e^{4} + 24 \, e^{4}\right )} e^{x} + 3 \, e^{\left (x + 4\right )} + 8 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.24, size = 23, normalized size = 1.05 \begin {gather*} 8\,{\mathrm {e}}^x+{\mathrm {e}}^x\,\ln \relax (x)\,\left (\frac {{\mathrm {e}}^4\,x^5}{2}+3\,{\mathrm {e}}^4\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 26, normalized size = 1.18 \begin {gather*} \frac {\left (x^{5} e^{4} \log {\relax (x )} + 6 x e^{4} \log {\relax (x )} + 16\right ) e^{x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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