Optimal. Leaf size=25 \[ \left (x-e^{-3+x} x\right ) \left (6 (4+x)+\frac {8}{\log (2)}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(25)=50\).
time = 0.12, antiderivative size = 98, normalized size of antiderivative = 3.92, number of steps
used = 28, number of rules used = 4, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {12, 2227,
2225, 2207} \begin {gather*} -36 e^{x-3} x^3+36 x^3-288 e^{x-3} x^2+288 x^2-\frac {96 e^{x-3} x^2}{\log (2)}-576 e^{x-3} x+576 x-\frac {64 e^{x-3} x}{\log ^2(2)}+\frac {64 x}{\log ^2(2)}-\frac {384 e^{x-3} x}{\log (2)}+\frac {96 (x+2)^2}{\log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )\right ) \, dx}{e^3 \log ^2(2)}\\ &=\frac {64 x}{\log ^2(2)}+\frac {96 (2+x)^2}{\log (2)}+\frac {\int e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right ) \, dx}{e^3 \log ^2(2)}+\int \left (576+576 x+108 x^2\right ) \, dx\\ &=576 x+288 x^2+36 x^3+\frac {64 x}{\log ^2(2)}+\frac {96 (2+x)^2}{\log (2)}+\frac {\int \left (-64 e^x-64 e^x x-96 e^x \left (4+6 x+x^2\right ) \log (2)-36 e^x \left (16+32 x+11 x^2+x^3\right ) \log ^2(2)\right ) \, dx}{e^3 \log ^2(2)}\\ &=576 x+288 x^2+36 x^3+\frac {64 x}{\log ^2(2)}+\frac {96 (2+x)^2}{\log (2)}-\frac {36 \int e^x \left (16+32 x+11 x^2+x^3\right ) \, dx}{e^3}-\frac {64 \int e^x \, dx}{e^3 \log ^2(2)}-\frac {64 \int e^x x \, dx}{e^3 \log ^2(2)}-\frac {96 \int e^x \left (4+6 x+x^2\right ) \, dx}{e^3 \log (2)}\\ &=576 x+288 x^2+36 x^3-\frac {64 e^{-3+x}}{\log ^2(2)}+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}+\frac {96 (2+x)^2}{\log (2)}-\frac {36 \int \left (16 e^x+32 e^x x+11 e^x x^2+e^x x^3\right ) \, dx}{e^3}+\frac {64 \int e^x \, dx}{e^3 \log ^2(2)}-\frac {96 \int \left (4 e^x+6 e^x x+e^x x^2\right ) \, dx}{e^3 \log (2)}\\ &=576 x+288 x^2+36 x^3+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}+\frac {96 (2+x)^2}{\log (2)}-\frac {36 \int e^x x^3 \, dx}{e^3}-\frac {396 \int e^x x^2 \, dx}{e^3}-\frac {576 \int e^x \, dx}{e^3}-\frac {1152 \int e^x x \, dx}{e^3}-\frac {96 \int e^x x^2 \, dx}{e^3 \log (2)}-\frac {384 \int e^x \, dx}{e^3 \log (2)}-\frac {576 \int e^x x \, dx}{e^3 \log (2)}\\ &=-576 e^{-3+x}+576 x-1152 e^{-3+x} x+288 x^2-396 e^{-3+x} x^2+36 x^3-36 e^{-3+x} x^3+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}-\frac {384 e^{-3+x}}{\log (2)}-\frac {576 e^{-3+x} x}{\log (2)}-\frac {96 e^{-3+x} x^2}{\log (2)}+\frac {96 (2+x)^2}{\log (2)}+\frac {108 \int e^x x^2 \, dx}{e^3}+\frac {792 \int e^x x \, dx}{e^3}+\frac {1152 \int e^x \, dx}{e^3}+\frac {192 \int e^x x \, dx}{e^3 \log (2)}+\frac {576 \int e^x \, dx}{e^3 \log (2)}\\ &=576 e^{-3+x}+576 x-360 e^{-3+x} x+288 x^2-288 e^{-3+x} x^2+36 x^3-36 e^{-3+x} x^3+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}+\frac {192 e^{-3+x}}{\log (2)}-\frac {384 e^{-3+x} x}{\log (2)}-\frac {96 e^{-3+x} x^2}{\log (2)}+\frac {96 (2+x)^2}{\log (2)}-\frac {216 \int e^x x \, dx}{e^3}-\frac {792 \int e^x \, dx}{e^3}-\frac {192 \int e^x \, dx}{e^3 \log (2)}\\ &=-216 e^{-3+x}+576 x-576 e^{-3+x} x+288 x^2-288 e^{-3+x} x^2+36 x^3-36 e^{-3+x} x^3+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}-\frac {384 e^{-3+x} x}{\log (2)}-\frac {96 e^{-3+x} x^2}{\log (2)}+\frac {96 (2+x)^2}{\log (2)}+\frac {216 \int e^x \, dx}{e^3}\\ &=576 x-576 e^{-3+x} x+288 x^2-288 e^{-3+x} x^2+36 x^3-36 e^{-3+x} x^3+\frac {64 x}{\log ^2(2)}-\frac {64 e^{-3+x} x}{\log ^2(2)}-\frac {384 e^{-3+x} x}{\log (2)}-\frac {96 e^{-3+x} x^2}{\log (2)}+\frac {96 (2+x)^2}{\log (2)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(25)=50\).
time = 0.11, size = 89, normalized size = 3.56 \begin {gather*} -\frac {4 \left (-\frac {1}{2} e^3 x \left (2 x^2 \log ^2(8)+x \log (4096) (4+\log (4096))+2 (4+\log (4096))^2\right )+e^x x \left (16+6 \log ^2(8)+x^2 \log ^2(8)+21 \log (2) (4+\log (64))+\log (4096)-\log (8) \log (4096)+x \log (8) (8+\log (16777216))\right )\right )}{e^3 \log ^2(2)} \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. \(114\) vs.
