Optimal. Leaf size=25 \[ -36 x^2 \left (-e^x+x\right )^2+\frac {x}{\frac {1}{4}+\log (4)} \]
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Rubi [B] time = 0.40, antiderivative size = 79, normalized size of antiderivative = 3.16, number of steps used = 35, number of rules used = 5, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.063, Rules used = {6, 12, 2196, 2176, 2194} \begin {gather*} -36 x^4+\frac {72 e^x x^3}{1+\log (256)}+\frac {288 e^x x^3 \log (4)}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}-\frac {144 e^{2 x} x^2 \log (4)}{1+\log (256)}+\frac {4 x}{1+\log (256)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+x^3 (-144-576 \log (4))+e^{2 x} \left (-72 x-72 x^2+\left (-288 x-288 x^2\right ) \log (4)\right )+e^x \left (216 x^2+72 x^3+\left (864 x^2+288 x^3\right ) \log (4)\right )}{1+4 \log (4)} \, dx\\ &=\frac {\int \left (4+x^3 (-144-576 \log (4))+e^{2 x} \left (-72 x-72 x^2+\left (-288 x-288 x^2\right ) \log (4)\right )+e^x \left (216 x^2+72 x^3+\left (864 x^2+288 x^3\right ) \log (4)\right )\right ) \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}+\frac {\int e^{2 x} \left (-72 x-72 x^2+\left (-288 x-288 x^2\right ) \log (4)\right ) \, dx}{1+\log (256)}+\frac {\int e^x \left (216 x^2+72 x^3+\left (864 x^2+288 x^3\right ) \log (4)\right ) \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}+\frac {\int \left (-72 e^{2 x} x-72 e^{2 x} x^2-288 e^{2 x} x (1+x) \log (4)\right ) \, dx}{1+\log (256)}+\frac {\int \left (216 e^x x^2+72 e^x x^3+288 e^x x^2 (3+x) \log (4)\right ) \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}-\frac {72 \int e^{2 x} x \, dx}{1+\log (256)}-\frac {72 \int e^{2 x} x^2 \, dx}{1+\log (256)}+\frac {72 \int e^x x^3 \, dx}{1+\log (256)}+\frac {216 \int e^x x^2 \, dx}{1+\log (256)}-\frac {(288 \log (4)) \int e^{2 x} x (1+x) \, dx}{1+\log (256)}+\frac {(288 \log (4)) \int e^x x^2 (3+x) \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}-\frac {36 e^{2 x} x}{1+\log (256)}+\frac {216 e^x x^2}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}+\frac {36 \int e^{2 x} \, dx}{1+\log (256)}+\frac {72 \int e^{2 x} x \, dx}{1+\log (256)}-\frac {216 \int e^x x^2 \, dx}{1+\log (256)}-\frac {432 \int e^x x \, dx}{1+\log (256)}-\frac {(288 \log (4)) \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx}{1+\log (256)}+\frac {(288 \log (4)) \int \left (3 e^x x^2+e^x x^3\right ) \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {18 e^{2 x}}{1+\log (256)}+\frac {4 x}{1+\log (256)}-\frac {432 e^x x}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}-\frac {36 \int e^{2 x} \, dx}{1+\log (256)}+\frac {432 \int e^x \, dx}{1+\log (256)}+\frac {432 \int e^x x \, dx}{1+\log (256)}-\frac {(288 \log (4)) \int e^{2 x} x \, dx}{1+\log (256)}-\frac {(288 \log (4)) \int e^{2 x} x^2 \, dx}{1+\log (256)}+\frac {(288 \log (4)) \int e^x x^3 \, dx}{1+\log (256)}+\frac {(864 \log (4)) \int e^x x^2 \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {432 e^x}{1+\log (256)}+\frac {4 x}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}-\frac {144 e^{2 x} x \log (4)}{1+\log (256)}+\frac {864 e^x x^2 \log (4)}{1+\log (256)}-\frac {144 e^{2 x} x^2 \log (4)}{1+\log (256)}+\frac {288 e^x x^3 \log (4)}{1+\log (256)}-\frac {432 \int e^x \, dx}{1+\log (256)}+\frac {(144 \log (4)) \int e^{2 x} \, dx}{1+\log (256)}+\frac {(288 \log (4)) \int e^{2 x} x \, dx}{1+\log (256)}-\frac {(864 \log (4)) \int e^x x^2 \, dx}{1+\log (256)}-\frac {(1728 \log (4)) \int e^x x \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}+\frac {72 e^{2 x} \log (4)}{1+\log (256)}-\frac {1728 e^x x \log (4)}{1+\log (256)}-\frac {144 e^{2 x} x^2 \log (4)}{1+\log (256)}+\frac {288 e^x x^3 \log (4)}{1+\log (256)}-\frac {(144 \log (4)) \int e^{2 x} \, dx}{1+\log (256)}+\frac {(1728 \log (4)) \int e^x \, dx}{1+\log (256)}+\frac {(1728 \log (4)) \int e^x x \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}+\frac {1728 e^x \log (4)}{1+\log (256)}-\frac {144 e^{2 x} x^2 \log (4)}{1+\log (256)}+\frac {288 e^x x^3 \log (4)}{1+\log (256)}-\frac {(1728 \log (4)) \int e^x \, dx}{1+\log (256)}\\ &=-36 x^4+\frac {4 x}{1+\log (256)}-\frac {36 e^{2 x} x^2}{1+\log (256)}+\frac {72 e^x x^3}{1+\log (256)}-\frac {144 e^{2 x} x^2 \log (4)}{1+\log (256)}+\frac {288 e^x x^3 \log (4)}{1+\log (256)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 34, normalized size = 1.36 \begin {gather*} 4 \left (-9 e^{2 x} x^2+18 e^x x^3-9 x^4+\frac {x}{1+\log (256)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.56, size = 58, normalized size = 2.32 \begin {gather*} -\frac {4 \, {\left (72 \, x^{4} \log \relax (2) + 9 \, x^{4} + 9 \, {\left (8 \, x^{2} \log \relax (2) + x^{2}\right )} e^{\left (2 \, x\right )} - 18 \, {\left (8 \, x^{3} \log \relax (2) + x^{3}\right )} e^{x} - x\right )}}{8 \, \log \relax (2) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 58, normalized size = 2.32 \begin {gather*} -\frac {4 \, {\left (72 \, x^{4} \log \relax (2) + 9 \, x^{4} + 9 \, {\left (8 \, x^{2} \log \relax (2) + x^{2}\right )} e^{\left (2 \, x\right )} - 18 \, {\left (8 \, x^{3} \log \relax (2) + x^{3}\right )} e^{x} - x\right )}}{8 \, \log \relax (2) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 1.36
method | result | size |
norman | \(-36 x^{4}+\frac {4 x}{8 \ln \relax (2)+1}+72 \,{\mathrm e}^{x} x^{3}-36 \,{\mathrm e}^{2 x} x^{2}\) | \(34\) |
risch | \(-\frac {288 x^{4} \ln \relax (2)}{8 \ln \relax (2)+1}-\frac {36 x^{4}}{8 \ln \relax (2)+1}+\frac {4 x}{8 \ln \relax (2)+1}-36 \,{\mathrm e}^{2 x} x^{2}+72 \,{\mathrm e}^{x} x^{3}\) | \(57\) |
default | \(\frac {4 x -36 \,{\mathrm e}^{2 x} x^{2}-288 x^{2} \ln \relax (2) {\mathrm e}^{2 x}+72 \,{\mathrm e}^{x} x^{3}+576 x^{3} \ln \relax (2) {\mathrm e}^{x}-36 x^{4}-288 x^{4} \ln \relax (2)}{8 \ln \relax (2)+1}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 52, normalized size = 2.08 \begin {gather*} \frac {4 \, {\left (18 \, x^{3} {\left (8 \, \log \relax (2) + 1\right )} e^{x} - 72 \, x^{4} \log \relax (2) - 9 \, x^{4} - 9 \, x^{2} {\left (8 \, \log \relax (2) + 1\right )} e^{\left (2 \, x\right )} + x\right )}}{8 \, \log \relax (2) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 51, normalized size = 2.04 \begin {gather*} \frac {4\,x-x^4\,\left (288\,\ln \relax (2)+36\right )+x^3\,{\mathrm {e}}^x\,\left (576\,\ln \relax (2)+72\right )-x^2\,{\mathrm {e}}^{2\,x}\,\left (288\,\ln \relax (2)+36\right )}{8\,\ln \relax (2)+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 1.28 \begin {gather*} - 36 x^{4} + 72 x^{3} e^{x} - 36 x^{2} e^{2 x} + \frac {4 x}{1 + 8 \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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