\(2(23)=46\).
time = 0.10, size = 115, normalized size = 4.60
method | result | size |
risch | \(36 x^{3}+288 x^{2}+576 x +\frac {96 x^{2}}{\ln \left (2\right )}+\frac {384 x}{\ln \left (2\right )}+\frac {64 x}{\ln \left (2\right )^{2}}+\frac {\left (-36 x^{3} \ln \left (2\right )^{2}-288 x^{2} \ln \left (2\right )^{2}-576 x \ln \left (2\right )^{2}-96 x^{2} \ln \left (2\right )-384 x \ln \left (2\right )-64 x \right ) {\mathrm e}^{x -3}}{\ln \left (2\right )^{2}}\) | \(88\) |
norman | \(\frac {\left (288 \ln \left (2\right )+96\right ) x^{2}+36 x^{3} \ln \left (2\right )+\frac {64 \left (9 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) x}{\ln \left (2\right )}-96 \left (3 \ln \left (2\right )+1\right ) {\mathrm e}^{-3} x^{2} {\mathrm e}^{x}-36 \ln \left (2\right ) {\mathrm e}^{-3} x^{3} {\mathrm e}^{x}-\frac {64 \left (9 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) {\mathrm e}^{-3} x \,{\mathrm e}^{x}}{\ln \left (2\right )}}{\ln \left (2\right )}\) | \(98\) |
default | \(\frac {{\mathrm e}^{-3} \left (-64 \,{\mathrm e}^{x} x -36 \ln \left (2\right )^{2} {\mathrm e}^{x} x^{3}-288 \ln \left (2\right )^{2} {\mathrm e}^{x} x^{2}-576 x \ln \left (2\right )^{2} {\mathrm e}^{x}-384 x \ln \left (2\right ) {\mathrm e}^{x}-96 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+96 \ln \left (2\right ) {\mathrm e}^{3} x^{2}+384 \ln \left (2\right ) {\mathrm e}^{3} x +36 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x^{3}+288 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x^{2}+576 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x +64 x \,{\mathrm e}^{3}\right )}{\ln \left (2\right )^{2}}\) | \(115\) |
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. 90 vs.
\(2 (25) = 50\).
time = 0.47, size = 90, normalized size = 3.60 \begin {gather*} \frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, x^{3} \log \left (2\right )^{2} + 24 \, {\left (3 \, \log \left (2\right )^{2} + \log \left (2\right )\right )} x^{2} + 16 \, {\left (9 \, \log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 1\right )} x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \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. 84 vs.
\(2 (25) = 50\).
time = 0.37, size = 84, normalized size = 3.36 \begin {gather*} \frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right ) + 16 \, x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (20) = 40\).
time = 0.11, size = 99, normalized size = 3.96 \begin {gather*} 36 x^{3} + \frac {x^{2} \cdot \left (96 + 288 \log {\left (2 \right )}\right )}{\log {\left (2 \right )}} + \frac {x \left (64 + 384 \log {\left (2 \right )} + 576 \log {\left (2 \right )}^{2}\right )}{\log {\left (2 \right )}^{2}} + \frac {\left (- 36 x^{3} \log {\left (2 \right )}^{2} - 288 x^{2} \log {\left (2 \right )}^{2} - 96 x^{2} \log {\left (2 \right )} - 576 x \log {\left (2 \right )}^{2} - 384 x \log {\left (2 \right )} - 64 x\right ) e^{x}}{e^{3} \log {\left (2 \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (25) = 50\).
time = 0.40, size = 92, normalized size = 3.68 \begin {gather*} \frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, x^{3} \log \left (2\right )^{2} + 72 \, x^{2} \log \left (2\right )^{2} + 24 \, x^{2} \log \left (2\right ) + 144 \, x \log \left (2\right )^{2} + 96 \, x \log \left (2\right ) + 16 \, x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 90, normalized size = 3.60 \begin {gather*} \frac {36\,x^3\,{\ln \left (2\right )}^2+x\,\left (384\,\ln \left (2\right )+576\,{\ln \left (2\right )}^2+64\right )-64\,x\,{\mathrm {e}}^{x-3}\,\left (\ln \left (64\right )+9\,{\ln \left (2\right )}^2+1\right )-36\,x^3\,{\mathrm {e}}^{x-3}\,{\ln \left (2\right )}^2-96\,x^2\,{\mathrm {e}}^{x-3}\,\left (\ln \left (2\right )+3\,{\ln \left (2\right )}^2\right )+96\,x^2\,\ln \left (2\right )\,\left (3\,\ln \left (2\right )+1\right )}{{\ln \left (2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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